(******************************************************************* This file was generated automatically by the Mathematica front end. It contains Initialization cells from a Notebook file, which typically will have the same name as this file except ending in ".nb" instead of ".m". This file is intended to be loaded into the Mathematica kernel using the package loading commands Get or Needs. Doing so is equivalent to using the Evaluate Initialization Cells menu command in the front end. DO NOT EDIT THIS FILE. This entire file is regenerated automatically each time the parent Notebook file is saved in the Mathematica front end. Any changes you make to this file will be overwritten. ***********************************************************************) Off[General::"spell",General::"spell1",Part::"partd",Syntax::"com", Less::"nord"](*That allows similar names and nulls to be used*) BeginPackage["GroupLoopHoop`"]; GroupLoopHoop::usage = "GroupLoopHoop is a package that creates a database of over 800 loops, including all groups with up to 73 elements. It generalizes signs & creates & manipulates Groups & 'Moufang Loops' (ordered division quasigroups) as index-Tables (Cayley tables with indices as elements) and signed tables (e.g. the Quaternion multiplication table). It also sets up conservative partial-fraction division algebras (Hoops) based on them, and provides procedures & tests for division, powers, roots, polar-dual formulations & loop properties."; a::usage="Symbol in HoopTbl etc.";b::usage="";c::usage="";d::usage=""; e::usage="";f::usage="";g::usage="";h::usage="";i::usage="";j::usage=""; k::usage="";l::usage="";m::usage="";n::usage="";o::usage="";p::usage=""; q::usage="";r::usage="";s::usage="";t::usage="";u::usage="";v::usage=""; w::usage="";x::usage="";y::usage=""; z::usage="";\[Alpha]::usage="";\[Beta]::usage="";\[Gamma]::usage="";\[Delta]\ ::usage="";\[Epsilon]::usage="";\[CurlyEpsilon]::usage="";\[Zeta]::usage="";\ \[Eta]::usage="";\[Theta]::usage="";\[CapitalTheta]::usage="";\[Iota]::usage=\ "";\[Kappa]::usage="";\[Lambda]::usage="";\[Mu]::usage="";\[Nu]::usage="";\ \[Xi]::usage="";\[CapitalXi]::usage="";\[CurlyPi]::usage="";\[CapitalPi]\ ::usage="";\[Rho]::usage="";\[CurlyRho]::usage="";\[Sigma]::usage="";\ \[CapitalSigma]::usage="";\[Tau]::usage=""; \!\(TraditionalForm\`\(abbr::usage = \*"\"\\""; abbrFactors::usage = \*"\"\\""; abelGrassmanQ::usage = "\"; abelianQ::usage = "\";\n allFactorGroups::usage = \ "\"; alph::usage = "\"; alpha::usage = "\"; alternativeQ::usage = "\"; amQ::usage = "\"; any::usage = "\"; arcTanh::usage = "\"; as::usage = "\";\ \[IndentingNewLine] associator::usage = "\"; AssociativeQ::usage = "\"; associativeQ::usage = "\";\)\) bolQ::usage="bolQ([gg_:glo,o_] tests the left & right Bol identities (PMA \ p97-8) (z.yz)x ==z(y.zx) & x(zy.z) ==(xz.y)z for all x,y,z in 'gg' and \ reports the error type if 'o'\[NotEqual]0."; c2simeq::usage="c2simeq[{a_,b_,c_]solves simultaneous linear equations Ax+B=C \ in the C2 algebra. Wherever the solution is indeterminate, it includes a free \ x$ variable";ca::usage="ca[m_Integer?Positive,a_Integer] or ca[{m_,a_:1}] checks that GCD[m,k=Abs[a]] is 1 before creating an \ m\[Times]m quasigroup indexTable with first row and first column containing \ the 'unity' {1,2,...,m}, and with every element occurring once in each row \ and column (i.e. it is a sorted quasigroup or Latin Square). The result is \ the Cyclic group 'Cm' if 'a' is 0 or absent, the Dihedral group 'D[m/2]' if \ 'k'='m'-1 and the generalised Quaternion group 'Q[m]' if 'k'='m'/2-1; some \ other values of 'k' give compound Dihedral or Quaternion groups. Some other \ cases give Hamiltonian quasigroups. If 'a' is negative, non-associativity is \ reported. The indexTable is output. A skeleton version is supplied."; caylindex::usage = "caylindex[gg_] converts a square symbolic table 'gg' to \ an indexTable on the assumption that the first row is the unity."; cd::usage= "cd[{gamma__:IntegerQ}] (generalized Cayley-Dickson) takes a list of none \ or more \[PlusMinus]1's and recursively builds signed-tables (including \ P,R,C,H,O,sedenions) by the generalized Cayley-Dickson doubling procedure. \ See Lounesto, Clifford Algebras & Spinors, 2nd. ed., p285. 'm' is the table (of size q\[Times]q) obtained by dropping the last \ \[PlusMinus]1; the starting case (with no \[PlusMinus]1's) is the primal \ multiplication table {{1}}. The top-left quarter of the new table repeats \ 'm'; the top right quarter is 'm' transposed and shifted by 'q' (using \ 'modqr' to give a signed index); the bottom left quarter is 'm' shifted by \ 'q', with all but the first column multiplied by the current \[PlusMinus]1. \ The bottom right quarter is 'm' transposed with all but the first row & \ column multiplied by the current \[PlusMinus]1. An extension takes any of {2, \ -2, 3, 3., -3, 4, 4. -4, -4.,'incantation'} in place of the first \ \[PlusMinus]1, and also allows '\[PlusMinus]1.' (i.e. real) elsewhere. This \ generates more signed hoop algebras and lots of rubbish. ('incantation' is \ any string that creates a hoop.) If the last gamma is {list}, e.g. {2} or \ {3}, the last stage is the composition of the penultimate result with the \ group ca[list]. md is a related Moufang Doubling operation."; centralizer::usage="standard group function"; center::usage="standard group function"; centre::usage="standard group function"; cl::usage="cl[p,q:0] creates Clifford[p,q]; cl[p,list] creates Clifford-like \ algebras with p binary generators, those in list having negative squares. See \ Lounesto[13] ch21. cl[p_,q_,gr_:{}] creates a Grassmann-like algebra by creating a \ Clifford-like algebra (using cl[p,q]) and then zeroing the squares of elements listed in 'gr'. \ As the result is not a groupoid, most hoop operations fail."; cliff::usage="cliff is a substitution list for Clifford-like algebras. 'e' is \ the unit, 'a', 'b', 'c', 'd', 'f' the generators (univectors), 'ab' etc are \ bivectors, 'abc' is a trivector, etc"; co::usage="co[gg_?indexTableQ,hh_?indexTableQ,a_:0] outputs the direct \ composition of 'g' & 'h' if 'a' is 0 or absent; some values of 'a' create \ quasigroups or groups. If 'a' is 1 or negative, non-groupoid & \ non-associative tables will be reported. 'gg' (but not hh) may be a signed \ table. co[g_,h_,gg_,hh_,a_,o_:0,p_:0] and co[g_,gg_,hh_,a_,o_:0] (NB first \ parameter 'a' can be 0, but must be supplied) generate the multiple \ compositions by iteratively composing the inner tables using the leading \ parameters. co[g_,h_,a_,c_] is a skeleton, omitting all tests."; \!\(collect::usage\ = \ "\"; conservativeQ::usage = "\"; cosorbit::usage = \ "\"; cQ::usage = "\"; det::usage = \*"\"\\""; det1::usage = "\"; dico::usage = "\";\) dozalcons::usage= "dozalcons[12vec] calculates 39 functions that are conserved in some \ Dozal operations. Their names are given in ds"; dozaltest::usage="dozaltest[A0_,B0_,n_:5] multiplies A0 by B0 using the first \ n 12 element loops and tabulates the conservation of the dozalcons \ functions. If 5\"";\)\) hoopShape::usage = "hoopShape[X_,o_:0] accepts either a valid Protoloop \ identifier ({mm,nn} or mnemonic) or a loop table. If the loop is symbolic, it \ creates a rule that will apply the symbols to the final result. If the \ specific shape[X] is not available, the indexTable (up to 11\[Cross]11) is \ found, put into 'glo' and used to set 'gi' for use by 'hoopInverse' and to \ find the generalised shape. The output is converted to the loop symbols if \ necessary. Non-conservative loops are rejected unless o<0"; genSin::usage="genSin[x_,m_:4,j_:0] creates the j'th phase of the generalized \ symmetric sinusoid or helix with m phases; generalised Cosh and Sinh \ functions are generated if m=0 or 2."; gi::usage="Data required by hoopInverse & set up by sh, shape or hoopFactor, \ consisting of the location of the signed 1's in the indexTable rows, \ multiplied by the inverse sign if the 1 is signed, & followed by the \ multiplicity of the factors (which is zero for factored quartics)."; gInverse::usage="finds the element inverses"; glo::usage="Set by 'ge', 'ma', 'pe', 'hoopTest', 'hoopTimes', & \ 'makeProtoloop' as the (global) indexTable of the loop being created or \ investigated; the default argument in many operations. See 'globa', 'gmn'."; globa::usage="Set by 'subgroups'. The (global) indexTable of the loop being \ analyzed. See 'glo', gmn'"; gmap::usage = "gmap[gg_,v_] maps the elements of the list 'v' onto the \ indexTable 'g', which may include signed elements."; gmn::usage = "Set by 'id' & 'makeProtoloop'; the (global) mnemonic of the \ current loop,used widely as a default argument. See 'globa', glo'"; gp::usage="Size multiplicity data needed by 'hoopPower', set up for the \ target loop by 'sh', 'hoopFactor' & 'toPolar'"; gpd::usage="gpd[gg_] is used on function arguments that should be groupoids \ (i.e. magmas, loops, quasigroups). It removes any TraditionalForm wrapper and \ rejects non-groupoids by 'Abort[]'. The first row must contain the elements, \ which may be matrices; signed elements are permitted in the body of the \ table."; grayinv::usage="grayinv[a_] calculates the inverse Graycode function of 'a', \ which must be an IntegerDigits list. Used in 'cl'"; groupFromRelators::usage="groupFromRelators[gp_,post_:{}] is a skeleton \ version of ge, which only handles pre-checked relators and only creates \ Cayley tables for groups."; \!\(\* RowBox[{\(hamiltonianQ::usage = "\"; hmin::usage = "\"; hoopFactor::usage = "\"; hoopInverse::usage = "\";\), "\[IndentingNewLine]", RowBox[{\(hoopList::usage = "\"\), ";", StyleBox[\(hoopPower::usage = \*"\"\\""\), ShowStringCharacters->True, NumberMarks->True], StyleBox[";", ShowStringCharacters->True, NumberMarks->True], StyleBox[\(hoopRoot::usage = "\"\), ShowStringCharacters->True, NumberMarks->True], StyleBox[";", ShowStringCharacters->True, NumberMarks-> True], \(hoopTimes::usage\ = \ "\"\), ";", \(hoopTest::usage = "\"\), ";", \(id::usage\ = \ "\"\), ";", \(idc::usage\ = \ "\"\), ";"}]}]\) indexTableQ::usage="indexTableQ[gg_:glo]] checks that the first row and \ column are Range[Length[gg]] and that every row and column contains each of \ these (possibly signed) integers once"; jacobiQ::usage="jacobiQ[gg_:glo,o_:0] tests whether xy=-yx & x.yz+y.zx+z.xy=0 \ for all x,y,z in glo"; jordanQ::usage="jordanQ[gg_:glo,o_:0] tests whether (xy).xx=x(y.xx) up to a \ sign";lcoset::usage = "lcoset[h_,g_:glo] finds the left coset for the \ subgrouplist 'h' in the indexTable 'g' or in 'glo'"; linearFactor::usage="linearFactor[f_,l_] finds the symbolic 'alph' form of a \ linear factor 'f' generated by 'fp' for a group of length 'l'"; lmatrix::usage = "lmatrix[gg_?indexTableQ] converts 'gg' into the \ corresponding matrix with 1's on the diagonal and {1,2,3..,m} in the first \ column. If an ordered set of symbols or numbers is mapped onto an lmatrix, \ the groupoid product is given by the left matrix dot product of this matrix \ and any other ordered set. Cf rmatrix."; KS::"usage"="KS[l_List,k_Integer?Positive,gap_:{},lef_:{}] is an adaptation \ of k-subsets in Skiena p44, to create subsets, of length 'k' of group indices \ that give groupoids from 'globa'. It creates a list 'gap' of rejected indices \ and a list 'lef' of elements left in, adding to 'lef' those new elements that \ do not have any products in 'gap'."; loop::"usage"="loop[[mm,nn]] is the data file. 'mm' is the number of \ elements, and 'nn' is the position in the extended GAP Atlas entry. For each \ loop, the first entry is the mnemonic, the second is 'dico', the third \ specifies a test (0 if 'dico' is unique), the fourth is a prescription, the \ 5th & 6th define the generators. If matrix formulations are known they are \ supplied as indices for 2\[Cross]2, 3\[Cross]3, & 4\[Cross]4 standard \ matrices u2, u3, u4. In a few cases, extra table creation procedures are \ appended."; loopQ::usage= "loopQ[gg_?MatrixQ,o_:0] tests whether the first row and first column are \ the same set of different elements (i.e. are the unit) and every row and \ column contain every element once (up to a sign). If 'o' \[NotEqual] 0 the \ reason for rejection will be output."; ma::"usage"="ma[{mm__},maxel_:72,o_:0] takes a list of one or more unitary \ monomial signed square matrices (all of the same size) and prepends the unit \ matrix. It then creates the dot products of these matrices and appends any \ that are not already present, until the size exceeds 'maxel' or a group has \ been completed. The group is then created as (global) element list 'matel' \ which is output if o\[NotEqual]0; the indexTable is put into 'glo' and is \ output.";matel::usage="Global. The elements of the latest matrix (ma or ms) \ or relator (ge) group. (The last relator acts as a sign in a signed table.)"; makeProtoloop::usage ="makeProtoloop[g_:{mm,nn},o_:0] is the full name for \ 'mp', q.v.";"makeProtoloop[] or makeProtoloop[mm,nn] uses the first available \ prescription in gd[mm,nn] to create the protoloop. makeProtoloop[G_:MatrixQ,o_:0] recognises three formats for G - a string \ (mnemonic) as a name, indices {mm,nn} in 'loop' (i.e. the extended GAP Atlas \ identifiers), or a matrix. It then attempts to make the protoloop, returning \ 'Unknown' or 'No protoloop' and clearing matel as appropriate. If 'o'=0 it \ uses the first available formulation. If 'o' is 1, 'ge' is used. If 'o' is \ 2, 3, or 4, the corresponding matrix formulation is used in mg. ms is used \ for signed tables If 'o' is 5 or 6, different protoloops may be generated (only P16a, P16i, \ g2003, Alt12n so far). The indexTable is output."; matstrep::usage = "matstrep[l_,rul_,sm_:smax] performs up to 'sm' string \ replace repetitions on the matrix."; md::usage="md[g_,l_:{}] (Moufang doubling) creates a non-associative Moufang \ loop from a non-abelian group. l={} uses gInverse to implement Chein \ doubling; 3 cases use l as a reordering."; mg::usage="mg[v_?VectorQ,u_] (Matrix Group) creates a matrix by sending the u \ matrices given in 'v' to 'ma'.u may be u2,u3, or u4. The size is limited to \ 'mm'+1";minsign::usage="minsign[g_] finds any signed elements of 'g' and sets \ up the appropriate simplification list from 'unitSigns'"; mm::usage="Global. mm is the table size of entries in 'loop', set by 'id' & \ 'gd', used as {mm,nn} in many procedures"; modqr::usage="The 'modqr' function shifts a signed index 'r' by 'q' whilst \ keeping in the range \[PlusMinus]{1,2q}. e.g. modqr[3,-2] = -5. Used in \ 'cd'.";MoufangQ::usage="MoufangQ tests if zx.yz=(z.xy)z, the 3rd Moufang \ Identity. See PMA p92. As this fails for many conservative signed tables such \ as Qr, unsigned tests can be performed by moufangQ."; moufangQ::usage="Tests if |zx.yz|=|(z.xy)z|, the (unsigned) 3rd Moufang \ Identity. See PMA p92"; moufangLQ::usage="Tests if |(zy.z)x|=|z(y.zx)|, the (unsigned) Left Moufang \ Identity. See PMA p93"; moufangRQ::usage="Tests if |x(z.yz)|=|(xz.y)z|, the (unsigned) Right Moufang \ Identity. See PMA p93"; mp::usage="mp[] or mp[mm,nn] uses the first available prescription in \ gd[mm,nn] to create the protoloop. mp[G_:MatrixQ,o_:0] recognises three formats for G - a string (mnemonic) \ as a name, indices {mm,nn} in 'loop' (i.e. the extended GAP Atlas \ identifiers), or a matrix. It then attempts to make the protoloop, returning \ 'Unknown' or 'No protoloop' and clearing matel as appropriate. If 'o'=0 it \ uses the first available formulation. If 'o' is 1, 'ge' is used. If 'o' is \ 2, 3, or 4, the corresponding matrix formulation is used in mg. ms is used \ for signed tables. If 'o' is 5 or 6, different protoloops may be generated (only P16a, P16i, \ g2003, Alt12n so far). The indexTable is output."; ms::usage="ms[v__?ListQ,l_:u2,maxel_:33] makes signed tables that may be \ hoops, from the specified unitary monomial matrices. It puts the unit matrix \ and either the matrices in v (or if v is a list of indices, possibly signed, \ the listed matrices from l, where l is u2,u3 or u4), into matel. It then \ creates a table from their dot products, adding any that do not match signed \ existing ones, up to a limit maxel. ms is used in place of mg in mp for \ signed tables"; msi::usage="msi[v__?ListQ,l_:u2,si={1,-1,\[ImaginaryI],-\[ImaginaryI]},maxel_:\ 64] is a variation on ms that handles P4 and P16c"; mstst::usage="mstst[m_,s_] tests whether matel contains a version of the \ matrix m, multiplied by the signs in s, returning 0 or the corresponding sign \ for use in ms."; ncm::usage="ncm performs some simplifications on NonCommutative polynomials \ of up to 3 variables. Should be replaced by instructions from \ NCAlgebra@math.ucsd.edu."; negatableQ::usage="negatableQ[gg_:glo,f_:2] tests whether the table 'gg' has \ the same elements in each 1/f \[Times] 1/f fraction with elements shifted by \ 'm/f' (allowing for signs). If so, the top left fraction may describe a \ signedTable."; \!\(nn::usage = "\"; normalSubgroup::usage\ = \ \*"\"\\""; orderCount::usage = "\"; pe::usage = "\"; period::usage = "\"; permute::usage = "\";\) pick::usage="pick[g_,m_List,n_List] picks out the rows 'm' & columns 'n' from \ 'g'";plex::usage="The vec used to give a plex-conjugate to {a,b,...}; set up \ by some shapes. If it does not give the standard shape in an obvious way, the \ plex form of the shape is supplied as a comment. Conjugates are a \ meta-mathematical trick that provide a quadratic size in some algebras; the \ more general approach is via the explicit 'shape' functions that give all the \ conserved sizes."; polsort::usage="creates a sorted list of elements in the polynomial P, for \ use in 'frag'"; pos::usage="pos[a_] negates 'a' if the first term is negated, so that 'frag2' \ squared terms match."; pow2::usage="A list {a,1,2,4,8..} used as elements of Cayley Tables to allow \ fast determinant evaluation via fp & 'linearFactor'. (May have problems with \ some Clifford algebras - use fp[G,power2 or power3].)"; power2::usage="A list {a,3,7,1,2,4,8..} used as elements of Cayley Tables to \ allow fast determinant evaluation via 'fp'"; power3::usage="A list {a,1,2,3,9,27,81..} used as elements of Cayley Tables \ to allow fast determinant evaluation via 'fp'."; QPower::usage= "QPower[{a,b,c,d},r] raises the quaternion (or biquaternion, as the \ coefficients can be complex) to the power or root r, giving the square root \ if r is omitted"; rcoset::usage = "rcoset[h_,g_] finds the right coset for the subgrouplist 'h' \ in the indexTable'g' or in 'glo'";reductiveQ::usage="tests xy.xx=x(y.xx)"; reim::usage = "Replaces Derivative[Re] by Re, etc"; reorder::usage = "reorder[gg_?indexTableQ,p_] generates the indexTable of \ 'gg' as reordered by the permutation 'p'; short permutations give \ sub-tables.";riffQ::usage="Tests if xy(z.xy)=((x.yz).x)y"; r3::usage="1/Sqrt[3]"; Rl::usage="The left remainder from the last P=hoopTimes[A,B], containing the \ sizes of A that are zero in P"; Rr::usage="The right remainder from the last P=hoopTimes[A,B], containing the \ sizes of B that are zero in P"; rmatrix::usage = "rmatrix[gg_?indexTableQ] converts 'gg' into the \ corresponding matrix with 1's on the diagonal and {1,2,3..,m) in the first \ row. If an ordered set of symbols or numbers is mapped onto an rmatrix, the \ groupoid product is given by the right matrix dot product of this matrix and \ any other ordered set. Cf lmatrix"; rotateQOP::usage="rotateQOP[{vecs},axis,angle,hoop:gmn] rotates the vecs \ (truncated or padded with zeroes if necessary) through angle about axis \ (which is also truncated or padded) The hoop must be Qr,Octr, or P4. gmn is \ set for further use. The classic Euler defnition is used, multiplying twice \ by half angles."; ruleExpand::usage = "ruleExpand[a] applies 'stringExpand' to a list of \ compressed rules e.g a4b2\[ShortRightArrow]aaaabb."; select::usage="select[l_] finds all elements appearing as products in the \ subarray 'l' of 'globa'"; s3::usage="s3 is the list of permutations that define S3"; s4::usage="s4 is the list of permutations that define S4"; s5::usage="s5 is the list of permutations that define S5"; sg::usage="sg is the (global) list of subgroup indices created by \ 'subgroups'; used by 'subgroupTypes'"; sh::usage="sh[hoop] is the list of factors of the symbolic hoop determinant."; shape::usage="shape[vec_,mnemonic_] obtains specific shape data (where known) \ from sh for the current or specified loop. The real determinant factors (i.e. \ those that are conserved) are supplied, and the lists gi & gp are set up for \ use by hoopInverse & toPolar. Plex conjugates are also supplied (where known) \ for completeness."; sgroup::usage = "sgroup[m_?IntegerQ] is adapted from Skiena p17. It creates \ the indexTable for S[n]"; sh12::usage="sh12[{12vec}] is the standard shape data for Dozal, the \ 12-element groups C3C4, C3K, Q12, D3C2, A4. It includes functions that are \ conserved on general multiplication but are not determinant factors, and \ others that are only conserved in some multiplications of orbits"; \!\(signlist::usage = "\"; \[DoubleStruckD]::usage = \ \*"\"\\""; \ \[DoubleStruckG]::usage = \*"\"\\ \""; \[DoubleStruckH]::usage = \*"\"\\""; \[DoubleStruckCapitalI]::usage = \ \*"\"\\""; \ \[DoubleStruckCapitalJ]::usage = \*"\"\\""; \[DoubleStruckK]::usage = \ \*"\"\\""; \ \[DoubleStruckL]::usage = \*"\"\\ \""; \[DoubleStruckM]::usage = \*"\"\\""; \[DoubleStruckN]::usage = \*"\"\\""; \[DoubleStruckO]::usage = \ \*"\"\\""; \ \[DoubleStruckP]::usage = \*"\"\\ \""; \[DoubleStruckS]::usage = \*"\"\\"";\n \(\(\[DoubleStruckCapitalY]::usage = \*"\"\\"";\);\)\) signedTableQ::usage="tests whether the group is an indexTable with signed \ elements"; simC2::usage="simc2[a_,b_,o_:0] solves a set of simultaneous equations ax+b=0 \ for x in the C2 algebra. The determinant factorises into p and n (i.e. \ \[PlusMinus]); if either is zero, the indeterminate solution has a degree of \ freedom $x. If o\[NotEqual]0 the solution is collapsed to real numbers"; strep::usage = "created by ge[...,\[PlusMinus]1] as string replacement \ rules.";streprpt::usage = "streprpt[str_,rul_,imax_] performs up to 'imax' \ string replace repetitions using the list 'rul'."; stringExpand::usage = "stringExpand[t] replaces 'letter integer' by 'integer' \ repetitions of 'letter' in the string 't'."; subgroupTypes::usage = "subgroupTypes[] or subgroupTypes[%] may be used after \ subgroups[g] to identify them in order."; subgrpPerm::usage="subgrpPerm[G_,a_:0] & subgrpPerm[G_,a_,b_,c_] are \ implementations of RandomPermutation from Skiena p7. They permute the group \ to increase the chance of pruning, and usually speed up the determination of \ subgroups for groups with 'm'>47. a, b, & c are arguments that are passed \ through."; subgroups::usage = "subgroups[g_,o_,i1_,i2_,p_:associativeQ] finds all the \ subloops, in the IndexTable 'g' that pass the 'p' test, so the default finds \ subgroups, within the optional length range {i1,i2}, as sublists of indices. \ 'globa' is set to 'g'. If 'o'=0, the number of subloops is output; if 'o'=1 \ the sublist is put into 'sg', for use by 'id' and is output; o=2 gives both. subgroups[g_,-n,i1_] or subgroups[g_,any,i1_,any,{n__}] test whether the \ subgroup identified as {i1,n} or {i1,n__} is a subgroup of 'g', returning 0 \ or the first matching n__ found. Used by 'id'"; tf::usage="TraditionalForm"; tidy::usage = "tidy[g_:globa,n_:-1,o_:1,p_:{}] uses the list 'n' (or the n'th subgroup \ if 'n' is an integer; the default is the last found subgroup) in the top-left \ corner of a reordering of 'g'; 'o' is the (optional) element at the beginning \ of the next column. 'p' is the (optional) starting element (or list) for \ the remainder. The reorder list and the new table are output. The default \ group is 'globa', as set up by 'subgroups'. It is faster to specify a \ subgroup as an element-list than to specify a position in the subgroup list, \ as the latter has to proceed via 'subgroups', which is slow for large \ groups"; toCycles::"usage"="toCycles[perm_?VectorQ] generates the cycles from the \ permutation."; toIndexTable::usage = "toIndexTable[g_?MatrixQ] rearranges a shuffled \ indexTable to make the first row and column into {1,2,...m}"; tohyPol::usage="tohyPol[{radius,angle}__,mnemonic_] contains the hyperbolic \ forms";tohyPolar::usage="tohyPolar[vec_,mnemonic_] obtains the hyperbolic \ form for the current groupoid"; tohyVec::usage="tohyVec[{ulna,angle}__,mnemonic_] contains the hyperbolic \ forms";tohyVector::usage="tohyVector[hpf_,mnemonic_] reverts the hyperbolic \ form for the current groupoid"; toPol::usage="toPol[vec_,mnemonic_] gives the polar form of vec in the \ specified hoop."; toPolar::usage="toPolar[vec_,mnemonic_] obtains the polar form from the \ vector form for the current groupoid"; toSignedTable::usage = "toSignedTable[g_?MatrixQ,r_:2,reo_:{},o_:0] \ (abbreviation ts) attempts to collapse 'g' to a signed table of length m/|r|, \ after reordering if 'reo' is provided. Non-conservative tables are rejected \ unless 'r'<0. Collapse is a skeleton,omitting reordering and all tests."; toVec::usage="toVec[mnemonic_] gives the vector form, using 'alph' \ variables."; toVector::usage="toVector[pf_,mnemonic_] reverts the polar form to the vector \ form for the current groupoid"; trico::usage="trico[g_] is the count of elements of 'g' having cube roots, \ with the unit roots first "; ts::usage = "ts[a___] is an abbreviation, used in data, for \ toSignedTable[g_?MatrixQ,r_:2,reo_:{},o_:0]. It attempts to collapse 'g' to a \ signed table of length m/|r|, after reordering if 'reo' is provided. \ Non-conservative tables are rejected unless 'r'<0."; u2::usage="u2, u3, & u4 are lists of 2\[Times]2, 3\[Times]3, & 4\[Cross]4 \ monomial unit matrices that include the Pauli Sigma and Dirac Gamma and Tau \ matrices. These allow development of matrix groups with different symmetries. \ Only some root matrices are included; dot multiplying them together generates \ many more. In combination with 'loop' & 'makeProtoloop', they allow the \ generation of many small groups with sets of matrices as elements. Further \ work is needed to increase the number of groups handled.";u3::usage="See u2"; u4::usage="See u2"; union::usage="Finds the unsorted union. Same as UnionNoSort of EAAM"; \!\(\* RowBox[{ RowBox[{ RowBox[{\(unitSigns::usage\), "=", "\"\<\[DoubleStruckD], \[DoubleStruckG], \[DoubleStruckH], \ \[DoubleStruckCapitalI], \[DoubleStruckCapitalJ], \[DoubleStruckL], \ \[DoubleStruckK], \[DoubleStruckM], \[DoubleStruckN], \[DoubleStruckO], \ \[DoubleStruckP], \[DoubleStruckS] and \[DoubleStruckCapitalY] are set up to \ act as unit signs related to \[ImaginaryI], with \ \[DoubleStruckD]=\!\(\@\(+1\)\%12\), \[DoubleStruckG]=\!\(\@\(+1\)\%7\), \ \[DoubleStruckCapitalI]=\!\(\@\(+1\)\%4\) (and distinct from \[ImaginaryI] \ and I), \[DoubleStruckCapitalJ]=\!\(\@\(+1\)\%3\) and \ \[DoubleStruckCapitalY]=\!\(\@\(+1\)\%3\)(both distinct from \ \!\(TraditionalForm\`\[ImaginaryJ]\) which is synonymous with \[ImaginaryI] \ and I in Mathematica), \[DoubleStruckH]=\!\(\@\(+1\)\), \[DoubleStruckK]=\!\(\ \@\(+1\)\), \[DoubleStruckM]=\!\(\@\(+1\)\) (all being extra negations), \ \[DoubleStruckN]=\!\(\@\(+1\)\%9\), \[DoubleStruckO]=\!\(\@\(+1\)\%8\), \ \[DoubleStruckP]=\!\(\@\(+1\)\%5\), \[DoubleStruckS]=\!\(\@\(+1\)\%\ \[DoubleStruckR]\). They are all non-Positive, with NumberQ=True, Abs=1, \ Re=0, Im=0 and equal their own Sign, so that 'gmap' treats them correctly. \ 'unitSigns' reduces their powers to their primary range, whilst minsign \ creates a specific reduction rule. \[DoubleStruckS] is general \ \[DoubleStruckR]'th root of unity.\>\""}], ";", \(unsign::usage = "\"\), ";", \(walsh::usage = "\"\), ";"}], "\n", \(\[Omega]::usage = "\<\[Omega]=2\[Pi]/3 used in many \ orbits\>";\)}]\) (*2,3 elements *) C2::usage="indexTable for C2";C3::usage="indexTable for C3"; C3n::usage="non-Moufang 3-element table"; C3C3j::usage="conservative \[DoubleStruckCapitalJ]-signedTable from C3C3, \ though conservativeQ and idc fail to simplify and so fail"; C9J::usage="conservative \[DoubleStruckCapitalJ]-signedTable from C9, though \ conservativeQ and idc fail to simplify and so fail"; C9j::usage="conservative \[DoubleStruckCapitalJ]-signedTable from a C9 \ isomorph, though conservativeQ and idc fail to simplify and so fail"; C4C3c::usage="conservative \[ImaginaryI]-signedTable"; C3i::usage="conservative \[ImaginaryI]-signedTable"; C12c::usage="conservative \[ImaginaryI]-signedTable"; (* 4 elements *) C4::usage="indexTable for C4";K::usage="indexTable for K \[Congruent] C2C2"; P4::usage="complexTable from P16 \[Congruent] Pauli algebra"; P16c::usage="complexTable from P16";C4C3c::usage="signedTable from C4C3"; Q4n::usage="non-conservative signedTable related to Q4 & P4";(* 5elements *) C5::usage="indexTable for C5"; C5n::usage="The smallest non-associative Jacobi-identity loop \ (arXiv:hep-th/0111292 v1)"; (*6 elements *) C3C2::usage="indexTable for C3C2"; D3::usage="indexTable for D3 \[Congruent] S3"; D3C2c::usage="signed indexTable from D3C2"; C6n::usage="A table with the repeated quadratic size of D3 replaced by two \ signed sums of 6 squares. Passes the hamiltonian test (only)"; \!\(\(\( (*\ 8, \ 9\ elements\ *) \)\(\[IndentingNewLine]\)\(C4C2::usage = "\"; C8::usage = "\"; KC2::usage = \*"\"\\""; D4::usage = "\"; Q8::usage = "\"; CL12::usage = "\"; P8::usage = "\";\[IndentingNewLine] Octi::usage = "\<'isoOctonion' signedTable from M(16,2)\>"; Octr::usage = "\"; O4::usage = "\"; O4k::usage = "\"; Q8C2r::usage = "\"; Q8n::usage = "\"; C8an::usage = "\";\[IndentingNewLine] (*\ 9\ elements\ *) \[IndentingNewLine] \(C3C3::usage = "\";\)\)\)\) (* 12 elements *) C3C4::usage="indexTable for C3C4"; C3K::usage="indexTable for C3K \[Congruent] C3C2C2"; Q12::usage="indexTable for Q12";A4::usage="indexTable for A4"; A4a::usage="second indexTable for A4";D3C2::usage="indexTable for D3C2"; C3P4::usage="conservative 12-element complexTable"; D3Mn::usage="indexTable for M1201, 12-element non-associative Moufang Loop \ from D3";Alt12n::usage="non-conservative alternative 12-element indexTable"; \!\(\(\( (*\ 16\ elements\ *) \)\(\[IndentingNewLine]\)\(Q16::usage = "\"; D4C2::usage = "\"; C4C4::usage = "\"; Q8C2::usage = "\"; C8pC2::usage = "\"; KK::usage = \*"\"\\""; C4K::usage = "\"; KiC4::usage = "\"; C4iC4::usage = "\"; P16::usage = "\"; P16a::usage = "\"; P16i::usage = "\"; Oct::usage = "\"; r31r::usage = "\<16 element signedTable from g3249\>"; CL4::usage = "\<16 element signedTable from g3250\>"; D4M1n::usage = "\"; Q8M2::usage = "\"; D4M4n::usage = "\"; D4M5n::usage = "\"; Sed::usage = "\";\)\)\) (* 20+ elements *) D5Mn::usage="indexTable for 20-element non-associative Moufang Loop from D5, \ GAP M2001"; D3C2Mn::usage="indexTable for 24-element non-associative Moufang Loop from \ D3C2, GAP M2401"; A4Mn::usage="indexTable for 24-element non-associative Moufang Loop from A4, \ GAP M402"; Q12M3n::usage="indexTable for 24-element non-associative Moufang Loop from \ Q12, GAP M2403"; Q12M4n::usage="indexTable for 24-element non-associative Moufang Loop from \ Q12, GAP M2405"; Q12M5n::usage="indexTable for 24-element non-associative Moufang Loop from \ Q12, GAP M2405";SL23::usage="indexTable for 24-element group SL[2,3]"; OctrC3::usage="conservative alternative 24-element signedTable"; Alt12nC2::usage="non-conservative alternative 24-element indexTable"; P27::usage="indexTable for group P27"; g2704::usage="indexTable for group 27-element without prescription"; OctC2M::usage="conservative alternative 32-element indexTable, GAP M3216"; P16M::usage="indexTable for 32-element non-associative Moufang Loop from P16, \ GAP M3221"; C8pC2M::usage="indexTable for 32-element non-associative Moufang Loop from \ C8pC2 GAP M3235"; KiC4M::usage="indexTable for 32-element non-associative Moufang Loop from \ KiC4, GAP M3254"; C4iC4M::usage="indexTable for 32-element non-associative Moufang Loop from \ C4iC4, GAP M4365"; Alt12nC3::usage="non-conservative alternative 36-element indexTable"; OctC3::usage="conservative alternative 48-element indexTable"; Begin["`Private`"]; protected = Unprotect[NumberQ, Positive, Abs, Sign, Re, Im, Interval];(* no longer needed?*) \!\(arcTanh[adj_, opp_] := Module[{x = Chop[adj], y = Chop[opp], b}, If[\((x \[Equal] y || x \[Equal] \(-y\))\), If[x \[NotEqual] 0, \[Infinity]\ Sign[y]/Sign[\ x], 0, 0], b = Log[\((x + y)\)/\@\(x\^2 - y\^2\) // N]; Chop[Re[b] + \[ImaginaryI]\ Chop[Im[b]]]]]; arcTanh[{x_, y_}] := arcTanh[x, y];\) \!\(\* RowBox[{ StyleBox[\(Interval[{a_, a_}] := a;\), ShowStringCharacters->True, NumberMarks->True], "\[IndentingNewLine]", \(pos[a_] := If[Head[a] === TraditionalForm, If[Position[a[\([1, 1]\)], \(-_\)] === {{}}, \(-a[\([1]\)]\), a[\([1]\)]], If[Position[a[\([1]\)], \(-_\)] === {{}}, \(-a\), a]]; (*used\ in\ frag*) \[IndentingNewLine]symdiff[a_Real, b_Real: 0. ] := If[a < b, b - a, a - b, a - b]; symdiff[a_Integer, b_Integer: 0] := If[a < b, b - a, a - b, a - b]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"reim", "=", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["Re", TagBox[\((any_)\), Derivative], MultilineFunction->None], "\[Rule]", "Re"}], ",", RowBox[{ SuperscriptBox["Im", TagBox[\((any_)\), Derivative], MultilineFunction->None], "\[Rule]", "Im"}]}], "}"}]}], ";"}]}]\) det[(a_)?MatrixQ,m_:2]:= Module[{am=Abs[m],b=a,c,d,i,J,ln=Length[a],lm=Length[a[[1]]],t=Times}, Which[ln\[Equal]1,d=Sum[J^(i-1)*a[[1,i]],{i,lm}],lm\[Equal]1, d=Sum[J^(i-1)*a[[i,1]],{i,ln}],True,If[ln>lm,b=Transpose[a];lm=ln]; c=J*b[[1]];b=Transpose[Drop[b,1]]; d=Sum[If[c[[i]]===0,0,J^i* t[c[[i]],det1[Drop[b,{i}],J,t]]],{i, lm}]];CoefficientList[ Expand[d]//.{J^any_\[Rule]J^Mod[any,am]},J]/.J\[Rule] Switch[am,1,1,2,-1,3,\[DoubleStruckJ],4,\[DoubleStruckI]]];; det1[a_,J_,t_]:= Module[{ln=Length[a],lm=Length[a[[1]]],b,c}, If[lm\[Equal]1,Sum[J^(i-1)*a[[i,1]],{i,ln}],b=Transpose[a];c=J*b[[1]]; b=Transpose[Drop[b,1]]; Sum[If[c[[i]]===0,0,J^i*t[c[[i]],det1[Drop[b,{i}],J,t]]],{i,ln}]]]; walsh[a_]:=If[EvenQ[Length[Position[a,1]]],1,-1]; walsh[a_,n_]:=If[EvenQ[Length[Position[a[[n]],1]]],1,-1]; grayinv[a_]:=Table[Mod[Plus@@Take[a,i],2],{i,Length[a]}] Distribution[l_List]:=Distribution[l,Union[l]];(* From Skiena p106 *) Distribution[l_List,set_List]:=Map[(Count)[l,#]&,set]; select[l_?ListQ] := Union[Flatten[ Table[globa[[l[[i]],l[[j]]]], {i, Length[l]}, {j, Length[l]}]]]; union[lst_]:=(* Union No Sort as in EEAM. Available as UnsortedUnion in Functions*) Apply[Head[lst], Fold[If[MemberQ[#1,#2],#1,Append[#1,#2]]&,{First[lst]},Drop[lst,1]]] permfromQ[p_List,n_Integer]:= Length[Complement[p,Range[n]]]\[Equal]0&&Sort[p]\[Equal]Union[p]; permute[P_List,p_List]:= If[permfromQ[p,Length[P]],P[[p]],Print[{P,p," error"}]]; toCycles[perm_?VectorQ]:= Module[{c,i,m,n,p=perm}, If[Sort[p]==Range[Max[p]], Select[Table[m=n=p\[LeftDoubleBracket]i\[RightDoubleBracket];c={}; While[p\[LeftDoubleBracket]n\[RightDoubleBracket]\[NotEqual]0, AppendTo[c,m=n];n=p\[LeftDoubleBracket]n\[RightDoubleBracket]; p\[LeftDoubleBracket]m\[RightDoubleBracket]=0]; c,{i,Length[p]}],#1=!={}&],"Not a Permutation"]]; fromCycles[cyc_List]:= Module[{p=Range[Max[Flatten[cyc]]],pos}, Scan[(pos=Last[#1]; Scan[Function[c, pos=p\[LeftDoubleBracket]pos\[RightDoubleBracket]=c],#1])&,cyc]; p];as[A_]:=(*Relace alph dummy variables by the coefficients in A *) Table[alph[[ii]]\[Rule]A[[ii]],{ii, Min[mm,Length[A]]}];tf=TraditionalForm;r3=1/Sqrt[3]; \!\(\* RowBox[{ RowBox[{\(\[DoubleStruckR] =. \), ";", RowBox[{"unitSigns", "=", RowBox[{"{", RowBox[{\(\[DoubleStruckD]\^12 \[ShortRightArrow] 1\), ",", \(\[DoubleStruckD]\^any_ \[Rule] \[DoubleStruckD]\^Mod[any, \ 12]\), ",", \(\[DoubleStruckG]\^7 \[Rule] 1\), ",", \(\[DoubleStruckG]\^any_ \[Rule] \[DoubleStruckG]\^Mod[any, \ 7]\), ",", \(\[DoubleStruckH]\^6 \[Rule] 1\), ",", \(\[DoubleStruckH]\^any_ \[Rule] \[DoubleStruckH]\^Mod[any, \ 6]\), ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalI]", FontWeight->"Plain"], "4"], "\[Rule]", "1"}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalI]", FontWeight->"Plain"], "any_"], "\[Rule]", SuperscriptBox[ StyleBox["\[DoubleStruckCapitalI]", FontWeight->"Plain"], \(Mod[any, 4]\)]}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], "3"], "\[Rule]", "1"}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], "any_"], "\[Rule]", SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], \(Mod[any, 3]\)]}], ",", \(\[DoubleStruckK]\^any_?OddQ \[Rule] \[DoubleStruckK]\), ",", \(\[DoubleStruckK]\^any_?EvenQ \[Rule] 1\), ",", \(\[DoubleStruckL]\^any_?OddQ \[Rule] \[DoubleStruckL]\), ",", \(\[DoubleStruckL]\^any_?EvenQ \[Rule] 1\), ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckM]", FontWeight->"Plain"], \(any_?OddQ\)], "\[Rule]", StyleBox["\[DoubleStruckM]", FontWeight->"Plain"]}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckM]", FontWeight->"Plain"], \(any_?EvenQ\)], "\[Rule]", "1"}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckN]", FontWeight->"Plain"], "9"], "\[Rule]", StyleBox["1", FontWeight->"Plain"]}], ",", RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckN]", FontWeight->"Plain"], "any_"], "\[Rule]", StyleBox[\(\[DoubleStruckN]\^Mod[any, 9]\), FontWeight->"Plain"]}], ",", \(\[DoubleStruckO]\^8 \[Rule] 1\), ",", \(\[DoubleStruckO]\^any_ \[Rule] \[DoubleStruckO]\^Mod[any, \ 8]\), ",", \(\[DoubleStruckP]\^5 \[Rule] 1\), ",", \(\[DoubleStruckP]\^any_ \[Rule] \[DoubleStruckP]\^Mod[any, \ 5]\), ",", \(\[DoubleStruckS]\^\[DoubleStruckR] \[Rule] 1\), ",", \(\[DoubleStruckS]\^any_ \[Rule] \[DoubleStruckS]\^Mod[any, \ \[DoubleStruckR]]\), ",", \(\[DoubleStruckCapitalY]\^3 \[Rule] 1\), ",", \(\[DoubleStruckCapitalY]\^any_ \[Rule] \ \[DoubleStruckCapitalY]\^Mod[any, 3]\)}], "}"}]}], ";"}], "\n", "\[IndentingNewLine]", \(signlist = {\[DoubleStruckD], \[DoubleStruckG], \ \[DoubleStruckH], \[DoubleStruckCapitalI], \[DoubleStruckCapitalJ], \ \[DoubleStruckK], \[DoubleStruckL], \[DoubleStruckM], \[DoubleStruckN], \ \[DoubleStruckO], \[DoubleStruckP], \[DoubleStruckS], \ \[DoubleStruckCapitalY]};\)}]\) NumberQ[\[DoubleStruckD]]=True;NumberQ[\[DoubleStruckG]]=True; NumberQ[\[DoubleStruckH]]=True;NumberQ[\[DoubleStruckCapitalI]]=True; NumberQ[\[DoubleStruckCapitalJ]]=True;NumberQ[\[DoubleStruckK]]=True; NumberQ[\[DoubleStruckL]]=True;NumberQ[\[DoubleStruckM]]=True; NumberQ[\[DoubleStruckN]]=True;NumberQ[\[DoubleStruckO]]=True; NumberQ[\[DoubleStruckP]]=True;NumberQ[\[DoubleStruckS]]=True; NumberQ[\[DoubleStruckCapitalY]]=True; Positive[\[DoubleStruckD]]=False;Positive[-\[DoubleStruckD]]=False; Positive[\[DoubleStruckG]]=False;Positive[-\[DoubleStruckG]]=False; Positive[\[DoubleStruckH]]=False;Positive[-\[DoubleStruckH]]=False; Positive[\[DoubleStruckCapitalI]]=False; Positive[-\[DoubleStruckCapitalI]]=False; Positive[\[DoubleStruckCapitalJ]]=False; Positive[-\[DoubleStruckCapitalJ]]=False;Positive[\[DoubleStruckK]]=False; Positive[-\[DoubleStruckK]]=False;Positive[\[DoubleStruckL]]=False; Positive[-\[DoubleStruckL]]=False;Positive[\[DoubleStruckM]]=False; Positive[-\[DoubleStruckM]]=False;Positive[\[DoubleStruckN]]=False; Positive[-\[DoubleStruckN]]=False;Positive[\[DoubleStruckO]]=False; Positive[-\[DoubleStruckO]]=False;Positive[\[DoubleStruckP]]=False; Positive[-\[DoubleStruckP]]=False;Positive[\[DoubleStruckS]]=False; Positive[-\[DoubleStruckS]]=False;Positive[\[DoubleStruckCapitalY]]=False; Positive[-\[DoubleStruckCapitalY]]=False; Abs[\[DoubleStruckD]]=1;Abs[\[DoubleStruckG]]=1;Abs[\[DoubleStruckH]]=1; Abs[\[DoubleStruckCapitalI]]=1;Abs[\[DoubleStruckCapitalJ]]=1; Abs[\[DoubleStruckK]]=1;Abs[\[DoubleStruckL]]=1;Abs[\[DoubleStruckM]]=1; Abs[\[DoubleStruckN]]=1;Abs[\[DoubleStruckO]]=1;Abs[\[DoubleStruckP]]=1; Abs[\[DoubleStruckS]]=1;Abs[\[DoubleStruckCapitalY]]=1; Sign[\[DoubleStruckD]]=\[DoubleStruckD]; Sign[\[DoubleStruckG]]=\[DoubleStruckG]; Sign[\[DoubleStruckH]]=\[DoubleStruckH]; Sign[\[DoubleStruckCapitalI]]=\[DoubleStruckCapitalI]; Sign[\[DoubleStruckCapitalJ]]=\[DoubleStruckCapitalJ]; Sign[\[DoubleStruckK]]=\[DoubleStruckK]; Sign[\[DoubleStruckL]]=\[DoubleStruckL]; Sign[\[DoubleStruckM]]=\[DoubleStruckM]; Sign[\[DoubleStruckN]]=\[DoubleStruckN]; Sign[\[DoubleStruckO]]=\[DoubleStruckO]; Sign[\[DoubleStruckP]]=\[DoubleStruckP]; Sign[\[DoubleStruckS]]=\[DoubleStruckS]; Sign[\[DoubleStruckCapitalY]]=\[DoubleStruckCapitalY]; Re[\[DoubleStruckD]]=0;Re[\[DoubleStruckG]]=0;Re[\[DoubleStruckH]]=0; Re[\[DoubleStruckCapitalI]]=0;Re[\[DoubleStruckCapitalJ]]=0; Re[\[DoubleStruckK]]=0;Re[\[DoubleStruckL]]=0;Re[\[DoubleStruckM]]=0; Re[\[DoubleStruckN]]=0;Re[\[DoubleStruckO]]=0;Re[\[DoubleStruckP]]=0; Re[\[DoubleStruckS]]=0;Re[\[DoubleStruckCapitalY]]=0;Im[\[DoubleStruckD]]=0; Im[\[DoubleStruckG]]=0;Im[\[DoubleStruckH]]=0;Im[\[DoubleStruckCapitalI]]=0; Im[\[DoubleStruckCapitalJ]]=0;Im[\[DoubleStruckK]]=0;Im[\[DoubleStruckL]]=0; Im[\[DoubleStruckM]]=0;Im[\[DoubleStruckN]]=0;Im[\[DoubleStruckO]]=0; Im[\[DoubleStruckP]]=0;Im[\[DoubleStruckS]]=0;Im[\[DoubleStruckCapitalY]]=0; Re[ a_ \[DoubleStruckCapitalJ]^b_]:=0; Im[ a_ \[DoubleStruckCapitalJ]^b_]:=0;Re[ \[DoubleStruckCapitalJ]^b_]:=0; Im[\[DoubleStruckCapitalJ]^b_]:=0;Re[a_ \[DoubleStruckCapitalJ]]=0; Im[a_ \[DoubleStruckCapitalJ]]=0;Re[ a_ \[DoubleStruckCapitalY]^b_]:=0; Im[ a_ \[DoubleStruckCapitalY]^b_]:=0;Re[ \[DoubleStruckCapitalY]^b_]:=0; Im[\[DoubleStruckCapitalY]^b_]:=0;Re[a_ \[DoubleStruckCapitalY]]=0; Im[a_ \[DoubleStruckCapitalY]]=0; minsign[g_]:=Module[{a=Intersection[Variables[g],signlist], s={}}, Do[Do[If[unitSigns[[i]][[1,1]]===a[[j]],AppendTo[s,unitSigns[[i]]]],{i, Length[unitSigns]}],{j,Length[a]}];s] hoopList={"C2","C4c","C3","C3C2c","C12c","C3i","C4C3c","C3C3j","C9j","C9J", "C4","K","Qr","P4","Dav","O2","C8r","C3C4j","CL2","C4C2r","P16c","C8c", "C5","C3C2","D3","C3C4c","Q12c","Q12r","Q12r","g2401c","D6r","D3C2c", "KC3c","C3Kr","Cl3r","Clm3r","C8","KC2","C4C2","D4","Q8","Octr","KiC4c", "O4","O4a","O4b","C4C4c","C4C4r","Q8C2r","C2D4r","P8","CL3","C4iC4c", "QC2r","KiC4r","CL21","CL03","Octi","C3C3","C9","C6C3c","C3C4","C3K", "Q12","D3C2","A4","C3Qr","C3P4","C6Kr","C3KC2r","Q12C2c","D8","Q16", "D4C2","C4C4","Q8C2","KK","C4K","KiC4","C4iC4","P16","Oct","Q8M2","CL4", "CL13","CL22","CL31","r31r","CL04","O8","C3C4C2","CL31C2","CL4C2","Q4n", "C6n","Q8n","D3Mn"}; \!\(\* RowBox[{\(C2 = ca[2]; C3 = ca[3]; C3n = {{a, b, c}, {b, c, a}, {c, \(-a\), b}};\), "\[IndentingNewLine]", RowBox[{\(C3C3j = {{1, 2, 3}, {2, 3\ \[DoubleStruckCapitalJ]\^2, \[DoubleStruckCapitalJ]}, {3, \ \[DoubleStruckCapitalJ], 2\ \[DoubleStruckCapitalJ]\^2}}\), ";", \(C9j = {{1, 2, 3}, {2, 3\ \[DoubleStruckCapitalJ]\^2, 1}, {3, 1, 2\ \[DoubleStruckCapitalJ]}}\), ";", \(C9J = {{1, 2, 3}, {2, 3, \[DoubleStruckCapitalJ]}, {3, \[DoubleStruckCapitalJ], 2\ \[DoubleStruckCapitalJ]}}\), ";", FormBox[\(C3i = {{1, 2, 3}, {2, 3, \[ImaginaryI]}, {3, \[ImaginaryI], 2\ \[ImaginaryI]}}\), "TraditionalForm"], ";"}], "\[IndentingNewLine]", \(C12c = {{1, 2, 3}, {2, 3, \(-\[ImaginaryI]\)}, {3, \(-\[ImaginaryI]\), \(-2\)\ \ \[ImaginaryI]}};\), "\[IndentingNewLine]", \(C4 = ca[4]; K = ca[4, 3]; P4 = {{1, 2, 3, 4}, {2, 1, \(-4\)\ \[ImaginaryI], 3\ \[ImaginaryI]}, {3, 4\ \[ImaginaryI], 1, \(-2\)\ \[ImaginaryI]}, {4, \(-3\)\ \[ImaginaryI], 2\ \[ImaginaryI], 1}}; C4C3c = {{1, 2, 3}, {2, 3\ \[ImaginaryI], \(-1\)}, {3, \(-1\), 2\ \[ImaginaryI]}};\), "\[IndentingNewLine]", \(P16c = {{1, 2, 3, 4}, {2, \(-1\), \(-4\)\ \[ImaginaryI], \(-3\)\ \[ImaginaryI]}, {3, 4\ \[ImaginaryI], \(-1\), 2\ \[ImaginaryI]}, {4, 3\ \[ImaginaryI], \(-2\)\ \[ImaginaryI], 1}}; Q4n = {{1, 2, 3, 4}, {2, 1, 4, \(-3\)}, {3, \(-4\), 1, 2}, {4, 3, \(-2\), 1}}; C5 = ca[5]; C5n = {{1, 2, 3, 4, 5}, {2, 1, 4, 5, 3}, {3, 5, 1, 2, 4}, {4, 3, 5, 1, 2}, {5, 4, 2, 3, 1}};\)}]\) C3C2=co[C3,C2];D3=ca[6,5]; D3C2c=ms[{6,7}, u3];C6n={{1,2,3,4,5,6},{2,3,1,6,4,5},{3,1,2,5,6,4},{4,5,6,2,1,3},{5,6,4,1, 3,2},{6,4,5,3,2,1}}; C4C2=co[C4,C2];KC2=co[K,C2];D4=ca[8,7];C8=ca[8];Q8=ca[8,3];Oct= cd[{-1,-1,-1,{2}}]; Q8C2r={{1,2,3,4,5,6,7,8},{2,-5,1,8,-3,-4,6,-7},{3,1,-5,-6,-2,7,-8,4},{4,-6, 8,-5,7,-3,-1,2},{5,-3,-2,7,1,8,4,6},{6,7,-4,-2,8,-5,3,-1},{7,-8,6,-1, 4,2,-5,-3},{8,4,-7,3,6,-1,-2,-5}}; P8={{1,2,3,4,5,6,7,8},{2,1,-4 \[ImaginaryI],3 \[ImaginaryI],6, 5,-8 \[ImaginaryI],7 \[ImaginaryI]},{3,4 \[ImaginaryI], 1,-2 \[ImaginaryI],7,8 \[ImaginaryI], 5,-6 \[ImaginaryI]},{4,-3 \[ImaginaryI],2 \[ImaginaryI],1, 8,-7 \[ImaginaryI],6 \[ImaginaryI],5},{5,6,7,8,1,2,3,4},{6, 5,-8 \[ImaginaryI],7 \[ImaginaryI],2,1,-4 \[ImaginaryI], 3 \[ImaginaryI]},{7,8 \[ImaginaryI],5,-6 \[ImaginaryI],3, 4 \[ImaginaryI],1,-2 \[ImaginaryI]},{8,-7 \[ImaginaryI], 6 \[ImaginaryI],5,4,-3 \[ImaginaryI],2 \[ImaginaryI],1}}; Q8n={{1,2,3,4,5,6,7,8},{2,1,4,7,6,5,8,3},{3,8,1,2,7,4,5,6},{4,3,6,1,8,7,2, 5},{5,6,7,8,1,2,3,4},{6,5,8,3,2,1,4,7},{7,4,5,6,3,8,1,2},{8,7,2,5,4,3, 6,1}}; CL12=cl[1,2];(*CL3 isomorph*) C3C3=co[C3,C3];C3C4=co[C3,C4];C3K=co[C3,K];Q12=co[C3,C4,2];D3C2=co[C3,K,2];A4= ge[{3,2,2},{"ba"\[Rule]"abc","ca"\[Rule]"ab"}];A4a=Transpose[A4];D3C4r= ms[{3,15}]; \!\(\*FormBox[ RowBox[{ RowBox[{"C3P4", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12"}, {"2", "3", "1", "5", "6", "4", "8", "9", "7", "11", "12", "10"}, {"3", "1", "2", "6", "4", "5", "9", "7", "8", "12", "10", "11"}, {"4", "5", "6", "1", "2", "3", 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25}}};\)\[IndentingNewLine] \(loop[\([31]\)] = {{"\", {1. , 1\_30}, 0, "\", {31}, {}, {}, {}, {}}};\)\) \!\(\(loop[\([32]\)] = {{"\", {2. , 2\_15, 1}, 0, "\", {32}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {8, 8\_3, 19}, 11, "\", {4, 4, 2}, {"\" -> "\"}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {4. , 4\_7, 7}, 0, "\", {8, 4}, {}, {12, 22}, {15, 27}, {12, 22}}, \[IndentingNewLine]{"\", {4, 4\_7, 7}, 0, "\", {8, 4}, {"\" \[Rule] "\"}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {8, 4\_4, 8, 11}, 14, "\", {8, 2, 2}, {"\" -> "\"}, {2, 35}, {3, 25}, {}}, {"\", {12, 4\_5, 23}, 0, "\", {4, 2, 2, 2}, {"\" \[Rule] "\", "\" \[Rule] "\"}, {}, \ {}, {5, 13}}, \[IndentingNewLine]{"\", {12, 4\_5, 15}, 0, "\", {8, 2, 2}, {"\" \[Rule] "\", "\" \[Rule] "\"}, {}, \ {}, {7, 37}}, \[IndentingNewLine]{"\", {4, 4\_4, 12, 7}, 0, "\", {2, 2, 2, 2, 2}, {"\" -> "\", "\" -> "\", "\" -> \ "\", "\" \[Rule] "\", "\" -> "\", "\" -> \ "\"}, {}, {}, {}}, \[IndentingNewLine]{"\", {12, 4\_3, 8, 19}, 0, "\", {8, 2, 2}, 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{34, 1\_32}, 0, "\", {33, 2}, {"\" \[Rule] "\"}, {}, {}, {}}, {"\", {2. \ , 2\_32}, 0, "\", {66}, {}, {}, {}, {}}};\)\[IndentingNewLine] \(loop[\([67]\)] = {{"\", {1. , 1\_66}, 0, "\", {67}, {}, {}, {}, {}}};\)\[IndentingNewLine] \(loop[\([68]\)] = {{"\", {2, 2\_16, 34}, 0, "\", {17, 4}, {"\" \[Rule] "\"}, {}, {}, {}}, {"\", {2. , 2\_33}, 0, "\", {68}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {18, 1\_16, 2\_17}, 0, "\", {}, {}, {}, {}, {}}, {"\", {36, 2\_16}, 0, "\", {34, 2}, {"\" \[Rule] "\"}, {}, {}, {}}, {"\", {4. \ , 4\_16}, 0, "\", {17, 2, 2}, {}, {}, {}, {}}};\)\[IndentingNewLine] \(loop[\([69]\)] = {{"\", {1. , 1\_68}, 0, "\", {69}, {}, {}, {}, {}}};\)\[IndentingNewLine] \(loop[\([70]\)] = {{"\", {6, 1\_28, 6\_6}, 0, "\", {}, {}, {}, {}, {}}, {"\", {8, 1\_30, 8\_4}, 0, "\", {}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {36, 1\_34}, 0, "\", {}, {}, {3, 18, 21}, {2, 16, 24}, {}}, \[IndentingNewLine]{"\", {2. , 2\_34}, 0, "\", {70}, {}, {\(-18\), 21}, {16, 30}, {20, 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"\[IndentingNewLine]", \({"\", {40, 4\_8, 4}, 0, "\", {2, 3, 3, 2, 2}, {"\" \[Rule] "\", "\" \[Rule] "\"}, {\ \(-4\), 5, 19}, {2, 3, 18, 32}, {}}\), ",", "\[IndentingNewLine]", \({\*"\"\<\!\(C3\^2\)\!\(C2\^3\)\>\"", {8. , 8\_8, 4}, 0, "\", {3, 3, 2, 2, 2}, {}, {1, 19, 20}, {3, 18, 22}, {1, \(-8\), 19}}\)}], "}"}]}], ";"}]\) \!\(loop[\([73]\)] = {{"\", {1. , 1\_72}, 0, "\", {}, {}, {}, {}, {}}}; loop[\([74]\)] = {{"\", {38, 1\_36}, 0, "\", {}, {}, {}, {}, {}}};\[IndentingNewLine] \({"\", {2. , 2\_36}, 0, "\", {}, {}, {}, {}, {}};\)\[IndentingNewLine] loop[\([75]\)] = {{"\", {1. , 1\_74}, 4, "\", {}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {1. , 1\_74}, 4, "\", {}, {}, {}, {}, {}}}; loop[\([80]\)] = {{"\", {2, 2\_4, 14\_5}, 0, "\", {}, {}, {}, {}, {}}, {"\", {10, 70}, 0, "\", {}, {}, {}, {}, {}}, {"\", {2, 2\_6, 4\_6, 14\_3}, 0, "\", {}, {}, {}, {}, {}}};\[IndentingNewLine] \(loop[\([96]\)] = {{"\", {32, 8\_4, 32}, 0, "\", {3, 2, 2, 2, 2, 2}, {"\" \[Rule] "\", "\" \[Rule] "\", \ "\" \[Rule] "\", "\" \[Rule] "\", "\" \[Rule] \ "\"}, {}, {}, {}}};\)\[IndentingNewLine] \(loop[\([104]\)] = {{"\", {2, 2\_38, 26}, 0, "\", {13, 8}, {"\" \[Rule] "\"}, {}, {}, {}}};\)\ \[IndentingNewLine] \(loop[\([120]\)] = {{"\", {26, 1\_24, 2\_35}, 0, "\", {}, {}, {}, {}, {}}, {"\", {2, 2\_44, 30}, 0, "\", \ {}, {}, {}, {}, {}}};\)\[IndentingNewLine] loop[\([144]\)] = {{"\", {52, 4\_4, 6\_4, 10\_4, 12}, 0, "\", {}, {}, {}, {}, {}}}; loop[\([168]\)] = {{"\", {22, 1\_104, 2\_21}, 0, "\", {}, {}, {}, {}, {}}}; loop[\([256]\)] = {{"\", {\(-120\), 136, 32512}, 0, "\", {}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {4, 28\_2, 196}, 0, "\", {}, {}, {}, {}, {}}, \ \[IndentingNewLine]{"\", {4, 16\_2, 28, 32\_2, 44, 84}, 0, "\", {}, {}, {}, {}, {}}}; loop[\([257]\)] = {{"\", {}, 0, "\", {}, {}, {}, {}, {}}};\) \!\(sh["\"] := Module[{}, gi = {1, 2, 1, 1}; gp = {1, 1}; plex = {a, \(-b\)}; {a - b, a + b}]\n toPol[{x_, y_}, "\"] := {x - y, x + y}\n toVec[{x_, y_}, "\"] := {x + y, y - x}/2\n tohyPol[{a_, b_}, "\"] := Chop[{a\^2 - b\^2, arcTanh[a, b]}]\n tohyVec[{uu_, \[Phi]_}, "\"] := Chop[\@uu\ {Cosh[\[Phi]], Sinh[\[Phi]]}]\) \!\(sh["\"] := Module[{}, gi = {1, \(-2\), 1}; gp = {1, 0. }; plex = {a, \(-b\)}; {\(-a\^2\) - b\^2}]\[IndentingNewLine] toPol[{a_, b_}, "\"] := {a\^2 + b\^2, ArcTan[a, b]}\[IndentingNewLine] toVec[{x_, y_}, "\"] := {\@x\ Cos[y], \@x\ Sin[y]}\) sh["C3"]:=Module[{},gi={1,3,2,1,1};gp={1,1,0.};plex={a,c,b}; (*strictly, shape should be symdiff^2, not (a-b)^2 etc.*){a+b+c,((a-b)^2+(b-c)^2+(c-a)^2)/2}] toPol[{a_,b_,c_},"C3"]:={a+b+c,((a-b)^2+(b-c)^2+(c-a)^2)/2, ArcTan[2*a-b-c,-Sqrt[3]*(b-c)]} toVec[{\[Alpha]_,rr_,\[Theta]_},"C3"]:= Chop[{\[Alpha]+2*Sqrt[rr]*Cos[\[Theta]],\[Alpha]+2*Sqrt[rr]* Cos[\[Theta]+2*Pi/3], \[Alpha]+2*Sqrt[rr]*Cos[\[Theta]-2*Pi/3]}/3] \!\(sh["\"] := Module[{}, gi = {1, \(-3\), \(-2\), 1, 1}; gp = {1, 1, 0. }; plex = {a, \(-c\), \(-b\)}; {a - b + c, \((\((a + b)\)^2 + \((b + c)\)^2 + \((c - a)\)^2)\)/2}]; toPol[{a_, b_, c_}, "\"] := {a - b + c, \((\((a + b)\)^2 + \((b + c)\)^2 + \((c - a)\)^2)\)/ 2, \[IndentingNewLine]ArcTan[ 2*a + b - c, \(-Sqrt[3]\)*\((b + c)\)]};\n \(toVec[{\[Alpha]_, rr_, \[Theta]_}, "\"] := Chop[{\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta]], \(-\[Alpha]\) - 2*Sqrt[rr]* Cos[\[Theta] - 2*Pi/3], \[IndentingNewLine]\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] + 2*Pi/3]}/3];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, 3\ \[ImaginaryI], 2\ \[ImaginaryI], 1, 1}; gp = {1, 1, 0. }; plex =. ; {a + \[ImaginaryI]\ b - c, \ \((\((a - \[ImaginaryI]\ b)\)\^2 + \((a + c)\)\^2 + \((\ \[ImaginaryI]\ b + c)\)\^2)\)/2}];\)\n toPol[{a_, b_, c_}, "\"] := {a + \[ImaginaryI]\ b - c, \ \((\((a - \[ImaginaryI]\ b)\)\^2 + \((a + c)\)\^2 + \((\ \[ImaginaryI]\ b + c)\)\^2)\)/2, \[IndentingNewLine]ArcTan[ 2*a - \[ImaginaryI]\ b + c, Sqrt[3]*\((\[ImaginaryI]\ b + c)\)]}\[IndentingNewLine] \(toVec[{\[Alpha]_, rr_, \[Theta]_}, "\"] := Chop[{\[Alpha] + 2*Sqrt[rr]* Cos[\[Theta]], \[IndentingNewLine]\(-\[ImaginaryI]\) \((\ \[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] - 2*Pi/3])\), \(-\((\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] + 2*Pi/3])\)\)}/ 3];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, \(-3\)\ \[ImaginaryI], \(-2\)\ \[ImaginaryI], 1, 1}; gp = {1, 1, 0. }; plex =. ; {a - \[ImaginaryI]\ b - c, \ \((\((a + \[ImaginaryI]\ b)\)\^2 + \((\[ImaginaryI]\ b - \ c)\)\^2 + \((c + a)\)\^2)\)/2}];\)\[IndentingNewLine] toPol[{a_, b_, c_}, "\"] := {a - \[ImaginaryI]\ b - c, \ \((\((a + \[ImaginaryI]\ b)\)\^2 + \((\[ImaginaryI]\ b - c)\)\^2 \ + \((c + a)\)\^2)\)/2, \[IndentingNewLine]ArcTan[2*a + \[ImaginaryI]\ b + c, Sqrt[3]*\((\(-\[ImaginaryI]\)\ b + c)\)]}\[IndentingNewLine] toVec[{\[Alpha]_, rr_, \[Theta]_}, "\"] := \[IndentingNewLine]Chop[{\[Alpha] + 2*Sqrt[rr]* Cos[\[Theta]], \[ImaginaryI] \((\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] - 2*Pi/3])\), \(-\((\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] + 2*Pi/3])\)\)}/3]\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, \(-3\), \(-2\), 1, 1}; gp == {1, 1, 0. }; plex =. ; {a + \[ImaginaryI]\ b + \[ImaginaryI]\ c, \((\((a - \ \[ImaginaryI]\ b)\)^2 + \((a - \[ImaginaryI]\ c)\)^2 - \((b - c)\)^2)\)/ 2}];\)\[IndentingNewLine] toPol[{a_, b_, c_}, "\"] := {a - \[ImaginaryI]\ b - c, \ \((\((a + \[ImaginaryI]\ b)\)\^2 + \((\[ImaginaryI]\ b - c)\)\^2 \ + \((c + a)\)\^2)\)/2, \[IndentingNewLine]ArcTan[2*a + \[ImaginaryI]\ b + c, Sqrt[3]*\((\(-\[ImaginaryI]\)\ b + c)\)]}\[IndentingNewLine] toVec[{\[Alpha]_, rr_, \[Theta]_}, "\"] := Chop[{\[Alpha] + 2*Sqrt[rr]* Cos[\[Theta]], \[ImaginaryI] \((\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] - 2*Pi/3])\), \(-\((\[Alpha] + 2*Sqrt[rr]*Cos[\[Theta] + 2*Pi/3])\)\)}/3]\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 1}; gp = {}; plex =. ; {a\^3 + b\^3\ \[DoubleStruckCapitalJ]\^2 + c\^3\ \[DoubleStruckCapitalJ]\ - 3\ a\ b\ c}];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, 3 \[DoubleStruckCapitalJ]\^2, 2 \[DoubleStruckCapitalJ]\^2, 1, 1}; gp = {}; plex =. ; {\((a + b\ \[DoubleStruckCapitalJ]\^2 + c\ \[DoubleStruckCapitalJ]\^2)\), \((\((a - b\ \ \[DoubleStruckCapitalJ]\^2)\)^2 + \((b\ \ \[DoubleStruckCapitalJ]\^2 - c\ \[DoubleStruckCapitalJ]\^2)\)^2 + \((c\ \ \[DoubleStruckCapitalJ]\^2 - a)\)^2)\)/2}];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, 3 \[DoubleStruckCapitalJ]^2, 2 \[DoubleStruckCapitalJ]^2, 1}; gp = {}; plex =. ; {a\^3 + b\^3\ \[DoubleStruckCapitalJ] + c\^3\ \[DoubleStruckCapitalJ]\^2 - 3\ a\ b\ c\ \[DoubleStruckCapitalJ]}];\)\) \!\(sh["\"] := Module[{}, gi = {1, 4, 3, 2, 1, 1, 1}; gp = {1, 1, 1, 0. }; \[IndentingNewLine]plex = {a, \ \(-b\), \ c, \(-\ d\)}; {a + b + c + d, a - b + c - d, \((a - c)\)\^2 + \((b - d)\)\^2}\[IndentingNewLine] (*\((a\^2 + \ c\^2 - 2\ b\ d)\)\^2 - \((b\^2 + d\^2 - 2\ a\ c)\)\^2*) ]\[IndentingNewLine] toPol[{a_, b_, c_, d_}, "\"] := {a + b + c + d, a - b + c - d, \((a - c)\)\^2 + \((b - d)\)\^2, \[IndentingNewLine]ArcTan[a - c, b - d]}\[IndentingNewLine] toVec[{\[Alpha]_, \[Beta]_, \[Epsilon]\[Epsilon]_, \[Sigma]_}, "\"] := \ {\[Alpha] + \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]*Cos[\[Sigma]], \[Alpha] - \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]*Sin[\[Sigma]], \[Alpha] + \[Beta] - 2*Sqrt[\[Epsilon]\[Epsilon]]* Cos[\[Sigma]], \[IndentingNewLine]\[Alpha] - \[Beta] - 2*Sqrt[\[Epsilon]\[Epsilon]]*Sin[\[Sigma]]}/4\[IndentingNewLine] tohyPol[{a_, b_, c_, d_}, "\"] := Chop[{\((a + c)\)^2 - \((b + d)\)^2, arcTanh[a + c, b + d], \((a - c)\)^2 + \((b - d)\)^2, ArcTan[a - c, b - d]}]\[IndentingNewLine] tohyVec[{uu_, \[Phi]_, rr_, \[Theta]_}, "\"] := Module[{r = Sqrt[rr], u = Sqrt[uu]}, Chop[r*{Cos[\[Theta]], Sin[\[Theta]], \(-Cos[\[Theta]]\), \(-Sin[\[Theta]]\)}/2 + u*{Cosh[\[Phi]], Sinh[\[Phi]], Cosh[\[Phi]], Sinh[\[Phi]]}/2]]\) \!\(sh["\"] := Module[{}, gi = {1, 2, 3, 4, 1, 1, 1, 1}; gp = {1, 1, 1, 1}; plex = {a, \(-\ b\), \ \(-c\), d}; \[IndentingNewLine]{a + b - c - d, a - b + c - d, a - b - c + d, a + b + c + d} (*\((a\^2 - b\^2 - c\^2 + d\^2)\)\^2 - 4 \((a\ d - b\ c)\)\^2*) ]\[IndentingNewLine] \(toPol[{a_, b_, c_, d_}, "\"] := {a + b - c - d, a - b + c - d, a - b - c + d, a + b + c + d};\)\[IndentingNewLine] toVec[{o1_, o2_, o3_, o4_}, "\"] := {o1 + o2 + o3 + o4, o1 - o2 - o3 + o4, \(-o1\) + o2 - o3 + o4, \(-o1\) - o2 + o3 + o4}/ 4\[IndentingNewLine] tohyPol[{a_, b_, c_, d_}, "\"] := Chop[{\((a + c)\)^2 - \((b + d)\)^2, arcTanh[a + c, b + d], \[IndentingNewLine]\((a - c)\)^2 - \((b - d)\)^2, arcTanh[a - c, b - d]}]\[IndentingNewLine] tohyVec[{uu_, \[Phi]_, vv_, \[Psi]_}, "\"] := Module[{v = Sqrt[vv], u = Sqrt[uu]}, \[IndentingNewLine]Chop[ v*{Cosh[\[Psi]], Sinh[\[Psi]], \(-Cosh[\[Psi]]\), \(-Sinh[\[Psi]]\)}/ 2 + u*{Cosh[\[Phi]], Sinh[\[Phi]], Cosh[\[Phi]], Sinh[\[Phi]]}/ 2]]\) \!\(\* RowBox[{ StyleBox[\(sh["\"] (*Signature\ \((4, 0)\)*) := Module[{}, gi = {1, \(-2\), \(-3\), \(-4\), 2}; gp = {1, 0. , 1, 0. }; plex = {a, \(-\ b\), \ \(-c\), \(-d\)}; {a\^2 + b\^2 + c\^2 + d\^2}]\), FormatType->StandardForm], "\n", RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ StyleBox[\(a\^2 + b\^2 + c\^2 + d\^2\), FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], StyleBox["[", FormatType->StandardForm], \(a, b\), "]"}], ",", StyleBox[\(c\^2 + d\^2\), FormatType->StandardForm], ",", RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], "[", \(c, d\), "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], "\n", StyleBox[\(toVec[{rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := {\@\(rr - ss\)\ Cos[\[Rho]], \@\(rr \ - ss\)\ Sin[\[Rho]], \@ss\ Cos[\[Sigma]], \@ss\ Sin[\[Sigma]]}\), FormatType->StandardForm], "\n", StyleBox[\(sh["\"] := Module[{}, gi = {1, \(-4\), \(-3\), \(-2\), 1, 1}; \ {a\^2 + b\^2 + c\^2 + d\^2 + \(\@2\) \((a\ b + b\ c - a\ d + c\ d)\), a\^2 + b\^2 + c\^2 + d\^2 - \(\@2\) \((a\ b + b\ c - a\ d + c\ d)\)} (*{\((a\^2 + b\^2 + c\^2 + d\^2)\)\^2 - 2\ \((a\ b + b\ c - a\ d + c\ d)\)\^2}*) ]\), FormatType->StandardForm]}]\) \!\(\* RowBox[{ RowBox[{ StyleBox[\(sh["\"]\), FormatType->StandardForm], \( (*Signature\ \((1, 3)\)*) \), StyleBox[":=", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], RowBox[{ StyleBox["Module", FormatType->StandardForm], StyleBox["[", FormatType->StandardForm], StyleBox[\({}, gi = {1, 2, 3, 4, 2}; gp = {1, 0. , 1, 0. }; plex = {a, \(-\ b\), \ \(-c\), \(-d\)}; {\ a\^2 - b\^2 - c\^2 - d\^2}\), FormatType->StandardForm], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], RowBox[{ StyleBox["Module", FormatType->StandardForm], StyleBox["[", FormatType->StandardForm], RowBox[{ StyleBox[\({r = \[Sqrt]\((\((\((b - c)\)\^2 + \((c - d)\)\^2 + \ \((d - b)\)\^2)\)/2)\)}\), FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ StyleBox["r", FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], StyleBox["[", FormatType->StandardForm], RowBox[{"a", "/", StyleBox["r", FormatType->StandardForm]}], "]"}], ",", \(\((b + c + d)\)/3\), ",", RowBox[{"If", "[", RowBox[{\(c \[Equal] d\), ",", "0", ",", RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], "[", \(2 b - c - d, \(-\@3\) \((c - d)\)\), "]"}]}], "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], StyleBox["]", FormatType->StandardForm]}]}], StyleBox[";", FormatType->StandardForm]}], "\[IndentingNewLine]", StyleBox[\(toVec[{r_, \[Sigma]_, s_, \[Theta]_}, "\"] := {r\ Tan[\[Sigma]], s + 2*r*Cos[\[Theta]]/3, s + 2*r*Cos[\[Theta] + 2*Pi/3]/3, \[IndentingNewLine]s + 2*r*Cos[\[Theta] + 4*Pi/3]/3};\), FormatType->StandardForm], "\[IndentingNewLine]", \(tohyPol[{a_, b_, c_, d_}, "\"] := Chop[{a\^2 - b\^2, arcTanh[a, b], c\^2 + d\^2, ArcTan[c, d]}];\), "\n", \(tohyVec[{uu_, \[Phi]_, vv_, \[Psi]_}, "\"] := Module[{v = \@vv, u = \@uu}, Chop[\ v\ {0, 0, Cos[\[Psi]], Sin[\[Psi]]} + u\ {Cosh[\[Phi]], Sinh[\[Phi]], 0, 0}]]\)}]\) \!\(\* RowBox[{ StyleBox[ RowBox[{ RowBox[{\(sh["\"]\), ":=", RowBox[{"Module", "[", RowBox[{\({}\), ",", RowBox[{\(gi = {1, \(-2\), \(-3\), 4, 1, 1}\), ";", \(gp = {1, 0. , 1, 0. }\), ";", \(plex = {a, \(-\ b\), \ \(-c\), d}\), ";", " ", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], \(a - d\), ")"}], "2"], "+", SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], \(b + c\), ")"}], "2"]}], ",", RowBox[{ SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], RowBox[{"a", StyleBox["+", FormatType->StandardForm], "d"}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], RowBox[{"b", StyleBox["-", FormatType->StandardForm], "c"}], ")"}], "2"]}]}], "}"}]}]}], "\[IndentingNewLine]", \( (*L4\ equivalent\ is \((a\^2 + b\^2 + c\ \^2 + d\^2)\)\^2 + 4 \((a\ d - b\ c)\)\^2*) \), "]"}]}], ";"}], FormatType->StandardForm], "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], \(a - d\), ")"}], "2"], "+", SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], \(b + c\), ")"}], "2"]}], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], StyleBox["[", FormatType->StandardForm], \(a - d, b + c\), "]"}], ",", "\[IndentingNewLine]", StyleBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"a", StyleBox["+", FormatType->StandardForm], "d"}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], RowBox[{"b", StyleBox["-", FormatType->StandardForm], "c"}], ")"}], "2"]}], FormatType->StandardForm], ",", RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], "[", \(a + d, b - c\), "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm]}], "\n", StyleBox[\(toVec[{rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := \[IndentingNewLine]{\ \@rr\ Cos[\ \[Rho]] + \@ss\ Cos[\[Sigma]], \ \@rr\ Sin[\[Rho]] + \@ss\ Sin[\[Sigma]], \ \ \@rr\ Sin[\[Rho]] - \@ss\ Sin[\[Sigma]], \(-\@rr\)\ Cos[\[Rho]] + \@ss\ Cos[\ \[Sigma]]}/2;\), FormatType->StandardForm]}]\) \!\(\* RowBox[{ RowBox[{\(sh["\"]\), ":=", RowBox[{ "Module", "[", \({}, gi = {1, \(-2\), 3, \(-4\), 1, 1}; gp = {1, 0. , 1, 0. }; plex = {a, \(-\ b\), \(-c\), \(-d\)}; \[IndentingNewLine]\ {\((a - c)\)\^2 + \ \((b - d)\)\^2, \((a + c)\)\^2 + \((b + d)\)\^2}\), "\[IndentingNewLine]", RowBox[{"(*", RowBox[{\(\((a\^2 + b\^2 + c\^2 + d\^2)\)\^2\), "-", RowBox[{"4", SuperscriptBox[ RowBox[{"(", StyleBox[\(a\ c + b\ d\), FormatType->StandardForm], ")"}], "2"]}]}], "*)"}], "]"}]}], ";", RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], RowBox[{"a", "-", StyleBox["c", FormatType->StandardForm]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{ StyleBox["(", FormatType->StandardForm], RowBox[{"b", StyleBox["-", FormatType->StandardForm], StyleBox["d", FormatType->StandardForm]}], ")"}], "2"]}], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox["ArcTan", FormatType->StandardForm], StyleBox["[", FormatType->StandardForm], \(a - c, b - d\), "]"}], ",", "\[IndentingNewLine]", StyleBox[\(\((a + c)\)\^2 + \((b + d)\)\^2\), FormatType->StandardForm], ",", RowBox[{ StyleBox["ArcTan", FormatType->StandardForm], "[", \(a + c, b + d\), "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], StyleBox[";", FormatType->StandardForm], StyleBox[\(toVec[{rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := {\ \ \@rr\ Cos[\[Rho]] + \@ss\ Cos[\ \[Sigma]], \ \@rr\ Sin[\[Rho]] + \@ss\ Sin[\[Sigma]], \ \[IndentingNewLine]\(-\@rr\)\ Cos[\[Rho]] + \(\@ss\) Cos[\[Sigma]], \ \(-\@rr\)\ Sin[\[Rho]] + \@ss\ \ Sin[\[Sigma]]}/2\), FormatType->StandardForm], StyleBox[";", FormatType->StandardForm]}]\) \!\(\(sh["\"] := Module[{}, gi = {1, 4, \(-3\), 2, 1, 1}; plex = {a, \(-\ b\), c, \(-d\)}; (*\({\((a\^2 + b\^2 + c\^2 + d\^2)\)\^2 - 2 \((a\ b - b\ c + a\ d + c\ d)\)\^2} {\((a\^2 - b\^2 + c\^2 \ - d\^2)\)\^2 + 2 \((a\ b + b\ c - a\ d + c\ d)\)\^2}\)\(,\)*) {a\^2 + b\^2 + c\^2 + d\^2 + \(\@2\) \((a\ b - b\ c + a\ d + c\ d)\), a\^2 + b\^2 + c\^2 + d\^2 - \(\@2\) \((a\ b - b\ c + a\ d + c\ d)\)}];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gi = {1, \(-4\), 3, \(-2\), 1, 1, 1}; gp = {1, 1, 1, 0. }; plex = {a, \ \(-b\), \ c, \(-d\)}; {a + b - c - d, a - b - c + d, \((a + c)\)\^2 + \((b + d)\)\^2} (*\((a\^2 + c\^2 + 2 b\ d)\)\ \^2 + \((b\^2 + 2\ ac + d\^2)\)\^2*) ];\)\[IndentingNewLine] \(toPol[{a_, b_, c_, d_}, "\"] := {a + b - c - d, a - b - c + d, \((a + c)\)\^2 + \((b + d)\)\^2, ArcTan[a + c, b + d]};\)\[IndentingNewLine] \(toVec[{\[Alpha]_, \[Beta]_, \[Epsilon]\[Epsilon]_, \[Sigma]_}, \ "\"] := {\[Alpha] + \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]* Cos[\[Sigma]], \[IndentingNewLine]\[Alpha] - \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]* Sin[\[Sigma]], \[IndentingNewLine]\(-\[Alpha]\) - \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]* Cos[\[Sigma]], \[IndentingNewLine]\(-\[Alpha]\) + \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]*Sin[\[Sigma]]}/4;\)\) \!\(\* RowBox[{\(sh["\"] (*Signature\ \((2, 2)\), \ matching\ \(Twistors!\)\ *) := Module[{}, gi = {1, 2, \(-3\), 4, 2}; gp = {1, 0. , 1, 0. }; plex = {a, \(-\ b\), \ \(-c\), \(-d\)}; \[IndentingNewLine]\ {a\^2 - b\^2 + c\^2 - d\^2}]\), "\[IndentingNewLine]", RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ RowBox[{ StyleBox[\(a\^2\), FormatType->StandardForm], StyleBox["+", FormatType->StandardForm], StyleBox[\(c\^2\), FormatType->StandardForm], StyleBox["-", FormatType->StandardForm], StyleBox[\(b\^2\), FormatType->StandardForm], StyleBox["-", FormatType->StandardForm], \(d\^2\)}], StyleBox[",", FormatType->StandardForm], StyleBox[\(ArcTan[a, c]\), FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox[\(b\^2\), FormatType->StandardForm], StyleBox["+", FormatType->StandardForm], \(d\^2\)}], ",", RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], "[", \(b, d\), "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], "\n", StyleBox[\(toVec[{rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := {\ \ \@\(rr + ss\)\ Cos[\[Rho]], \ \ \@ss\ Cos[\[Sigma]], \@\(rr + ss\)\ Sin[\[Rho]], \@ss\ Sin[\[Sigma]]}\), FormatType->StandardForm]}]\) \!\(\* RowBox[{\(sh["\"] := Module[{}, gi = {1, \(-4\), 3, \(-2\), 1, 1, 1}; gp = {1, 1, 1, 0. }; plex = {a, \(-b\), c, \(-d\)}; \[IndentingNewLine]\ {a + b + c - d, a - b + c + d, \((a - c)\)\^2 + \((b + d)\)\^2}\[IndentingNewLine] (*\((a\^2 \ + c\^2 + 2\ b\ d)\)\^2 - \((b\^2 - 2\ a\ c + d\^2)\)\^2*) ]\), "\n", RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{\(a + b + c - d\), ",", \(a - b + c + d\), ",", \(\((a - c)\)\^2 + \((b + d)\)\^2\), StyleBox[",", FormatType->StandardForm], RowBox[{ StyleBox[ RowBox[{"A", StyleBox["rcTan", FormatType->StandardForm]}]], StyleBox["[", FormatType->StandardForm], \(a - c, b + d\), "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], "\n", StyleBox[\(toVec[{\[Alpha]_, \[Beta]_, \[Epsilon]\[Epsilon]_, \[Sigma]_}, \ "\"] := \[IndentingNewLine]{\[Alpha] + \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]* Cos[\[Sigma]], \[IndentingNewLine]\[Alpha] - \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]* Sin[\[Sigma]], \[IndentingNewLine]\[Alpha] + \[Beta] - 2*Sqrt[\[Epsilon]\[Epsilon]]* Cos[\[Sigma]], \[IndentingNewLine]\(-\[Alpha]\) + \[Beta] + 2*Sqrt[\[Epsilon]\[Epsilon]]*Sin[\[Sigma]]}/4\), FormatType->StandardForm], "\[IndentingNewLine]", \(tohyPol[{a_, b_, c_, d_}, "\"] := Chop[{\((a + c)\)^2 - \((b - d)\)^2, arcTanh[a + c, b - d], \((a - c)\)^2 + \((b + d)\)^2, ArcTan[a - c, b + d]}]\), "\[IndentingNewLine]", \(tohyVec[{uu_, \[Phi]_, rr_, \[Theta]_}, "\"] := Module[{r = Sqrt[rr], u = Sqrt[uu]}, Chop[r*{Cos[\[Theta]], Sin[\[Theta]], \(-Cos[\[Theta]]\), Sin[\[Theta]]}/2 + u*{Cosh[\[Phi]], Sinh[\[Phi]], Cosh[\[Phi]], \(-Sinh[\[Phi]]\)}/ 2]]\), "\[IndentingNewLine]", \(sh["\"] (*Signature\ \ \((3, 1)\)*) := Module[{}, gi = {1, \(-2\), \(-3\), 4, 2}; gp = {1, 0. , 1, 0. }; plex = {a, \(-\ b\), \ \(-c\), \(-d\)}; \[IndentingNewLine]\ {a\^2 + b\^2 + c\^2 - d\^2}]\), "\n", RowBox[{ StyleBox[\(toPol[{a_, b_, c_, d_}, "\"]\), FormatType->StandardForm], StyleBox[":=", FormatType->StandardForm], StyleBox[" ", FormatType->StandardForm], RowBox[{ StyleBox["{", FormatType->StandardForm], RowBox[{ StyleBox[\(a\^2 + b\^2\), FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], StyleBox[\(If[b \[NotEqual] 0, ArcTan[a, b], 0]\), FormatType->StandardForm], StyleBox[",", FormatType->StandardForm], \(c\^2 - d\^2\), ",", RowBox[{"If", "[", RowBox[{\(d \[NotEqual] 0\), ",", RowBox[{ StyleBox[ RowBox[{"a", StyleBox["rcTanh", FormatType->StandardForm]}]], "[", \(c, d\), "]"}], ",", "0"}], "]"}]}], StyleBox["}", FormatType->StandardForm]}]}], StyleBox[" ", FormatType->StandardForm], "\n", StyleBox[\(toVec[{rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := {\ \ \@rr\ Cos[\[Rho]], \ \@rr\ \ Sin[\[Rho]], \@ss\ Cosh[\[Sigma]], \@ss\ Sinh[\[Sigma]]}\), FormatType->StandardForm], "\[IndentingNewLine]", \(sh["\"] := Module[{}, gi = {1, 2, 3, 4, 1, 1}; gp =. ; plex =. ; \[IndentingNewLine]\ {a\^2 - b\^2 - c\^2 - d\^2, a\^2 + b\^2 + c\^2 + d\^2}]\)}]\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 5, 4, 3, 2, 1, 1}; plex =. ; \[IndentingNewLine]{a + b + c + d + e, \((\((a - b)\)^2 + \((b - c)\)^2 + \((c - d)\)^2 + \((d - e)\)^2 + \((e - a)\)^2 + \[IndentingNewLine]\((a - c)\)^2 + \((b - d)\)^2 + \((c - e)\)^2 + \((d - a)\)^2 + \((e - b)\)^2)\)^2/ 16 - \[IndentingNewLine]5\ \((a\ b - a\ c + b\ c - a\ d - b\ d + \ c\ d + a\ e - b\ e - c\ e + d\ e)\)\^2/4}]\n toVec[{\[Alpha]_, \[Epsilon]_, \[Sigma]_}, "\"] := \(\({\[Alpha] + \ \[Epsilon]*Sin[\[Sigma]], \[Alpha] + \[Epsilon]* Sin[\[Sigma] + 2 \[Pi]/5], \[IndentingNewLine]\[Alpha] + \[Epsilon]* Sin[\[Sigma] + 4 \[Pi]/5], \[Alpha] + \[Epsilon]* Sin[\[Sigma] + 6 \[Pi]/5], \[Alpha] + \[Epsilon]* Sin[\[Sigma] + 8 \[Pi]/5]}/5\)\(\ \)\)\) \!\(sh["\"] := Module[{}, gi = {1, 3, 2, 4, 6, 5, 1, 1, 1, 1}; gp = {1, 1, 1, 0. , 1, 0. }; plex = {a, c, b, d, f, e}; \[IndentingNewLine]{a + b + c + d + e + f, a + b + c - d - e - f, \((\((a - b + d - e)\)\^2 + \((a - c + d - f)\)\^2 + \((b - c + \ e - f)\)\^2)\)/ 2, \((\((a - b - d + e)\)\^2 + \((a - c - d + f)\)\^2 + \((b - c - \ e + f)\)\^2)\)/2}]\n toVec[{\[Alpha]_, \[Delta]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\[Kappa]_, \ \[Psi]_}, "\"] := Module[{\[Eta] = 2 \@ \[Eta]\[Eta], \[Kappa] = 2 \@ \[Kappa]\[Kappa]}, {\ \ \[Alpha] + \[Delta] + \[Eta]\ Cos[\ \[Chi]] + \[Kappa]\ Cos[\[Psi]], \[Alpha] + \[Delta] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/3] + \[Kappa]\ Cos[\[Psi] + 2 \[Pi]/3], \[Alpha] + \[Delta] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/3] + \[Kappa]\ Cos[\[Psi] - 2\ \[Pi]/ 3], \[IndentingNewLine]\[Alpha] - \[Delta] + \[Eta]\ \ Cos[\[Chi]] - \[Kappa]\ Cos[\[Psi]], \[Alpha] - \[Delta] + \[Eta]\ Cos[\[Chi] \ + 2\ \[Pi]/3] - \[Kappa]\ Cos[\[Psi] + 2 \[Pi]/3], \[Alpha] - \[Delta] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/3] - \[Kappa]\ Cos[\[Psi] - 2\ \[Pi]/3]}/6]\n toPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{a1 = Chop[2 a - b - c + 2 d - e - f], a2 = Chop[2 a - b - c - 2 d + e + f], b1 = Chop[Simplify[c - b - e + f]], b2 = Chop[Simplify[c - b + e - f]]}, {a + b + c + d + e + f, a + b + c - d - e - f, \((\((a - b + d - e)\)\^2 + \((b - c + e - f)\)\^2 + \((c - a + \ f - d)\)\^2)\)/2, ArcTan[a1, Sqrt[3] b1], \((\((a - b - d + e)\)\^2 + \((b - c - e + f)\)\^2 + \ \((c - a - f + d)\)\^2)\)/2, ArcTan[a2, Sqrt[3] b2]}]\n tohyPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{a1 = Chop[2 a - b - c + 2 d - e - f], a2 = Chop[2 a - b - c - 2 d + e + f], b1 = Chop[Simplify[c - b - e + f]], b2 = Chop[ Simplify[ c - b + e - f]]}, {\((a + b + c)\)\^2 - \((d + e + f)\)\^2, arcTanh[a + b + c, d + e + f], \((\((a - b + d - e)\)\^2 + \((b - c + e - f)\)\^2 + \ \((c - a + f - d)\)\^2)\)/2, ArcTan[a1, Sqrt[3] b1], \((\((a - b - d + e)\)\^2 + \((b - c - e + f)\)\^2 + \ \((c - a - f + d)\)\^2)\)/2, ArcTan[a2, Sqrt[3] b2]}]\n tohyVec[{\[Gamma]\[Gamma]_, \[Delta]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := Module[{\[Gamma] = \@\[Gamma]\[Gamma], \[Eta] = \@\[Eta]\[Eta], \[Kappa] \ = \@\[Kappa]\[Kappa]}, {\ \[Gamma]\ Cosh[\[Delta]] + \[Eta]\ Cos[\[Chi]] + \ \[Kappa]\ Cos[\[Psi]], \[Gamma]\ Cosh[\[Delta]] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/3] + \[Kappa]\ Cos[\[Psi] + 2 \[Pi]/ 3], \[Gamma]\ Cosh[\[Delta]] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/3] + \[Kappa]\ Cos[\[Psi] - 2\ \[Pi]/ 3], \[Gamma]\ Sinh[\[Delta]] + \[Eta]\ Cos[\[Chi]] - \ \[Kappa]\ Cos[\[Psi]], \[Gamma]\ Sinh[\[Delta]] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/3] - \[Kappa]\ Cos[\[Psi] + 2 \[Pi]/ 3], \[Gamma]\ Sinh[\[Delta]] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/3] - \[Kappa]\ Cos[\[Psi] - 2\ \[Pi]/3]}/3]\) \!\(\(sh["\"] := Module[{}, gi = {1, 2, 5, 4, 3, 6, 1, 1, 2}; gp = {1, 1, 1, 0. , 1, 0. }; plex =. ; \[IndentingNewLine]{a + b + c + d + e + f, a - b + c - d + e - f, \((\((a - c)\)\^2 + \((c - e)\)\^2 + \((e - a)\)\^2 - \((b - \ d)\)\^2 - \((d - f)\)\^2 - \((f - b)\)\^2)\)/2}];\)\n toPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{ace = \((\((a - c)\)\^2 + \((c - e)\)\^2 + \((e - a)\)\^2)\)/ 2, \[Chi] = ArcTan[2\ a - c - e, Sqrt[3] \((e - c)\)], bdf = \((\((b - d)\)\^2 + \((d - f)\)\^2 + \((f - b)\)\^2)\)/ 2, \[Psi] = ArcTan[2\ b - d - f, Sqrt[3] \((f - d)\)]}, {a + b + c + d + e + f, a - b + c - d + e - f, ace - bdf, \[Chi], ace, \[Psi]}]\n toVec[{\[Gamma]_, \[Delta]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\[Kappa]_, \ \[Psi]_}, "\"] := Module[{\[Eta] = 4 \@ \[Kappa]\[Kappa], \[Kappa] = 4 \@\( \[Kappa]\[Kappa] - \[Eta]\[Eta]\)}, {\ \[Gamma] + \[Delta] + \ \[Eta]\ Cos[\[Chi]]\ , \[Gamma] - \[Delta] + \[Kappa]\ \ Cos[\[Psi]], \ \[Gamma] + \[Delta] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/3], \[Gamma] - \[Delta] + \[Kappa]\ \ Cos[\[Psi] + 2 \[Pi]/3], \[Gamma] + \[Delta] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/3], \[Gamma] - \[Delta] + \[Kappa]\ \ Cos[\[Psi] - 2\ \[Pi]/3]}/6]\n tohyPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{\[Chi]}, {\((a + c + e)\)\^2 - \((b + d + f)\)\^2, arcTanh[a + c + e, b + d + f], \((\((a - c)\)\^2 + \((c - e)\)\^2 + \((e - a)\)\^2)\)/ 2, \[IndentingNewLine]\[Chi] = ArcTan[2\ a - c - e, Sqrt[3] \((e - c)\)], \((\((b - d)\)\^2 + \((d - f)\)\^2 + \((f - b)\)\^2)\ \)/2, \[IndentingNewLine]ArcTan[2\ b - d - f, Sqrt[3] \((f - d)\)] - \[Chi]}]\n tohyVec[{\[Gamma]\[Gamma]_, \[Phi]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := Module[{\[Gamma] = 2 \@ \[Gamma]\[Gamma], \[Eta] = 4 \@ \[Eta]\[Eta], \[Kappa] = 4 \@ \[Kappa]\[Kappa]}, {\[Gamma]\ Cosh[\[Phi]] + \[Eta]\ \ Cos[\[Chi]], \[Gamma]\ Sinh[\[Phi]] + \[Kappa]\ \ Cos[\[Chi] + \[Psi]], \ \[Gamma]\ Cosh[\[Phi]] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/ 3], \[IndentingNewLine]\[Gamma]\ Sinh[\[Phi]] + \ \[Kappa]\ \ Cos[\[Chi] + \[Psi] + 2 \[Pi]/3], \[Gamma]\ Cosh[\[Phi]] + \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/ 3], \[Gamma]\ Sinh[\[Phi]] + \[Kappa]\ \ Cos[\[Chi] + \ \[Psi] - 2\ \[Pi]/3]}/6]\) \!\(TraditionalForm\`sh["\"] := Module[{}, gi = {1, 3, 2, 5, 4, 6, 1, 1, 1, 1}; gp =. ; plex =. ; \[IndentingNewLine]{a + b + c - d - e - f, a + b + c + d + e + f, \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - a)\)\^2 - \((d - \ e)\)\^2 - \((e - f)\)\^2 - \((f - d)\)\^2)\)/ 2, \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - a)\)\^2 + \((d - \ e)\)\^2 + \((e - f)\)\^2 + \((f - d)\)\^2)\)/2}]\) \!\(sh["\"] := (*Revised\ \(8/1\)/6*) \[IndentingNewLine]Module[{}, gp = {1, 0. , 1, 0. , 1, 0. }; gi = {1, 3, 2, \(-4\), \(-6\), \(-5\), 1, 1, 1}; plex = {a, b, c, \(-d\), \(-e\), \(-f\)}; \[IndentingNewLine]{\((a + b + c)\)\^2 \ + \((d + e + f)\)\^2, 3 \((\((a - b + \((d + e - 2 f)\) r3)\)\^2 + \((\((a + b - 2 c)\) \ r3 - d + e)\)\^2)\)/4, 3 \((\((a - b - \((d + e - 2 f)\) r3)\)\^2 + \((\((a + b - 2 c)\) \ r3 + d - e)\)\^2)\)/4}]\) \!\(toPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{f1 = a - b + \((d + e - 2 f)\) r3, f2 = \((a + b - 2 c)\) r3 - d + e, f3 = a - b - \((d + e - 2 f)\) r3, f4 = \((a + b - 2 c)\)/r3 + d - e}, {\((a + b + c)\)\^2 + \((d + e + f)\)\^2 (*\[Epsilon]\ \[Epsilon]*) , If[d + e + f \[Equal] 0, If[a + b + c < 0, \[Pi], 0], ArcTan[a + b + c, d + e + f]] (*\[Sigma]*) , \[IndentingNewLine]Expand[ 3 \((f1\^2 + f2\^2)\)/4] (*\[Eta]\[Eta]*) , If[f2 \[Equal] 0, If[f1 < 0, \[Pi], 0], ArcTan[f1, f2]] - \[Pi]/ 6 (*\[CurlyKappa]*) , \[IndentingNewLine]Expand[ 3 \((f3\^2 + f4\^2)\)/4] (*\[Kappa]\[Kappa]*) , If[f4 \[Equal] 0, If[f3 < 0, \[Pi], 0], ArcTan[f3, f4]] - \[Pi]/ 6 (*\[Psi]*) }]\[IndentingNewLine] toVec[{\[Epsilon]\[Epsilon]_, \[Sigma]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := Module[{\[Epsilon] = \@\[Epsilon]\[Epsilon]/ 3, \[Eta] = \@\(\[Eta]\[Eta]/12\), \[Kappa] = \ \@\(\[Kappa]\[Kappa]/12\), x = \[Chi] + \[Pi]/6, y = \[Psi] + \[Pi]/ 6}, \[IndentingNewLine]{ (*a*) \[Epsilon]\ Cos[\[Sigma]] + \ \[Eta]\ Cos[x] + \[Eta]\ r3\ Sin[x] + \[Kappa]\ Cos[y] + \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine] (*b*) \[Epsilon]\ Cos[\[Sigma]] - \ \[Eta]\ Cos[x] + \[Eta]\ r3\ Sin[x] - \[Kappa]\ Cos[y] + \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine] (*c*) \[Epsilon]\ Cos[\[Sigma]] - \ \[Eta]\ 2 r3\ Sin[x] - \[Kappa]\ 2 r3\ Sin[y], \[IndentingNewLine] (*d*) \[Epsilon]\ Sin[\[Sigma]] + \ \[Eta]\ r3\ Cos[x] - \[Eta]\ Sin[x] - \[Kappa]\ r3\ Cos[y] + \[Kappa]\ Sin[ y], \[IndentingNewLine] (*e*) \[Epsilon]\ Sin[\[Sigma]] + \ \[Eta]\ r3\ Cos[x] + \[Eta]\ Sin[x] - \[Kappa]\ r3\ Cos[y] - \[Kappa]\ Sin[ y], \[IndentingNewLine] (*f*) \[Epsilon]\ Sin[\[Sigma]] - \ \[Eta]\ 2 r3\ Cos[x] + \[Kappa]\ 2 r3\ Cos[y]}]\) \!\(sh["\"] := Module[{}, gi = {1, 3, 2, \(-4\), \(-5\), \(-6\), 1, 2}; gp = {1, 2, 1, 0. , 1, 0. }; plex = {a, c, b, \(-d\), \(-e\), \(-f\)}; {\((a + b + c)\)\^2 + \((d + e + \ f)\)\^2, \((\((a - b)\)\^2 + \((a - c)\)\^2 + \((b - c)\)\^2 + \((d - e)\)\^2 \ + \((d - f)\)\^2 + \((e - f)\)\^2)\)/2}]\[IndentingNewLine] toPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{abc = \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - a)\)\^2)\)/2, def = \((\((d - e)\)\^2 + \((e - f)\)\^2 + \((f - d)\)\^2)\)/ 2}, {\((a + b + c)\)\^2 + \((d + e + f)\)\^2, If[d + e + f === 0, 0, ArcTan[a + b + c, d + e + f]], abc + def, If[c == b, 0, ArcTan[2\ a - b - c, Sqrt[3] \((c - b)\)]], def, If[e == f, 0, ArcTan[2\ d - e - f, Sqrt[3] \((f - e)\)]]}]\n toVec[{\[Epsilon]\[Epsilon]_, \[Sigma]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := Module[{\[Epsilon] = \@\[Epsilon]\[Epsilon]/3, \[Eta] = 2 \@\( \[Eta]\[Eta] - \[Kappa]\[Kappa]\)/3, \[Kappa] = 2 \@ \[Kappa]\[Kappa]/ 3}, {\ \[Epsilon]\ Cos[\[Sigma]] + \[Eta]\ Cos[\[Chi]]\ , \ \[IndentingNewLine]\[Epsilon]\ Cos[\[Sigma]] + \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/ 3], \[IndentingNewLine]\[Epsilon]\ Cos[\[Sigma]] + \[Eta]\ \ Cos[\[Chi] - 2\ \[Pi]/ 3], \[IndentingNewLine]\[Epsilon]\ Sin[\[Sigma]] + \ \[Kappa]\ \ Cos[\[Psi]], \[IndentingNewLine]\[Epsilon]\ Sin[\[Sigma]] + \ \[Kappa]\ \ Cos[\[Psi] + 2\ \[Pi]/ 3], \[IndentingNewLine]\[Epsilon]\ Sin[\[Sigma]] + \ \[Kappa]\ \ Cos[\[Psi] - 2\ \[Pi]/3]}]\) \!\(sh["\"] := Module[{}, gi = {1, \(-2\), \(-5\), \(-4\), \(-3\), \(-6\), 1, 2}; gp = {1, 1, 1, 0. , 1, 0. }; plex = {a, c, b, \(-d\), \(-e\), \(-f\)}; {\((a - c + e)\)\^2 + \((b - d + \ f)\)\^2, \((\((a + c)\)^2 + \((c + e)\)^2 + \((a - e)\)^2 + \((b + d)\)^2 + \((d + f)\)^2 + \((b - f)\)^2)\)/ 2}]\[IndentingNewLine] toPol[{a_, b_, c_, d_, e_, f_}, "\"] := Module[{ace = \((\((a + c)\)^2 + \((a - e)\)^2 + \((c + e)\)^2)\)/2, bdf = \((\((b + d)\)^2 + \((d + f)\)^2 + \((b - f)\)^2)\)/ 2}, {\((a - c + e)\)\^2 + \((b - d + f)\)\^2, If[b + f === d, 0, ArcTan[a - c + e, b - d + f]], \[IndentingNewLine]ace + bdf, If[c \[Equal] \(-e\), 0, ArcTan[2\ a - e + c, Sqrt[3] \((\(-c\) - e)\)]], \[IndentingNewLine]bdf, If[d \[Equal] \(-f\), 0, ArcTan[2\ b + d - f, Sqrt[3] \((\(-d\) - f)\)]]}]\n \(toVec[{\[Epsilon]\[Epsilon]_, \[Sigma]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := Module[{\[Epsilon] = \@\[Epsilon]\[Epsilon]/3, \[Eta] = 2 \@\( \[Eta]\[Eta] - \[Kappa]\[Kappa]\)/3, \[Kappa] = 2 \@ \[Kappa]\[Kappa]/ 3}, {\ \ \[Epsilon]\ Cos[\[Sigma]] + \[Eta]\ Cos[\[Chi]]\ , \ \[IndentingNewLine]\ \ \[Epsilon]\ Sin[\[Sigma]] + \[Kappa]\ Cos[\[Psi]], \ \[IndentingNewLine]\(-\[Epsilon]\)\ Cos[\[Sigma]] - \[Eta]\ Cos[\[Chi] - 2\ \[Pi]/ 3], \[IndentingNewLine]\(-\[Epsilon]\)\ Sin[\[Sigma]] - \ \[Kappa]\ Cos[\[Psi] - 2\ \[Pi]/ 3], \[IndentingNewLine]\ \[Epsilon]\ Cos[\[Sigma]] + \ \[Eta]\ Cos[\[Chi] + 2\ \[Pi]/ 3], \[IndentingNewLine]\ \[Epsilon]\ Sin[\[Sigma]] + \ \[Kappa]\ Cos[\[Psi] + 2\ \[Pi]/3]}];\)\) sh["g2401c"]:=Module[{},gi={1,3,2,-4*I,-5*I,-6*I,2,1};gp={1,1,1,0.,1,0.};;plex={a,c,b,-d,-e,-f}; {((a-b)^2+(a-c)^2+(b-c)^2-I*((d-e)^2+(d-f)^2+(e-f)^2))/2,(a+b+c)^2-I*(d+e+f)^2}] sh["D6r"]:=Module[{},gi={1,2,-5,4,-3,6,1,1,2}; gp={1,1,1,0.,1,0.};;plex=.;{a-b-c+d+e-f,a+b-c-d+e+f, ((a+c)^2+(c+e)^2+(a-e)^2-(b+d)^2-(d+f)^2-(b-f)^2)/2}] sh["D3C2c"]:=Module[{},gi={1,2,3,5,4,6,1,1,2};plex={a,c,b,-d,-e,-f}; gp={1,1,1,0.,1,0.};{(a-b+c-d-e-f),(a+b-c-d-e+f), ((a+d)^2+(a+e)^2+(d-e)^2-(b+c)^2-(b-f)^2-(c+f)^2)/ 2}] (*Two relabellings of C3C2*) sh["KC3c"]:=Module[{},gi={1,-3,-2,4,-6,-5,1,1,1,1};gp={1,1,1,0.,1,0.};;plex={a,b,c,-d,-e,-f};{a-b+c-d+e-f,a-b+c+d-e+f,((a+b-d-e)^2+(a-c-d+f)^2+(b+c-e-f)^2)/2,((a+b+d+e)^2+(a-c+d-f)^2+(b+c+e+f)^2)/2}]; sh["C3Kr"]:=Module[{},gi={1,3,2,4,6,5,1,1,1,1};gp={1,1,1,0.,1,0.};;{a-b-c-d+e+f,a-b-c+d-e-f,((a+b-d-e)^2+(a+c-d-f)^2+(b-c-e+f)^2)/2,((a+b+d+e)^2+(a+c+d+f)^2+(b-c+e-f)^2)/2}] \!\(sh["\"] := Module[{}, \(gp = {1, 1, 1, 0. , 1, 0. };\); gi = {1, \(-3\), \(-2\), \(-4\), 6, 5, 1, 1}; plex = {a, b, c, \(-d\), \(-e\), \(-f\)}; \[IndentingNewLine]{\((a - b + c)\)\^2 \ + \((d - e + f)\)\^2, \((\((\((a + b)\)^2 + \((\ b + c)\)^2 + \((a - \ c)\)^2)\) + \((d + e)\)^2 + \((e + f)\)^2 + \((\ f - d)\)^2)\)^2/4 - 3\ \((\(-b\)\ d - c\ d + a\ e - c\ e + a\ f + b\ f)\)\^2}]\) \!\(sh["\"] := Module[{}, \(gp = {1, 1, 1, 0. , 1, 0. };\); gi = {1, 3, 2, \(-4\), \(-6\), \(-5\), 1, 1}; plex = {a, b, c, \(-d\), \(-e\), \(-f\)}; \[IndentingNewLine]{\((a - b - c)\)\^2 \ + \((d - e - f)\)\^2, \((\((\((a + b)\)^2 + \((\ b - c)\)^2 + \((a + \ c)\)^2)\) + \((d + e)\)^2 + \((e - f)\)^2 + \((\ f + d)\)^2)\)^2/4 - 3\ \((\(-b\)\ d + c\ d + a\ e + c\ e - a\ f - b\ f)\)\^2}]\) \!\(\(sh["\"] := Module[{}, gi = {1, 8, 7, 6, 5, 4, 3, 2, 1, 1, 1, 1}; gp = {1, 1, 1, 1, 0. , 1, 0. (*?*) }; plex = {a, \(-b\), c, \(-d\), e, \(-f\), g, \(-h\)}; {\((a - b + c - d + e - f + g - h)\)\ , \((a + b + c + d + e + f + g + h)\), \((a - c + e - g)\)\^2 + \((b - d + f - h)\)\^2, \((\((a \ - e)\)\^2 + \((c - g)\)\^2)\)\^2 + \((\((b - f)\)\^2 + \((d - h)\)\^2)\)\^2 + 4\ \((\((a - e)\)\ \((d - h)\) + \((f - b)\)\ \((c - g)\))\)\ \((\((a - e)\)\ \((b - f)\) + \((c - g)\)\ \((d - h)\))\)}];\)\n \(sh["\"] := Module[{}, gp = {1, 1, 1, 1, 1, 1, 1, 1}; plex = {a, b, c, d, \(-e\), \(-f\), \(-g\), \(-h\)}; gi = {1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]{a + b + c + d - e - f - g - h, a + b - c - d + e + f - g - h, a - b + c - d + e - f + g - h, a - b - c + d - e + f + g - h, a - b - c + d + e - f - g + h, a - b + c - d - e + f - g + h, a + b - c - d - e - f + g + h, a + b + c + d + e + f + g + h}];\)\[IndentingNewLine] \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := \[IndentingNewLine]{a + b + c + d - e - f - g - h, a + b - c - d + e + f - g - h, \[IndentingNewLine]a - b + c - d + e - f + g - h, a - b - c + d - e + f + g - h, \[IndentingNewLine]a - b - c + d + e - f - g + h, a - b + c - d - e + f - g + h, \[IndentingNewLine]a + b - c - d - e - f + g + h, a + b + c + d + e + f + g + h};\)\n toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Epsilon]_, \[Sigma]_, \ \[Zeta]_, \[Theta]_}, "\"] := \[IndentingNewLine]{\ \[Alpha] + \[Beta] \ + \[Gamma] + \[Delta] + \[Epsilon]\ + \[Sigma] + \[Zeta]\ + \[Theta], \ \[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \[Epsilon]\ - \[Sigma] + \[Zeta]\ \ + \[Theta], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \ \[Epsilon] + \[Sigma] - \[Zeta] + \[Theta], \[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Epsilon] - \[Sigma] - \[Zeta] + \[Theta], \[IndentingNewLine]\(-\ \[Alpha]\) + \[Beta] + \[Gamma] - \[Delta] + \[Epsilon]\ - \[Sigma] - \ \[Zeta]\ + \[Theta], \(-\[Alpha]\) + \[Beta] - \[Gamma] + \[Delta] - \ \[Epsilon]\ + \[Sigma] - \[Zeta]\ + \[Theta], \ \[IndentingNewLine]\(-\[Alpha]\) - \[Beta] + \[Gamma] + \[Delta] - \[Epsilon]\ \ - \[Sigma] + \[Zeta]\ + \[Theta], \(-\[Alpha]\) - \[Beta] - \[Gamma] - \ \[Delta] + \[Epsilon]\ + \[Sigma] + \[Zeta]\ + \[Theta]}/8\) \!\(\(sh["\"] := Module[{}, gi = {1, 4, 3, 2, 5, 8, 7, 6, 1, 1, 1, 1, 1, 1}; gp = {1, 1, 1, 1, 1, 0. , 1, 0. }; plex = {a, \(-b\), c, \(-d\), e, \(-f\), g, \(-h\)}; \[IndentingNewLine]{a + b + c + d - e - f - g - h, a - b + c - d + e - f + g - h, a - b + c - d - e + f - g + h, a + b + c + d + e + f + g + h, \((a - c - e + g)\)\^2 + \((b - d - f + h)\)\^2, \((a - c + e \ - g)\)\^2 + \((b - d + f - h)\)\^2}];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := {a + b + c + d + e + f + g + h, a - b + c - d + e - f + g - h, a + b + c + d - e - f - g - h, a - b + c - d - e + f - g + h, \((a - c - e + g)\)\^2 + \((b - d - f + h)\)\^2, \ \[IndentingNewLine]ArcTan[a - c - e + g, b - d - f + h], \[IndentingNewLine]\((a - c + e - g)\)\^2 + \((b - d + f - h)\ \)\^2, \[IndentingNewLine]ArcTan[a - c + e - g, b - d + f - h]};\)\n toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Epsilon]\[Epsilon]_, \ \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := Module[{\[Epsilon] = 2 \@ \[Epsilon]\[Epsilon], \[Zeta] = 2 \@ \[Zeta]\[Zeta]}, \[IndentingNewLine]{\ \ \[Alpha] + \[Beta] + \ \[Gamma] + \[Delta] + \[Epsilon]\ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Theta]], \ \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Epsilon]\ Sin[\[Sigma]] + \ \[Zeta]\ Sin[\[Theta]], \[IndentingNewLine]\[Alpha] + \[Beta] + \[Gamma] + \ \[Delta] - \[Epsilon]\ Cos[\[Sigma]] - \[Zeta]\ Cos[\[Theta]], \[Alpha] - \ \[Beta] + \[Gamma] - \[Delta] - \[Epsilon]\ Sin[\[Sigma]] - \[Zeta]\ Sin[\ \[Theta]], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \ \[Epsilon]\ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Theta]], \[Alpha] - \[Beta] - \ \[Gamma] + \[Delta] - \[Epsilon]\ Sin[\[Sigma]] + \[Zeta]\ Sin[\[Theta]], \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Epsilon]\ \ Cos[\[Sigma]] - \[Zeta]\ Cos[\[Theta]], \[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Epsilon]\ Sin[\[Sigma]] - \[Zeta]\ Sin[\[Theta]]}/8]\) \!\(\(sh["\"] := Module[{}, gi = {1, 2, 7, 4, 5, 6, 3, 8, 1, 1, 1, 1, 2}; gp = {1, 1, 1, 1, 1, 0. , 1, 0. }; plex =. ; {a + b - c - d + e + f - g - h, a - b + c - d + e - f + g - h, a - b - c + d + e - f - g + h, a + b + c + d + e + f + g + h, \[IndentingNewLine]\((a - e)\)\^2 - \((b - f)\)\^2 + \((c - g)\ \)\^2 - \((d - h)\)\^2 (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := {a + b + c + d + e + f + g + h, a + b - c - d + e + f - g - h, a - b + c - d + e - f + g - h, a - b - c + d + e - f - g + h, \[IndentingNewLine]\((a - e)\)\^2 + \((c - g)\)\^2 - \((b - f)\)\ \^2 - \((d - h)\)\^2, ArcTan[a - e, c - g], \[IndentingNewLine]\((b - f)\)\^2 + \((d - h)\)\^2, ArcTan[b - f, d - h]};\)\n \(toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Epsilon]\[Epsilon]_, \ \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := Module[{\[Epsilon] = 4 \@\( \[Epsilon]\[Epsilon] + \[Zeta]\[Zeta]\), \[Zeta] = 4 \@ \[Zeta]\[Zeta]}, \[IndentingNewLine]{\[Alpha] + \[Beta] + \ \[Gamma] + \[Delta] + \[Epsilon]\ Cos[\[Sigma]], \[Alpha] + \[Beta] - \ \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Theta]], \[Alpha] - \[Beta] + \[Gamma] - \ \[Delta] + \[Epsilon]\ Sin[\[Sigma]], \[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Zeta]\ Sin[\[Theta]], \[Alpha] + \[Beta] + \[Gamma] + \[Delta] - \ \[Epsilon]\ Cos[\[Sigma]], \[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \[Zeta]\ \ Cos[\[Theta]], \[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \[Epsilon]\ Sin[\ \[Sigma]], \[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \[Zeta]\ \ Sin[\[Theta]]}]/8;\)\n \(tohyPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := {\((a + b + e + f)\)\^2 - \((c + d + g + h)\)\^2, arcTanh[a + b + e + f, c + d + g + h], \[IndentingNewLine]\((a - b + e - f)\)\^2 - \((c - d + g - h)\ \)\^2, arcTanh[a - b + e - f, c - d + g - h], \[IndentingNewLine]\((a - e)\)\^2 + \((c - g)\)\^2, ArcTan[a - e, c - g], \[IndentingNewLine]\((b - f)\)\^2 + \((d - h)\)\^2, ArcTan[b - f, d - h]};\)\n \(tohyVec[{\[Alpha]\[Alpha]_, \[Phi]_, \[Gamma]\[Gamma]_, \[Rho]_, \ \[Epsilon]\[Epsilon]_, \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := Module[{\[Alpha] = \@\[Alpha]\[Alpha], \[Gamma] = \@\[Gamma]\[Gamma], \ \[Epsilon] = 2 \@ \[Epsilon]\[Epsilon], \[Zeta] = 2 \@ \[Zeta]\[Zeta]}, \[IndentingNewLine]{\[Alpha]\ \ Cosh[\[Phi]] + \[Gamma]\ Cosh[\[Rho]] + \[Epsilon]\ Cos[\[Sigma]], \[Alpha]\ \ Cosh[\[Phi]] - \[Gamma]\ Cosh[\[Rho]] + \[Zeta]\ Cos[\[Theta]], \[Alpha]\ \ Sinh[\[Phi]] + \[Gamma]\ Sinh[\[Rho]] + \[Epsilon]\ Sin[\[Sigma]], \[Alpha]\ \ Sinh[\[Phi]] - \[Gamma]\ Sinh[\[Rho]] + \[Zeta]\ Sin[\[Theta]], \[Alpha]\ \ Cosh[\[Phi]] + \[Gamma]\ Cosh[\[Rho]] - \[Epsilon]\ Cos[\[Sigma]], \[Alpha]\ \ Cosh[\[Phi]] - \[Gamma]\ Cosh[\[Rho]] - \[Zeta]\ Cos[\[Theta]], \[Alpha]\ \ Sinh[\[Phi]] + \[Gamma]\ Sinh[\[Rho]] - \[Epsilon]\ Sin[\[Sigma]], \[Alpha]\ \ Sinh[\[Phi]] - \[Gamma]\ Sinh[\[Rho]] - \[Zeta]\ Sin[\[Theta]]}]/4;\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 6, 7, 8, 5, 2, 3, 4, 1, 1, 1, 1, 2}; gp = {1, 1, 1, 1, 1, 0. , 1, 0. }; plex = {a, \(-b\), \(-c\), \(-d\), e, \(-f\), \(-g\), \(-h\)}; \[IndentingNewLine]{a + b - c - d + e + f - g - h, a - b + c - d + e - f + g - h, a - b - c + d + e - f - g + h, a + b + c + d + e + f + g + h, \((a - e)\)\^2 + \((b - f)\)\^2 + \((c - g)\)\^2 + \((d - h)\)\ \^2 (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := {a + b + c + d + e + f + g + h, a - b + c - d + e - f + g - h, a - b - c + d + e - f - g + h, a + b - c - d + e + f - g - h, \((a - e)\)\^2 + \((b - f)\)\^2, ArcTan[a - e, b - f], \((c - g)\)\^2 + \((d - h)\)\^2, ArcTan[c - g, d - h]};\)\n \(toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Epsilon]\[Epsilon]_, \ \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := {\ \ \[Alpha] + \[Beta] \ + \[Gamma] + \[Delta] + 4 \@ \[Epsilon]\[Epsilon]\ Cos[\[Sigma]], \[Alpha] - \[Beta] - \ \[Gamma] + \[Delta] + 4 \@ \[Epsilon]\[Epsilon]\ Sin[\[Sigma]], \[Alpha] + \[Beta] - \ \[Gamma] - \[Delta] + 4 \@ \[Zeta]\[Zeta]\ Cos[\[Theta]], \[Alpha] - \[Beta] + \[Gamma] \ - \[Delta] + 4 \@ \[Zeta]\[Zeta]\ Sin[\[Theta]], \[Alpha] + \[Beta] + \[Gamma] \ + \[Delta] - 4 \@ \[Epsilon]\[Epsilon]\ Cos[\[Sigma]], \[Alpha] - \[Beta] - \ \[Gamma] + \[Delta] - 4 \@ \[Epsilon]\[Epsilon]\ Sin[\[Sigma]], \[Alpha] + \[Beta] - \ \[Gamma] - \[Delta] - 4 \@ \[Zeta]\[Zeta]\ Cos[\[Theta]], \[Alpha] - \[Beta] + \[Gamma] \ - \[Delta] - 4 \@ \[Zeta]\[Zeta]\ Sin[\[Theta]]}/8;\)\n \(tohyPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := {\((a + b + e + f)\)\^2 - \((c + d + g + h)\)\^2, arcTanh[a + b + e + f, c + d + g + h], \[IndentingNewLine]\((a - b + e - f)\)\^2 - \((c - d + g - h)\ \)\^2, arcTanh[a - b + e - f, c - d + g - h], \[IndentingNewLine]\((a - e)\)\^2 + \((b - f)\)\^2, ArcTan[a - e, b - f], \[IndentingNewLine]\((c - g)\)\^2 + \((d - h)\)\^2, ArcTan[c - g, d - h]};\)\n \(tohyVec[{\[Alpha]\[Alpha]_, \[Phi]_, \[Gamma]\[Gamma]_, \[Rho]_, \ \[Epsilon]\[Epsilon]_, \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := Module[{\[Alpha] = \@\[Alpha]\[Alpha], \[Gamma] = \@\[Gamma]\[Gamma], \ \[Epsilon] = 2 \@ \[Epsilon]\[Epsilon], \[Zeta] = 2 \@ \[Zeta]\[Zeta]}, \[IndentingNewLine]{\[Alpha]\ \ Cosh[\[Phi]] + \[Gamma]\ Cosh[\[Rho]] + \[Epsilon]\ Cos[\[Sigma]], \[Alpha]\ \ Cosh[\[Phi]] - \[Gamma]\ Cosh[\[Rho]] + \[Epsilon]\ Sin[\[Sigma]], \ \[IndentingNewLine]\[Alpha]\ Sinh[\[Phi]] + \[Gamma]\ Sinh[\[Rho]] + \[Zeta]\ \ Cos[\[Theta]], \[Alpha]\ Sinh[\[Phi]] - \[Gamma]\ Sinh[\[Rho]] + \[Zeta]\ \ Sin[\[Theta]], \[IndentingNewLine]\[Alpha]\ Cosh[\[Phi]] + \[Gamma]\ Cosh[\ \[Rho]] - \[Epsilon]\ Cos[\[Sigma]], \[Alpha]\ Cosh[\[Phi]] - \[Gamma]\ Cosh[\ \[Rho]] - \[Epsilon]\ Sin[\[Sigma]], \[IndentingNewLine]\[Alpha]\ \ Sinh[\[Phi]] + \[Gamma]\ Sinh[\[Rho]] - \[Zeta]\ Cos[\[Theta]], \[Alpha]\ \ Sinh[\[Phi]] - \[Gamma]\ Sinh[\[Rho]] - \[Zeta]\ Sin[\[Theta]]}]/4;\)\) \!\(\(sh["\"] := Module[{}, gi = {1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), \(-8\), 4}; gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), \(-h\)}; \ \[IndentingNewLine]{a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2}];\)\n toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{}, {a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2, ArcTan[a, b], c\^2 + d\^2, ArcTan[c, d], e\^2 + f\^2, ArcTan[e, f], g\^2 + h\^2, ArcTan[g, h]}]\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = \@\(pp - qq - rr - ss\), q = \@qq, r = \@rr, s = \@ss}, {\ \ p\ Cos[\[Phi]]\ , p\ Sin[\[Phi]], q\ Cos[\[Psi]]\ , q\ Sin[\[Psi]]\ , r\ Cos[\[Rho]]\ , r\ Sin[\[Rho]], s\ Cos[\[Sigma]], s\ Sin[\[Sigma]]}]\) \!\(sh["\"] := Module[{}, gi = {1, 2, 3, 4, \(-5\), \(-8\), \(-7\), \(-6\), 1, 1, 2}; gp = {1, 0. , 1, 0. , 1, 0, 1, 0. }; \[IndentingNewLine]{\((a - b + c - d)\)\^2 + \((e - f + g - \ h)\)\^2, \((a + b + c + d)\)\^2 + \((e + f + g + h)\)\^2, \((a - c)\)\^2 - \ \((b - d)\)\^2 + \((e - g)\)\^2 - \((f - h)\)\^2 (*repeated*) }]\n toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{}, { (*pp*) \((a - c)\)\^2 - \((b - d)\)\^2 + \((e - g)\)\^2 - \ \((f - h)\)\^2, (*\[Phi]*) If[e == g, 0, ArcTan[a - c, e - g]], \[IndentingNewLine] (*qq*) \((b - d)\)\^2 + \((f - \ h)\)\^2, \[IndentingNewLine] (*\[Psi]*) If[f == h, 0, ArcTan[b - d, f - h]], (*rr*) \((a - b + c - d)\)\^2 + \((e - f + g - h)\)\^2, \ (*\[Rho]*) If[e + g \[Equal] f + h, 0, ArcTan[a - b + c - d, e - f + g - h]], (*ss*) \((a + b + c + d)\)\^2 + \((e + f + g + h)\)\^2, \ (*\[Sigma]*) If[e + f \[Equal] g + h, 0, ArcTan[a + b + c + d, e + f + g + h]]}]\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = \@\(pp + qq\)/2, q = \@qq/2, r = \@rr/4, s = \@ss/4}, { (*a*) \ \ p\ Cos[\[Phi]]\ + r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine]\ \ \ (*b*) \ \(+q\)\ Cos[\ \[Psi]] - r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine] (*c*) \(-p\)\ Cos[\[Phi]] + r\ Cos[\[Rho]] + s\ Cos[\[Sigma]]\ , \[IndentingNewLine]\ \ \ (*d*) \(-q\)\ Cos[\ \[Psi]] - r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine] (*e*) \ \ p\ Sin[\[Phi]] + r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \[IndentingNewLine]\ \ \ (*f*) \(+q\)\ \ Sin[\[Psi]] - r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \[IndentingNewLine] (*g*) \(-p\)\ Sin[\[Phi]] + r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \[IndentingNewLine]\ \ \ (*h*) \ \(-q\)\ Sin[\ \[Psi]] - r\ Sin[\[Rho]] + s\ Sin[\[Sigma]]}]\) \!\(\(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, 2, 3, 4, \(-5\), \(-6\), \(-7\), \(-8\), 1, 1, 1, 1}; plex = {a, b, c, d, \(-e\), \(-f\), \(-g\), \(-h\)}; \[IndentingNewLine]{\((a - b \ - c + d)\)\^2 + \((e - f - g + h)\)\^2, \((a - b + c - d)\)\^2 + \((e - f + g \ - h)\)\^2, \[IndentingNewLine]\((a + b - c - d)\)\^2 + \((e + f - g - \ h)\)\^2, \((a + b + c + d)\)\^2 + \((e + f + g + h)\)\^2}];\)\n toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; {\((a - b - c + d)\)\^2 + \((e - f - g + h)\)\^2, If[e + h \[Equal] f + g, 0, ArcTan[a - b - c + d, e - f - g + h]], \((a - b + c - d)\)\^2 + \((e - f + g - h)\)\^2, If[e + g \[Equal] f + h, 0, ArcTan[a - b + c - d, e - f + g - h]], \((a + b - c - d)\)\^2 + \((e + f - g - h)\)\^2, If[e + f \[Equal] g + h, 0, ArcTan[a + b - c - d, e + f - g - h]], \((a + b + c + d)\)\^2 + \((e + f + g + h)\)\^2, If[e + f + g + h \[Equal] 0, 0, ArcTan[a + b + c + d, e + f + g + h]]}]\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, {\ \ p\ Cos[\[Phi]]\ + q\ Cos[\[Psi]]\ + r\ Cos[\[Rho]]\ + s\ Cos[\[Sigma]], \(-\ p\)\ Cos[\[Phi]] - q\ Cos[\[Psi]] + r\ Cos[\[Rho]]\ + s\ Cos[\[Sigma]], \(-p\)\ Cos[\[Phi]]\ + q\ Cos[\[Psi]]\ - r\ Cos[\[Rho]]\ + s\ Cos[\[Sigma]], \ p\ Cos[\[Phi]] - q\ Cos[\[Psi]]\ - r\ Cos[\[Rho]]\ + s\ Cos[\[Sigma]], \[IndentingNewLine]\ \ p\ Sin[\[Phi]]\ + q\ Sin[\[Psi]]\ + r\ Sin[\[Rho]]\ + s\ Sin[\[Sigma]], \(-p\)\ Sin[\[Phi]] - q\ Sin[\[Psi]]\ + r\ Sin[\[Rho]]\ + s\ Sin[\[Sigma]], \[IndentingNewLine]\(-p\)\ Sin[\[Phi]]\ + q\ Sin[\[Psi]]\ - r\ Sin[\[Rho]]\ + s\ Sin[\[Sigma]], \ p\ Sin[\[Phi]]\ - q\ Sin[\[Psi]] - r\ Sin[\[Rho]] + s\ Sin[\[Sigma]]}]/4\) \!\(\(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, \(-2\), 3, \(-4\), \(-5\), 6, \(-7\), 8, 1, 1, 1, 1}; plex = {a, \(-b\), c, \(-d\), \(-e\), f, \(-g\), h}; \[IndentingNewLine]{\((a - c - f + h)\)\^2 + \((b - d + e - \ g)\)\^2, \((a - c + f - h)\)\^2 + \((b - d - e + g)\)\^2, \[IndentingNewLine]\ \((a + c - f - h)\)\^2 + \((b + d + e + g)\)\^2, \((a + c + f + h)\)\^2 + \ \((b + d - e - g)\)\^2}];\)\n \(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, \(-2\), \(-3\), 4, \(-5\), 6, 7, \(-8\), 1, 1, 1, 1}; plex = {a, \(-b\), \(-c\), d, \(-e\), f, g, \(-h\)}; \[IndentingNewLine]{\((\(-a\) + d + f + g)\)\^2 + \ \((b + c + e - h)\)\^2, \((a - d + f + g)\)\^2 + \((b + c - e + h)\)\^2, \ \[IndentingNewLine]\((a + d - f + g)\)\^2 + \((b - c + e + h)\)\^2, \((a + d \ + f - g)\)\^2 + \((\(-b\) + c + e + h)\)\^2}];\)\) \!\(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, 4, 3, 2, \(-5\), \(-8\), \(-7\), \(-6\), 1, 1, 1, 1}; plex = {a, \(-b\), c, \(-d\), \(-e\), f, \(-g\), h}; {\((b - d + e - g)\)\^2 + \((a - c - f + h)\)\^2, \((b - d - e \ + g)\)\^2 + \((a - c + f - h)\)\^2, \[IndentingNewLine]\((a - b + c - d)\)\^2 \ + \((e - f + g - h)\)\^2, \((a + b + c + d)\)\^2 + \((e + f + g + h)\)\^2}]; toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{aa, bb, ss, w}, ss := aa^2 + bb^2; \[IndentingNewLine]w := Which[bb === 0 && aa < 0, \[Pi], bb \[Equal] 0, 0, aa \[Equal] 0, \[Pi]/2\ Sign[bb], True, ArcTan[aa, bb]]; {aa = a - c - f + h; bb = b - d + e - g; ss, w, aa = a - c + f - h; bb = b - d - e + g; ss, w, aa = a - b + c - d; bb = e - f + g - h; ss, w, aa = a + b + c + d; bb = e + f + g + h; ss, w}];\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, {\ \ \ \ p\ Cos[\[Phi]] + q\ Cos[\[Psi]] + r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine]\ p\ Sin[\[Phi]] + q\ Sin[\[Psi]] - r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine]\(-p\)\ Cos[\[Phi]] - q\ Cos[\[Psi]] + r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \(-p\)\ Sin[\[Phi]] - q\ Sin[\[Psi]] - r\ Cos[\[Rho]] + s\ Cos[\[Sigma]], \[IndentingNewLine]\ \ p\ Sin[\[Phi]] - q\ Sin[\[Psi]] + r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \(-p\)\ Cos[\[Phi]] + q\ Cos[\[Psi]] - r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \[IndentingNewLine]\(-p\)\ Sin[\[Phi]] + q\ Sin[\[Psi]] + r\ Sin[\[Rho]] + s\ Sin[\[Sigma]], \ \ p\ Cos[\[Phi]] - q\ Cos[\[Psi]] - r\ Sin[\[Rho]] + s\ Sin[\[Sigma]]}/4]\) \!\(\(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, \(-4\), 3, \(-2\), \(-5\), 8, \(-7\), 6, 1, 1, 1, 1}; plex = {a, \(-b\), c, \(-d\), \(-e\), f, \(-g\), h}; {\((b + d + e + g)\)\^2 + \((a + c - f - h)\)\^2, \((a + b - \ c - d)\)\^2 + \((e + f - g - h)\)\^2, \((a - b - c + d)\)\^2 + \((e - f - g + \ h)\)\^2, \((b + d - e - g)\)\^2 + \((a + c + f + h)\)\^2}];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, \(-7\), 5, \(-8\), \(-4\), \(-6\), 1, 1, 1, 1, 2}; gp = {1, 1, 1, 1, 1, 0. , 1, 0. }; plex = {a, \(-b\), \(-c\), \(-d\), e, \(-f\), \(-g\), \(-h\)}; {a - b - c - d - e - f + g + h, a + b + c - d - e + f + g - h, a + b + c + d - e - f - g + h, a - b - c + d - e + f - g - h, \((a + e)\)^2 + \((b - c)\)^2 + \((d + g)\)^2 + \((f + h)\)^2 (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := \[IndentingNewLine]{a - b - c - d - e - f + g + h, a + b + c - d - e + f + g - h, a + b + c + d - e - f - g + h, a - b - c + d - e + f - g - h, \((a + e)\)^2 + \((b - c)\)^2, ArcTan[a + e, b - c], \((d + g)\)^2 + \((f + h)\)^2\ , ArcTan[d + g, f + h]};\)\n \(toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Epsilon]\[Epsilon]_, \ \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_}, "\"] := Module[{\[Epsilon] = 4 \@ \[Epsilon]\[Epsilon], \[Zeta] = 4 \@ \[Zeta]\[Zeta]}, \[IndentingNewLine]{\ \ \ \ \ \[Alpha] + \ \[Beta] + \[Gamma] + \[Delta] + \[Epsilon]\ Cos[\[Sigma]], \(-\[Alpha]\) + \ \[Beta] + \[Gamma] - \[Delta] + \[Epsilon]\ Sin[\[Sigma]], \(-\[Alpha]\) + \ \[Beta] + \[Gamma] - \[Delta] - \[Epsilon]\ Sin[\[Sigma]], \(-\[Alpha]\) - \ \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Theta]], \(-\[Alpha]\) - \ \[Beta] - \[Gamma] - \[Delta] + \[Epsilon]\ Cos[\[Sigma]], \(-\[Alpha]\) + \ \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Sin[\[Theta]], \ \ \[Alpha] + \ \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Theta]], \ \ \ \[Alpha] - \ \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Sin[\[Theta]]}/8];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 2, 3, 4, \(-5\), \(-6\), 7, 8, 2, 2}; gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, b, \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), \(-h\)}; {\((a + \ b)\)\^2 - \((c + d)\)\^2 + \((e + f)\)\^2 - \((g + h)\)\^2 (*repeated*) , \ \((a - b)\)\^2 - \((c - d)\)\^2 + \((e - f)\)\^2 - \((g - h)\)\^2 \ (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := \[IndentingNewLine]{\((a + b)\)\^2 + \((e + f)\ \)\^2, If[e \[Equal] \(-f\), 0, ArcTan[a + b, e + f]], \[IndentingNewLine]\((c + d)\)\^2 + \((g + h)\)\^2, If[g \[Equal] \(-h\), 0, ArcTan[c + d, g + h]], \((a - b)\)\^2 + \((e - f)\)\^2, If[e \[Equal] f, 0, ArcTan[a - b, e - f]], \[IndentingNewLine]\((c - d)\)\^2 + \((g - h)\)\^2, If[g \[Equal] h, 0, ArcTan[c - d, g - h]]};\)\n \(toVec[{\[Epsilon]\[Epsilon]_, \[Sigma]_, \[Zeta]\[Zeta]_, \[Theta]_, \ \[Eta]\[Eta]_, \[Phi]_, \[Kappa]\[Kappa]_, \[Psi]_}, "\"] := Module[{\[Epsilon] = \@\[Epsilon]\[Epsilon], \[Zeta] = \ \@\[Zeta]\[Zeta], \[Eta] = \@\[Eta]\[Eta], \[Kappa] = \@\[Kappa]\[Kappa]}, \ \[IndentingNewLine]{\ \ \[Epsilon]\ Cos[\[Sigma]] + \[Eta]\ Cos[\[Phi]], \ \[Epsilon]\ Cos[\[Sigma]] - \[Eta]\ Cos[\[Phi]], \[IndentingNewLine]\[Zeta]\ \ Cos[\[Theta]] + \[Kappa]\ Cos[\[Psi]], \[Zeta]\ Cos[\[Theta]] - \[Kappa]\ \ Cos[\[Psi]], \[IndentingNewLine]\[Epsilon]\ Sin[\[Sigma]] + \[Eta]\ \ Sin[\[Phi]], \[Epsilon]\ Sin[\[Sigma]] - \[Eta]\ Sin[\[Phi]], \ \[IndentingNewLine]\[Zeta]\ Sin[\[Theta]] + \[Kappa]\ Sin[\[Psi]], \[Zeta]\ \ Sin[\[Theta]] - \[Kappa]\ Sin[\[Psi]]}/2];\)\) \!\(\(sh["\"] (*Signature\ \((1, 3)\)\ repeated*) := Module[{}, gi = {1, 2, 3, 4, 5, 6, 7, 8, 2, 2}; gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, \(-b\), \(-c\), \(-d\), e, \(-f\), \(-g\), \(-h\)}; \[IndentingNewLine]{\((a + e)\)\^2 - \ \((b + f)\)\^2 - \((c + g)\)\^2 - \((d + h)\)\^2 (*repeated*) , \((a - \ e)\)\^2 - \((b - f)\)\^2 - \((c - g)\)\^2 - \((d - h)\)\^2 (*repeated*) }];\)\ \n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; {\((a + e)\)\^2 - \((b + f)\)\^2, arcTanh[a + e, b + f], \[IndentingNewLine]\((c + g)\)\^2 + \((d + h)\)\^2, ArcTan[c + g, d + h], \[IndentingNewLine]\((a - e)\)\^2 - \((b - f)\)\^2, arcTanh[a - e, b - f], \[IndentingNewLine]\((c - g)\)\^2 + \((d - h)\)\^2, ArcTan[c - g, d - h]}];\)\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, {\ p\ Cosh[\[Phi]] + r\ Cosh[\[Rho]], \ p\ Sinh[\[Phi]] + r\ Sinh[\[Rho]]\ , q\ Cos[\[Psi]]\ \ + s\ Cos[\[Sigma]], q\ Sin[\[Psi]]\ \ + s\ Sin[\[Sigma]], \[IndentingNewLine]p\ Cosh[\[Phi]] - r\ Cosh[\[Rho]], p\ Sinh[\[Phi]] - r\ Sinh[\[Rho]]\ , q\ Cos[\[Psi]] - s\ Cos[\[Sigma]], q\ Sin[\[Psi]] - s\ Sin[\[Sigma]]}]/2\) \!\(\(sh["\"] := Module[{}, gp =. ; gi = {1, 2, 3, \(-4\), 5, \(-6\), \(-7\), \(-8\), 2}; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), h}; \[IndentingNewLine]{4\ \((a\ h - b\ g + c\ f - d\ e)\)\^2 + \ \((a\^2 - b\^2 - c\^2 + d\^2 - e\^2 + f\^2 + g\^2 - h\^2)\)\^2}];\)\ \[IndentingNewLine] \(sh["\"] := (*\ CL3\ isotope*) Module[{}, gp =. ; gi = {1, \(-2\), \(-3\), \(-4\), 5, 6, 7, \(-8\), 2}; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), h}; {4 \((a\ h - b\ g + c\ f - d\ e)\)\^2 + \((a\^2 + b\^2 + c\^2 \ + d\^2 - e\^2 - f\^2 - g\^2 - h\^2)\)\^2}];\)\[IndentingNewLine] sh["\"] := Module[{}, gi = {1, 4, 3, 2, \(-5\), \(-6\), \(-7\), \(-8\), 2, 1, 1}; gp = {1, 1, 2, 1, 0. , 1, 0. (*?*) }; plex = {a, \(-b\), c, \(-d\), \(-e\), \(-f\), \(-g\), \(-h\)}; {\((a - c)\)\^2 + \((b \ - d)\)\^2 + \((e - g)\)\^2 + \((f - h)\)\^2 (*repeated*) , \ \[IndentingNewLine]\((a - b + c - d)\)\^2 + \((e - f + g - h)\)\^2, \((a + b \ + c + d)\)\^2 + \((e + f + g + h)\)\^2}]\[IndentingNewLine] sh["\"] := Module[{}, gi = {1, 2, \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), \(-8\), 2, 2}; plex = {a, b, \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), \(-h\)}; gp = {1, 1, 1, 1, 0. , 1, 0. (*?*) }; {\((a - b)\)\^2 + \((c - d)\)\^2 + \((e - f)\)\^2 + \ \((g - h)\)\^2, \((a + b)\)\^2 + \((c + d)\)\^2 + \((e + f)\)\^2 + \((g + \ h)\)\^2}]\[IndentingNewLine] sh["\"] := Module[{}, gi = {1, 2, \(-3\), \(-8\), 5, 6, \(-7\), \(-4\), 2, 1, 1}; plex = {a, \(-b\), \(-c\), \(-d\), e, \(-f\), \(-g\), \(-h\)}; gp = {1, 1, 1, 1, 0. , 1, 0. (*?*) }; {\((a - e)\)\^2 - \((b - f)\)\^2 + \((c - g)\)\^2 - \ \((d - h)\)\^2 (*repeated*) , \((a + b + e + f)\)\^2 + \((c - d + g - \ h)\)\^2, \((a - b + e - f)\)\^2 + \((c + d + g + h)\)\^2}]\[IndentingNewLine] \(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, \(-2\), 3, 4, 5, 6, \(-7\), 8, 2, 2}; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), h}; \[IndentingNewLine]{\((a - h)\)\^2 + \((b + g)\)\^2 - \((c + \ f)\)\^2 - \((d - e)\)\^2 (*repeated*) , \((a + h)\)\^2 + \((b - g)\)\^2 - \ \((c - f)\)\^2 - \((d + e)\)\^2 (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := \[IndentingNewLine]{ (*p, \[Phi]*) \((a - \ h)\)\^2 + \((b + g)\)\^2, ArcTan[a - h, b + g], \[IndentingNewLine] (*q, \[Psi]*) \((c + f)\)\^2 + \((d - \ e)\)\^2, ArcTan[c + f, d - e], \[IndentingNewLine] (*r, \[Rho]*) \((a + h)\)\^2 + \((b - \ g)\)\^2, ArcTan[a + h, b - g], \[IndentingNewLine] (*s, \[Sigma]*) \((c - f)\)\^2 + \((d + \ e)\)\^2, ArcTan[c - f, d + e]};\)\n \(toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, { (*a*) \ p\ Cos[\[Phi]]\ + r\ Cos[\[Rho]], (*b*) \ p\ Sin[\[Phi]]\ + r\ Sin[\[Rho]], \[IndentingNewLine] (*c*) q\ Cos[\[Psi]] + s\ Cos[\[Sigma]], (*d*) q\ Sin[\[Psi]] + s\ Sin[\[Sigma]], \[IndentingNewLine] (*e*) \(-q\)\ Sin[\[Psi]] \ + s\ Sin[\[Sigma]], (*f*) \ q\ Cos[\[Psi]] - s\ Cos[\[Sigma]], \[IndentingNewLine] (*g*) \ p\ Sin[\[Phi]]\ - r\ Sin[\[Rho]], (*h*) \(-\ p\)\ Cos[\[Phi]] + r\ Cos[\[Rho]]}/2];\)\[IndentingNewLine] \(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), 8, 2, 2}; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), h}; \[IndentingNewLine]{\((a - h)\)\^2 + \((b + g)\)\^2 + \((c - \ f)\)\^2 + \((d + e)\)\^2 (*repeated*) , \((a + h)\)\^2 + \((b - g)\)\^2 + \ \((c + f)\)\^2 + \((d - e)\)\^2 (*repeated*) }];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := \[IndentingNewLine]{ (*p, \[Phi]*) \((a - \ h)\)\^2 + \((b + g)\)\^2, ArcTan[a - h, b + g], \[IndentingNewLine] (*q, \[Psi]*) \((c - f)\)\^2 + \((d + \ e)\)\^2, ArcTan[c - f, d + e], \[IndentingNewLine] (*r, \[Rho]*) \((a + h)\)\^2 + \((b - \ g)\)\^2, ArcTan[a + h, b - g], \[IndentingNewLine] (*s, \[Sigma]*) \((c + f)\)\^2 + \((d - \ e)\)\^2, ArcTan[c + f, d - e]};\)\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, { (*a*) \ p\ Cos[\[Phi]]\ + r\ Cos[\[Rho]], (*b*) \ p\ Sin[\[Phi]]\ + r\ Sin[\[Rho]], \[IndentingNewLine] (*c*) q\ Cos[\[Psi]] + s\ Cos[\[Sigma]], (*d*) q\ Sin[\[Psi]] + s\ Sin[\[Sigma]], \[IndentingNewLine] (*e*) q\ Sin[\[Psi]] - s\ Sin[\[Sigma]], (*f*) \(-q\)\ Cos[\[Psi]] + s\ Cos[\[Sigma]], \[IndentingNewLine] (*g*) \ p\ Sin[\[Phi]]\ - r\ Sin[\[Rho]], (*h*) \(-\ \(p\ Cos[\[Phi]]\)\) + r\ Cos[\[Rho]]}/2]\) \!\(\(sh["\"] := Module[{}, gi = {1, 2, \(-3\), 4, \(-5\), 6, \(-7\), 8, 4}; gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, \(-b\), \(-c\), \(-d\), \(-e\), \(-f\), \(-g\), \(-h\)}; \ \[IndentingNewLine]{a\^2 - b\^2 + c\^2 - d\^2 + e\^2 - f\^2 + g\^2 - h\^2}];\)\n toPol[{a_, b_, c_, d_, e_, f_, g_, h_}, "\"] := Module[{}, {a\^2 + c\^2, ArcTan[a, c], b\^2 + d\^2, ArcTan[b, d], e\^2 + g\^2, ArcTan[e, g], f\^2 + h\^2, ArcTan[f, h]}]\n toVec[{pp_, \[Phi]_, qq_, \[Psi]_, rr_, \[Rho]_, ss_, \[Sigma]_}, "\"] := Module[{p = Sqrt[pp], q = Sqrt[qq], r = Sqrt[rr], s = Sqrt[ss]}, {p\ Cos[\[Phi]]\ , q\ Cos[\[Psi]], p\ Sin[\[Phi]]\ , q\ Sin[\[Psi]]\ , r\ Cos[\[Rho]]\ , s\ Cos[\[Sigma]], r\ Sin[\[Rho]], s\ Sin[\[Sigma]]}]\) \!\(\(sh["\"] := Module[{}, (*L2\ shapes\ not\ conserved*) gp = {1, 1, 1, 1, 1, 0. , 1, 0. }; plex =. ; gi = {1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]{a + b + c + d + e + f + g + h, a + b - c - d + e + f - g - h, \[IndentingNewLine]a - b + c - d + e - f + g - h, a + b + c + d + e + f + g + h, \((a - e)\)\^2 - \((b - f)\)\^2 - \((c - g)\)\^2 - \((d - h)\)\ \^2, \((a - e)\)\^2 + \((b - f)\)\^2 + \((c - g)\)\^2 + \((d - h)\)\^2}];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 7, 9, 8, 4, 6, 5, 1, 1, 1, 1, 1}; gp = {1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, c, b, g, i, h, d, f, e}; {\((a + b + c + d + e + f + g + h + i)\)\ , \[IndentingNewLine]\((\((a - b - d + f + h - i)\)\^2 + \ \((b - c + d - e + i - g)\)\^2 + \((\(-a\) + c + e - f + g - h)\)\^2)\)/ 2, \((\((a - b + e - f - g + i)\)\^2 + \((b - c - d + f + g - \ h)\)\^2 + \((a - c - d + e - h + i)\)\^2)\)/ 2, \((\((a - b + d - e + g - h)\)\^2 + \((b - c + e - f + h - \ i)\)\^2 + \((c - a + f - d + i - g)\)\^2)\)/ 2, \((\((a + b + c - d - e - f)\)\^2 + \((d + e + f - g - h - \ i)\)\^2 + \((a + b + c - g - h - i)\)\^2)\)/2}];\)\) \!\(\(toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_}, "\"] := {a + b + c + d + e + f + g + h + i, \((\((a - b - d + f + h - i)\)\^2 + \((b - c + d - e + i - \ g)\)\^2 + \((\(-a\) + c + e - f + g - h)\)\^2)\)/2, \[IndentingNewLine]ArcTan[ 2\ a - b - c - d - e + 2\ f - g + 2\ h - i, \(-Sqrt[3]\) \((b - c + d - e + i - g)\)], \[IndentingNewLine]\((\((a - b + e - f - g + i)\)\^2 + \ \((b - c - d + f + g - h)\)\^2 + \((a - c - d + e - h + i)\)\^2)\)/ 2, \[IndentingNewLine]ArcTan[ 2\ a - b - c - d + 2\ e - f - g - h + 2\ i, \(-Sqrt[3]\) \((b - c - d + f + g - h)\)], \[IndentingNewLine]\((\((a - b + d - e + g - h)\)\^2 + \ \((b - c + e - f + h - i)\)\^2 + \((c - a + f - d + i - g)\)\^2)\)/ 2, \[IndentingNewLine]ArcTan[ 2\ a - b - c + 2\ d - e - f + 2\ g - h - i, \(-Sqrt[3]\) \((b - c + e - f + h - i)\)], \[IndentingNewLine]\((\((a + b + c - d - e - f)\)\^2 + \ \((d + e + f - g - h - i)\)\^2 + \((a + b + c - g - h - i)\)\^2)\)/ 2, \[IndentingNewLine]ArcTan[ 2\ a + 2\ b + 2\ c - d - e - f - g - h - i, \(-Sqrt[3]\) \((d + e + f - g - h - i)\)]};\)\n toVec[{\[Alpha]_, \[Zeta]2_, \[Theta]_, \[Eta]2_, \[CurlyTheta]_, \ \[Kappa]2_, \[CurlyKappa]_, \[Lambda]2_, \[Phi]_}, "\"] := Module[{\[Zeta] = \@\[Zeta]2, \[Eta] = \@\[Eta]2, \[Kappa] = \@\[Kappa]2, \ \[Lambda] = \@\[Lambda]2}, \[IndentingNewLine]{\ \ \[Alpha] + 2 \[Zeta]\ Cos[\[Theta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Eta]\ Cos[\[CurlyTheta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Kappa]\ Cos[\[CurlyKappa]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Lambda]\ Cos[\[Phi]], \ \[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 2 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta] + 2 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa] + 2 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi]], \[IndentingNewLine]\[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 4 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta] + 4 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa] + 4 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi]], \[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 2 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta] + 4 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Lambda]\ Cos[\[Phi] + 2 \[Pi]/3], \[IndentingNewLine]\[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 4 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Kappa]\ Cos[\[CurlyKappa] + 2 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi] + 2 \[Pi]/3], \ \[Alpha] + 2 \[Zeta]\ Cos[\[Theta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Eta]\ Cos[\[CurlyTheta] + 2 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa] + 4 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi] + 2 \[Pi]/3], \[IndentingNewLine]\[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 4 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta] + 2 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Lambda]\ Cos[\[Phi] + 4 \[Pi]/3], \ \[Alpha] + 2 \[Zeta]\ Cos[\[Theta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Eta]\ Cos[\[CurlyTheta] + 4 \[Pi]/3] + 2 \[Kappa]\ Cos[\[CurlyKappa] + 2 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi] + 4 \[Pi]/3], \ \[Alpha] + 2 \[Zeta]\ Cos[\[Theta] + 2 \[Pi]/3] + 2 \[Eta]\ Cos[\[CurlyTheta]]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2 \[Kappa]\ Cos[\[CurlyKappa] + 4 \[Pi]/3] + 2 \[Lambda]\ Cos[\[Phi] + 4 \[Pi]/3]}/9]\) \!\(\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 9, 8, 7, 6, 5, 4, 1, 1, 1, 1, 1}; gp = {1, 1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. (*?*) }; plex =. ; {a + b + c - d + e + f + g + h - i, \[IndentingNewLine]\ \((\((a + b + c + d - e - f)\)\^2 + \((a \ + b + c - g - h + i)\)\^2 + \((\(-d\) + e + f - g - h + i)\)\^2)\)/ 2, \ \((\((a - c + d + e - g + h)\)\^2 + \((b - c + d + f - g - \ i)\)\^2 + \((a - b + e - f + h + i)\)\^2)\)/ 2, \ \((\((a - b + d + f + g - h)\)\^2 + \((a - c - e + f + g + \ i)\)\^2 + \((b - c - d - e + h + i)\)\^2)\)/ 2, \((\((b - c + e - f + g - h)\)\^2 + \((a - b - d - e - g - \ i)\)\^2 + \((a - c - d - f - h - i)\)\^2)\)/2}];\)\(\n\) \( (*\(sh["\"] := Module[{}, gi = {1, 3, 2, \(-7\), \(-9\), \(-8\), \(-4\), \(-6\), \(-5\), 1, 1, 1, 1, 1}; gp = {1, 1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. (*?*) }; plex =. ; {a + b + c - d - e - f + g + h + i, \[IndentingNewLine]\((\((a - b - d + e + g - h)\)\^2 + \((a \ - c - d + f + g - i)\)\^2 + \((b - c - e + f + h - i)\)\^2)\)/ 2, \ \((\((b - c + d - f + g - h)\)\^2 + \((a - b - e + f - g + \ i)\)\^2 + \((a - c + d - e - h + i)\)\^2)\)/ 2, \ \((\((a - c + e - f - g + h)\)\^2 + \((b - c - d + e - g + \ i)\)\^2 + \((a - b + d - f + h - i)\)\^2)\)/ 2, \((\((a + b + c + d + e + f)\)\^2 + \((d + e + f + g + h + \ i)\)\^2 + \((a + b + c - g - h - i)\)\^2)\)/2}];\)*) \)\)\) \!\(\(sh["\"] := Module[{}, \[IndentingNewLine]gi = {1, 3, 2, 10, 12, 11, 7, 9, 8, 4, 6, 5, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]gp = {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, c, b, d, f, e, g, i, h, j, l, k}; {a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]\((a + b + c - g - h - i)\)\^2 + \((d + e + \ f - j - k - l)\)\^2, \n\t\((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c \ + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + \ l)\)\^2)\)/ 2, \[IndentingNewLine]\((\((a - b - d + e + g - h - j + k)\)\^2 + \ \((b - c - e + f + h - i - k + l)\)\^2 + \((\(-a\) + c + d - f - g + i + j - \ l)\)\^2)\)/ 2, \((\((\((a + b - 2 c - g - h + 2 i)\) r3 - d + e + j - \ k)\)\^2 + \((a - b - g + h + \((d + e - 2 f - j - k + 2 l)\) r3)\)\^2)\) 3/4, \((\((\((a + b - 2 c - g - h + 2 i)\) r3 + d - e - j + \ k)\)\^2 + \((a - b - g + h - \((d + e - 2 f - j - k + 2 l)\) r3)\)\^2)\) 3/4}];\)\) \!\(\(toVec[{\[Alpha]_, \[Beta]_, \[Epsilon]\[Epsilon]_, \[Sigma]_, \[Zeta]\ \[Zeta]_, \[Tau]_, \[Lambda]\[Lambda]_, \[Phi]_, \[Eta]\[Eta]_, \[Chi]_, \ \[Kappa]\[Kappa]_, \[Psi]_}, "\"] := \[IndentingNewLine]Module[{\ \[Epsilon] = 2 \@ \[Epsilon]\[Epsilon], \[Zeta] = 2 \@ \[Zeta]\[Zeta], \[Lambda] = 2 \@ \[Lambda]\[Lambda], \[Eta] = \@\(3 \[Eta]\[Eta]\), \[Kappa] \ = \@\(3 \[Kappa]\[Kappa]\), x = \[Chi] + \[Pi]/6, y = \[Psi] + \[Pi]/ 6}, \[IndentingNewLine]{\[Alpha] + \[Beta] + \[Epsilon]\ Cos[\ \[Sigma]] + \[Zeta]\ Cos[\[Tau]] + \[Lambda]\ Cos[\[Phi]] + \[Eta]\ Cos[ x] + \[Eta]\ r3\ Sin[x] + \[Kappa]\ Cos[ y] + \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] + \[Beta] + \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \ \[Omega]] - \[Eta]\ Cos[x] + \[Eta]\ r3\ Sin[x] - \[Kappa]\ Cos[ y] + \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] + \[Beta] + \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ Cos[\[Phi] - \ \[Omega]] - 2 \[Eta]\ r3\ Sin[x] - 2 \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau]] - \[Lambda]\ Cos[\[Phi]] + \[Eta]\ r3\ \ Cos[x] - \[Eta]\ Sin[x] - \[Kappa]\ r3\ Cos[y] + \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \ \[Omega]] + \[Eta]\ r3\ Cos[x] + \[Eta]\ Sin[x] - \[Kappa]\ r3\ Cos[ y] - \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ Cos[\[Phi] - \ \[Omega]] - 2 \[Eta]\ r3\ Cos[x] + 2 \[Kappa]\ r3\ Cos[ y], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau]] + \[Lambda]\ Cos[\[Phi]] - \[Eta]\ Cos[ x] - \[Eta]\ r3\ Sin[x] - \[Kappa]\ Cos[ y] - \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \ \[Omega]] + \[Eta]\ Cos[x] - \[Eta]\ r3\ Sin[x] + \[Kappa]\ Cos[ y] - \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ Cos[\[Phi] - \ \[Omega]] + 2 \[Eta]\ r3\ Sin[x] + 2 \[Kappa]\ r3\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau]] - \[Lambda]\ Cos[\[Phi]] - \[Eta]\ r3\ \ Cos[x] + \[Eta]\ Sin[x] + \[Kappa]\ r3\ Cos[y] - \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \ \[Omega]] - \[Eta]\ r3\ Cos[x] - \[Eta]\ Sin[x] + \[Kappa]\ r3\ Cos[ y] + \[Kappa]\ Sin[ y], \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ Cos[\[Phi] - \ \[Omega]] + 2 \[Eta]\ \ r3\ Cos[x] - 2 \[Kappa]\ r3\ Cos[y]}/12];\)\) \!\(\(toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}, "\"] := \[IndentingNewLine]Module[{\[Epsilon], r = \(-Sqrt[3]\), e1 = a + b + c - g - h - i, e2 = d + e + f - j - k - l, \[IndentingNewLine]a1 = a - b + d - e + g - h + j - k, a2 = a - b - d + e + g - h - j + k, \[IndentingNewLine]b1 = Chop[FullSimplify[b - c + e - f + h - i + k - l]], b2 = Chop[ FullSimplify[ b - c - e + f + h - i - k + l]], \[IndentingNewLine]c1 = c - a + f - d + i - g + l - j, c2 = c - a - f + d + i - g - l + j, f1 = a - b - g + h + \((d + e - 2\ f - j - k + 2 l)\) r3, \[IndentingNewLine]f2 = \((a + b - 2 c - g - h + 2 i)\) r3 - d + e + j - k, \[IndentingNewLine]f3 = a - b - g + h - \((d + e - 2\ f - j - k + 2 l)\) r3, \[IndentingNewLine]f4 = \((a + b - 2 c - g - h + 2 i)\) r3 + d - e - j + k}, \[IndentingNewLine]Chop[{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]\[Epsilon] = e1\^2 + e2\^2, If[\[Epsilon] == 0 && e2 \[Equal] 0, 0, ArcTan[e1, e2], ArcTan[e1, e2]], \[IndentingNewLine]\((a1\^2 + b1\^2 + c1\^2)\)/ 2, If[b1 \[Equal] 0, 0, ArcTan[a1 - c1, r\ b1]], \[IndentingNewLine]\((a2\^2 + b2\^2 + c2\^2)\)/2, If[b2 \[Equal] 0, 0, ArcTan[a2 - c2, r\ b2]], \[IndentingNewLine]\((f1\^2 + f2\^2)\) 3/4, If[f2 \[Equal] 0, \(-\[Pi]\)/6, ArcTan[f1, f2] - \[Pi]/6], \[IndentingNewLine]\((f3\^2 + f4\^2)\) 3/4, If[f4 \[Equal] 0, \(-\[Pi]\)/6, ArcTan[f3, f4] - \[Pi]/6]}]];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]gp = {1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, b, c, \(-d\), \(-e\), \(-f\), g, h, i, \(-j\), \(-k\), \(-l\)}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \n\t\((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - \ f + h - i + k - l)\)\^2 + \((c - a + f - d + i - g + l - j)\)\^2)\)/ 2, \((\((a - b - d + e + g - h - j + k)\)\^2 + \((b - c - e + f + \ h - i - k + l)\)\^2 + \((c - a - f + d + i - g - l + j)\)\^2)\)/ 2, \((\((a - b + d - e - g + h - j + k)\)\^2 + \((b - c + e - f - \ h + i - k + l)\)\^2 + \((c - a + f - d - i + g - l + j)\)\^2)\)/ 2, \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - \ h + i + k - l)\)\^2 + \((c - a - f + d - i + g + l - j)\)\^2)\)/2}];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}, "\"] := Module[{r = \(-Sqrt[3]\), a1 = a - b + d - e + g - h + j - k, a2 = a - b - d + e + g - h - j + k, \[IndentingNewLine]a3 = a - b + d - e - g + h - j + k, a4 = a - b - d + e - g + h + j - k, \[IndentingNewLine]b1 = Chop[FullSimplify[b - c + e - f + h - i + k - l]], b2 = Chop[ FullSimplify[ b - c - e + f + h - i - k + l]], \[IndentingNewLine]b3 = Chop[FullSimplify[b - c + e - f - h + i - k + l]], b4 = Chop[ FullSimplify[ b - c - e + f - h + i + k - l]], \[IndentingNewLine]c1 = c - a + f - d + i - g + l - j, c2 = c - a - f + d + i - g - l + j, \[IndentingNewLine]c3 = c - a + f - d - i + g - l + j, c4 = c - a - f + d - i + g + l - j}, \[IndentingNewLine]Chop[{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \[IndentingNewLine]\((a1\^2 + b1\^2 + c1\^2)\)/2, If[b1 \[Equal] 0, 0, ArcTan[a1 - c1, r\ b1]], \[IndentingNewLine]\((a2\^2 + b2\^2 + c2\^2)\)/2, If[b2 \[Equal] 0, 0, ArcTan[a2 - c2, r\ b2]], \[IndentingNewLine]\((a3\^2 + b3\^2 + c3\^2)\)/2, If[b3 \[Equal] 0, 0, ArcTan[a3 - c3, r\ b3]], \[IndentingNewLine]\((a4\^2 + b4\^2 + c4\^2)\)/2, If[b4 \[Equal] 0, 0, ArcTan[a4 - c4, r\ b4]]}]];\)\) \!\(toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Zeta]\[Zeta]_, \ \[Tau]_, \[Lambda]\[Lambda]_, \[Phi]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := \[IndentingNewLine]Module[{\[Zeta] = 2 \@ \[Zeta]\[Zeta], \[Lambda] = 2 \@ \[Lambda]\[Lambda], \[Eta] = 2 \@ \[Eta]\[Eta], \[Kappa] = 2 \@ \[Kappa]\[Kappa]}, \[IndentingNewLine]{\ \ \[Alpha] + \[Beta] \ + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ Cos[\ \[Phi]]\ \ \ \ \ + \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]], \ \[IndentingNewLine]\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\ \[Tau] + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \ \[Omega]] + \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[IndentingNewLine]\[Alpha] + \ \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ \ Cos[\[Phi] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]] + \[Kappa]\ \ Cos[\[Psi] - \[Omega]], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \ \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ + \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]]\ , \ \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Cos[\ \[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \ \[Omega]] - \[Kappa]\ Cos[\[Psi] + \[Omega]], \[IndentingNewLine]\[Alpha] - \ \[Beta] + \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ \ Cos[\[Phi] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]] - \[Kappa]\ \ Cos[\[Psi] - \[Omega]], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \ \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]]\ , \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\ \[Tau] + \[Omega]] + \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \ \[Omega]] - \[Kappa]\ Cos[\[Psi] + \[Omega]], \[IndentingNewLine]\[Alpha] + \ \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] + \[Lambda]\ \ Cos[\[Phi] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]] - \[Kappa]\ \ Cos[\[Psi] - \[Omega]], \[IndentingNewLine]\[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]]\ , \ \[IndentingNewLine]\[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Cos[\ \[Tau] + \[Omega]] - \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \ \[Omega]] + \[Kappa]\ Cos[\[Psi] + \[Omega]], \[IndentingNewLine]\[Alpha] - \ \[Beta] - \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Lambda]\ \ Cos[\[Phi] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]] + \[Kappa]\ \ Cos[\[Psi] - \[Omega]]}/12]\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 10, 11, 12, 7, 9, 8, 4, 5, 6, 1, 1, 1, 2, 2}; \[IndentingNewLine]gp = {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex =. ; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - \ l)\)\^2, \n\((\((\(-b\) + c - h + i)\)\^2 + \((a - c + g - i)\)\^2 + \((a - b \ + g - h)\)\^2 - \((\(-e\) + f - k + l)\)\^2 - \((d - f + j - l)\)\^2 - \((d - \ e + j - k)\)\^2)\)/ 2, \n\((\((\(-b\) + c + h - i)\)\^2 + \((a - c - g + i)\)\^2 + \ \((a - b - g + h)\)\^2 + \((\(-e\) + f + k - l)\)\^2 + \((d - f - j + l)\)\^2 \ + \((d - e - j + k)\)\^2)\)/2}];\)\n \(toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}, "\"] := Module[{\[Epsilon], r = \(-Sqrt[3]\), g1 = a + b + c - g - h - i, g2 = d + e + f - j - k - l, \[IndentingNewLine]a1 = a - b + g - h, a2 = a - b - g + h, \[IndentingNewLine]b1 = Chop[FullSimplify[b - c + h - i]], b2 = Chop[ FullSimplify[b - c - h + i]], \[IndentingNewLine]c1 = \(-a\) + c - g + i, c2 = \(-a\) + c + g - i, \[IndentingNewLine]d1 = d - e + j - k, d2 = d - e - j + k, \[IndentingNewLine]e1 = Chop[FullSimplify[e - f + k - l]], e2 = Chop[ FullSimplify[e - f - k + l]], \[IndentingNewLine]f1 = \(-d\) + f - j + l, f2 = \(-d\) + f + j - l}, \[IndentingNewLine]Chop[{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]\[Epsilon] = g1\^2 + g2\^2, If[\[Epsilon] == 0 && g2 \[Equal] 0, 0, ArcTan[g1, g2], ArcTan[g1, g2]], \[IndentingNewLine]\((a1\^2 + b1\^2 + c1\^2)\)/ 2, If[b1 \[Equal] 0, 0, ArcTan[a1 - c1, r\ b1]], \[IndentingNewLine]\((d1\^2 + e1\^2 + f1\^2)\)/2, If[e1 \[Equal] 0, 0, ArcTan[d1 - f1, r\ e1]], \[IndentingNewLine]\((a2\^2 + b2\^2 + c2\^2)\)/2, If[b2 \[Equal] 0, 0, ArcTan[a2 - c2, r\ b2]], \[IndentingNewLine]\((d2\^2 + e2\^2 + f2\^2)\)/2, If[e2 \[Equal] 0, 0, ArcTan[d2 - f2, r\ e2]]}]];\)\) \!\(\(toVec[{\[Alpha]_, \[Beta]_, \[Epsilon]\[Epsilon]_, \[Sigma]_, \[Zeta]\ \[Zeta]_, \[Tau]_, \[Lambda]\[Lambda]_, \[Phi]_, \[Eta]\[Eta]_, \[Chi]_, \ \[Kappa]\[Kappa]_, \[Psi]_}, "\"] := \[IndentingNewLine]Module[{\ \[Epsilon] = 2 \@ \[Epsilon]\[Epsilon], \[Zeta] = 4 \@ \[Zeta]\[Zeta], \[Lambda] = 4 \@ \[Lambda]\[Lambda], \[Eta] = 4 \@ \[Eta]\[Eta], \[Kappa] = 4 \@ \[Kappa]\[Kappa]}, {\ \ \[Alpha] + \[Beta] + \ \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ + \[Eta]\ Cos[\ \[Chi]], \[IndentingNewLine]\[Alpha] + \[Beta] + \[Epsilon]\ \ Cos[\[Sigma]] \ + \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]], \ \ \[IndentingNewLine]\[Alpha] + \[Beta] + \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ \ Cos[\[Tau] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]]\ , \ \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi] + \[Omega]]\ \ + \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] + \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi] - \[Omega]]\ \ + \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ \ Cos[\[Tau]]\ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ \ , \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ \ Cos[\[Tau] + \[Omega]] - \[Eta]\ Cos[\[Chi] + \[Omega]], \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Epsilon]\ \ Cos[\[Sigma]] + \[Zeta]\ \ Cos[\[Tau] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]]\ , \ \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi] + \[Omega]]\ \ - \[Kappa]\ Cos[\[Psi] + \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] - \[Epsilon]\ \ Sin[\[Sigma]]\ \ + \ \[Lambda]\ Cos[\[Phi] - \[Omega]]\ \ - \[Kappa]\ Cos[\[Psi] - \[Omega]]}/ 12];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11, 12, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]gp = {1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex =. ; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \n\t\((\((a - b + g - h)\)^2 + \((b - c + h - i)\)\^2 + \((\(-a\) + c - g + \ i)\)\^2 - \n\t\ \((\(-d\) + e - j + k)\)\^2 - \((\(-e\) + f - k + l)\)\^2 - \ \((d - f + j - l)\)\^2)\)/ 2, \n\t\((\((a - b - g + h)\)\^2 + \((b - c - h + i)\)\^2 + \ \((\(-a\) + c + g - i)\)\^2 - \n\t\ \ \ \((\(-d\) + e + j - k)\)\^2 - \((\(-e\ \) + f + k - l)\)\^2 - \((d - f - j + l)\)\^2)\)/2}];\)\) \!\(\(toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}, "\"] := Module[{r = \(-Sqrt[3]\), a1 = a - b + g - h, a2 = \(-d\) + e - j + k, a3 = a - b - g + h, a4 = \(-d\) + e + j - k, \[IndentingNewLine]b1 = Chop[FullSimplify[b - c + h - i]], b2 = Chop[ FullSimplify[\(-e\) + f - k + l]], \[IndentingNewLine]b3 = Chop[FullSimplify[b - c - h + i]], b4 = Chop[ FullSimplify[\(-e\) + f + k - l]], \[IndentingNewLine]c1 = \(-a\) + c - g + i, c2 = d - f + j - l, c3 = \(-a\) + c + g - i, c4 = d - f - j + l}, \[IndentingNewLine]Chop[{a + b + c + d + e + f + g + h + i + j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]a + b + c + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \[IndentingNewLine]\((a1\^2 + b1\^2 + c1\^2)\)/2, If[b1 \[Equal] 0, 0, ArcTan[a1 - c1, r\ b1]], \[IndentingNewLine]\((a2\^2 + b2\^2 + c2\^2)\)/2, If[b2 \[Equal] 0, 0, ArcTan[a2 - c2, r\ b2]], \[IndentingNewLine]\((a3\^2 + b3\^2 + c3\^2)\)/2, If[b3 \[Equal] 0, 0, ArcTan[a3 - c3, r\ b3]], \[IndentingNewLine]\((a4\^2 + b4\^2 + c4\^2)\)/2, If[b4 \[Equal] 0, 0, ArcTan[a4 - c4, r\ b4]]}]];\)\) \!\(toVec[{\[Alpha]_, \[Beta]_, \[Gamma]_, \[Delta]_, \[Zeta]\[Zeta]_, \ \[Tau]_, \[Lambda]\[Lambda]_, \[Phi]_, \[Eta]\[Eta]_, \[Chi]_, \[Kappa]\ \[Kappa]_, \[Psi]_}, "\"] := \[IndentingNewLine]Module[{\[Zeta] = 4 \@ \[Zeta]\[Zeta], \[Lambda] = 4 \@ \[Lambda]\[Lambda], \[Eta] = 4 \@ \[Eta]\[Eta], \[Kappa] = 4 \@ \[Kappa]\[Kappa]}, {\ \ \ \ \ \ \[IndentingNewLine]\[Alpha] + \ \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\[Tau]]\ \ \ \ \ \ + \[Eta]\ \ Cos[\[Chi]], \[IndentingNewLine]\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \ \[Zeta]\ Cos[\[Tau] + \[Omega]] + \[Eta]\ Cos[\[Chi] + \[Omega]], \ \ \[IndentingNewLine]\[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[Zeta]\ Cos[\ \[Tau] - \[Omega]] + \[Eta]\ Cos[\[Chi] - \[Omega]], \[IndentingNewLine]\ \[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \[Lambda]\ Cos[\[Phi]]\ \ \ \ \ \ \ - \[Kappa]\ Cos[\[Psi]]\ , \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \ \[Delta] - \[Lambda]\ Cos[\[Phi] + \[Omega]] - \[Kappa]\ Cos[\[Psi] + \ \[Omega]], \[IndentingNewLine]\[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \ \[Lambda]\ Cos[\[Phi] - \[Omega]] - \[Kappa]\ Cos[\[Psi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\ \[Tau]]\ \ \ \ \ \ - \[Eta]\ Cos[\[Chi]]\ , \[IndentingNewLine]\[Alpha] + \ \[Beta] - \[Gamma] - \[Delta] + \[Zeta]\ Cos[\[Tau] + \[Omega]] - \[Eta]\ \ Cos[\[Chi] + \[Omega]], \[IndentingNewLine]\[Alpha] + \[Beta] - \[Gamma] - \ \[Delta] + \[Zeta]\ Cos[\[Tau] - \[Omega]] - \[Eta]\ Cos[\[Chi] - \[Omega]], \ \[IndentingNewLine]\[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \[Lambda]\ Cos[\ \[Phi]]\ \ \ \ \ \ + \[Kappa]\ Cos[\[Psi]]\ , \[IndentingNewLine]\[Alpha] - \ \[Beta] - \[Gamma] + \[Delta] - \[Lambda]\ Cos[\[Phi] + \[Omega]] + \[Kappa]\ \ Cos[\[Psi] + \[Omega]], \[IndentingNewLine]\[Alpha] - \[Beta] - \[Gamma] + \ \[Delta] - \[Lambda]\ Cos[\[Phi] - \[Omega]] + \[Kappa]\ Cos[\[Psi] - \ \[Omega]]}/12]\) \!\(\(sh["\"] := Module[{}, gp =. ; plex =. ; gi = {1, 3, 2, 4, 9, 11, 7, 12, 5, 10, 6, 8, 1, 1, 3}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l, \n\t\((\((\(-a\) + c - d + f - g + i - j + l)\)\^2 + \((a \ - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2)\)/ 2, \n\t\((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - e - h + k)\) - \n\ \ \ \ \ \ \ \ \ \ \ \ \ \t\((a - d + g - j)\)\ \((b - e - h + k)\)\ \((c + f - i - l)\) - \((a - d - g + j)\)\ \((b + e - h - k)\)\ \((c - f + i - l)\) - \[IndentingNewLine]\ \ \ \((a + d - g - j)\)\ \((b - e + h - k)\)\ \((c - f - i + l)\) + \((c + f - i - l)\)\ \((c - f + i - l)\)\ \((c - f - i + l)\)}];\)\) \!\(\(sh["\"] := Module[{}, \[IndentingNewLine]gp = {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. (*?*) }; gi = {1, 3, 2, \(-4\), \(-6\), \(-5\), \(-7\), \(-9\), \(-8\), \(-10\), \(-12\ \), \(-11\), 2, 2}; {a, c, b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \(-l\), \ \(-k\)}; \[IndentingNewLine]{\((a + b + c)\)\^2 + \((d + e + f)\)\^2 + \((g + \ h + i)\)\^2 + \((j + k + l)\)\^2, \[IndentingNewLine]\((\((a - b)\)\^2 + \((b \ - c)\)\^2 + \((c - a)\)\^2 + \((d - e)\)\^2 + \((e - f)\)\^2 + \((f - d)\)\^2 \ + \n\t\ \ \ \ \((g - h)\)\^2 + \((h - i)\)\^2 + \((i - g)\)\^2 + \((j - \ k)\)\^2 + \((k - l)\)\^2 + \((l - j)\)\^2)\)^2/4 - 3 \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((h\ j - \ i\ j - g\ k + i\ k + g\ l - h\ l)\)\^2 + \n\t\((b\ j - c\ j - a\ k + c\ k + a\ \ l - b\ l)\)\^2 + \((b\ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 + \n\t\ \((e\ g - f\ g - d\ h + f\ h + d\ i - e\ i)\)\^2 + \((\(-e\)\ j + f\ j + d\ k \ - f\ k - d\ l + e\ l)\)\^2)\)}];\)\) \!\(\(sh["\"] := Module[{}, \[IndentingNewLine]gp = { (*?*) }; gi = {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 2, 2}; {a, c, b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \(-l\), \ \(-k\)}; \[IndentingNewLine]{\((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + \ h + i)\)\^2 - \((j + k + l)\)\^2, \[IndentingNewLine]\((\((a - b)\)\^2 + \((b \ - c)\)\^2 + \((a - c)\)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((d - f)\)\^2 \ - \n\t\ \ \ \ \((g - h)\)\^2 - \((h - i)\)\^2 - \((g - i)\)\^2 - \((j - \ k)\)\^2 - \((k - l)\)\^2 - \((j - l)\)\^2)\)^2/4 + 3 \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 - \((h\ j - \ i\ j - g\ k + i\ k + g\ l - h\ l)\)\^2 + \n\t\((b\ j - c\ j - a\ k + c\ k + a\ \ l - b\ l)\)\^2 + \((b\ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 - \n\t\ \((e\ g - f\ g - d\ h + f\ h + d\ i - e\ i)\)\^2 - \((\(-e\)\ j + f\ j + d\ k \ - f\ k - d\ l + e\ l)\)\^2)\)}];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]gp = {1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, c, b, d, f, e, g, i, h, j, l, k}; \[IndentingNewLine]{a - b - c + d - e - f + g - h - i + j - k - l, a - b - c - d + e + f - g + h + i + j - k - l, \[IndentingNewLine]a - b - c - d + e + f + g - h - i - j + k + l, a - b - c + d - e - f - g + h + i - j + k + l, \n\t\((\((a + b - d - e - g - h + j + k)\)\^2 + \((b - c - e + \ f - h + i + k - l)\)\^2 + \((a + c - d - f - g - i + j + l)\)\^2)\)/ 2, \((\((a + b - d - e + g + h - j - k)\)\^2 + \((b - c - e + f + \ h - i - k + l)\)\^2 + \((a + c - d - f + g + i - j - l)\)\^2)\)/ 2, \((\((a + b + d + e + g + h + j + k)\)\^2 + \((b - c + e - f + \ h - i + k - l)\)\^2 + \((a + c + d + f + g + i + j + l)\)\^2)\)/ 2, \[IndentingNewLine]\((\((a + b + d + e - g - h - j - k)\)\^2 + \ \((b - c + e - f - h + i - k + l)\)\^2 + \((a + c + d + f - g - i - j - \ l)\)\^2)\)/2}];\)\) \!\(\(sh["\"] := Module[{}, gi = {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]gp = {1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; plex = {a, c, b, d, f, e, g, i, h, j, l, k}; \[IndentingNewLine]{a + b + c + d + e + f + g - h - i - j - k - l, \[IndentingNewLine]a + b + c - d - e - f - g + h + i - j - k - l, \[IndentingNewLine]a + b + c - d - e - f + g - h - i + j + k + l, \[IndentingNewLine]a + b + c + d + e + f - g + h + i + j + k + l, \n\t\((\((a - b + d - e - g - h + j - k)\)\^2 + \((b - c + e - \ f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f + g + i - j + l)\)\^2)\)/ 2, \n\t\((\((a - b - d + e + g + h + j - k)\)\^2 + \((b - c - e + \ f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f - g - i - j + l)\)\^2)\)/ 2, \((\((a - b - d + e - g - h - j + k)\)\^2 + \((b - c - e + f + \ h - i - k + l)\)\^2 + \((\(-a\) + c + d - f + g + i + j - l)\)\^2)\)/ 2, \((\((a - b + d - e + g + h - j + k)\)\^2 + \((b - c + e - f - \ h + i - k + l)\)\^2 + \((\(-a\) + c - d + f - g - i + j - l)\)\^2)\)/ 2}];\)\[IndentingNewLine] \(sh["\"] := Module[{}, plex =. ; gi = {1, 6, 5, 4, 3, 2, \(-7\), \(-8\), \(-9\), \(-10\), \(-11\), \(-12\), 1, 2, 2, 1}; \[IndentingNewLine]gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. }; \n\t{\((a - b + c - d + e - f)\)\^2 + \((g - h + i - j + k \ - l)\)\^2, \((\((a - b + d - e)\)\^2 + \((a - c + d - f)\)\^2 + \((b - c + e \ - f)\)\^2 + \((g - h + j - k)\)\^2 + \((g - i + j - l)\)\^2 + \((h - i + k - \ l)\)\^2)\)/ 2, \((\((a + b - d - e)\)\^2 + \((b + c - e - f)\)\^2 + \((a - c \ - d + f)\)\^2 + \((g + h - j - k)\)\^2 + \((h + i - k - l)\)\^2 + \((g - i - \ j + l)\)\^2)\)/ 2, \((a + b + c + d + e + f)\)\^2 + \((g + h + i + j + k + \ l)\)\^2}];\)\) \!\(\(sh12[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}] := Module[{ab = a - b - g + h, ag = a - b + g - h, bc = b - c - h + i, bh = b - c + h - i, \[IndentingNewLine]ca = a - c - g + i, ci = a - c + g - i, de = d - e - j + k, dj = d - e + j - k, \[IndentingNewLine]ef = e - f - k + l, ek = e - f + k - l, fd = d - f - j + l, fl = d - f + j - l, f1 = a - b - g + h + \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f2 = \((a + b - 2 c - g - h + 2 i)\)/\@3 - d + e + j - k, \[IndentingNewLine]f3 = a - b - g + h - \((d + e - 2\ f - j - k + 2 l)\)/\@3, \[IndentingNewLine]f4 = \((a + b - 2 c - g - h + 2 i)\)/\@3 + d - e - j + k, abef, agek}, \[IndentingNewLine]abef = 3 \((ab\ ef - bc\ de)\)\^2; agek = 3 \((ag\ ek - bh\ dj)\)\^2; \[IndentingNewLine]{o1 = a + b + c + d + e + f + g + h + i + j + k + l, o2 = a + b + c - d - e - f + g + h + i - j - k - l, \[IndentingNewLine]o3 = a + b + c + d + e + f - g - h - i - j - k - l, o4 = a + b + c - d - e - f - g - h - i + j + k + l, \[IndentingNewLine] (*5*) l22a = 3 \((f1\^2 + f2\^2)\)/4, (*6*) l22b = 3 \((f3\^2 + f4\^2)\)/4, \[IndentingNewLine] (*7*) p22 = \((a + b + c + g + h + i)\)\^2 + \((d + e + f + j + k + \ l)\)\^2, (*8*) q22 = \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - \ l)\)\^2, (*9*) r22 = \((a + b + c + g + h + i)\)\^2 - \((d + e + f + j + k + \ l)\)\^2, (*10*) s22 = \((a + b + c - g - h - i)\)\^2 - \((d + e + f - j - k - \ l)\)\^2, \[IndentingNewLine] (*11*) p23 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + \ fl)\)\^2)\)/2, (*12*) q23 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - \ fl)\)\^2)\)/2, \[IndentingNewLine] (*13*) r23 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + \ fd)\)\^2)\)/2, (*14*) s23 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - \ fd)\)\^2)\)/2, \[IndentingNewLine] (*15*) q24 = \((a + b + c)\)\^2 + \((d + e + f)\)\^2 + \((g + h + \ i)\)\^2 + \((j + k + l)\)\^2, \[IndentingNewLine] (*16*) s24 = \((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + h + \ i)\)\^2 - \((j + k + l)\)\^2, \[IndentingNewLine] (*17*) p26 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)/ 2, (*18*) q26 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)/ 2, \[IndentingNewLine] (*19*) r26 = \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)/ 2, (*20*) s26 = \((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/ 2, \n\ \ \ \ \ \ \ \ (*21*) \ l3 = \((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - e - h + k)\) + \[IndentingNewLine]\((c + f - \ i\ - l)\)\ \((c - f + \ i\ - l)\)\ \((c - f - \ i\ + l)\) - \((a - d + g - j)\)\ \((b - e - h + k)\)\ \((c + f - \ i\ - l)\) - \[IndentingNewLine]\((a - d - g + j)\)\ \((b + e - h - k)\)\ \((c - f + i - l)\) - \((a + d - g - j)\)\ \((b - e + h - k)\)\ \((c - f - \ i\ + l)\), \[IndentingNewLine] (*22*) l4p = p26\^2 + agek, \[IndentingNewLine] (*23*) l4q = q26\^2 + abef, \[IndentingNewLine] (*24*) l4r = r26\^2 - 3 agek, \[IndentingNewLine] (*25*) l4s = l22a\ l22b (*s26\^2 - abef*) , (*26*) L4Q = \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((a - c)\)\^2 + \((d \ - e)\)\^2 + \((e - f)\)\^2 + \((d - f)\)\^2 + \((g - h)\)\^2 + \ \[IndentingNewLine]\((h - i)\)\^2 + \((g - i)\)\^2 + \((j - k)\)\^2 + \((k - \ l)\)\^2 + \((j - l)\)\^2)\)^2/4 - 3\ \((\((a\ \((f - e)\) + b\ \((d - f)\) + c\ \((e - d)\))\)\^2 \ + \((a\ \((i - h)\) + b\ \((g - i)\) + c\ \((h - g)\))\)\^2 + \((d\ \((h - i)\ \) + e\ \((i - g)\) + f\ \((g - h)\))\)\^2 + \((g\ \((k - l)\) + h\ \((l - j)\ \) + i\ \((j - k)\))\)\^2 + \[IndentingNewLine]\((\((b - c)\)\^2 + \((e - \ f)\)\^2)\)\ j\^2 + \((\((a - c)\)\^2 + \((d - f)\)\^2)\)\ k\^2 + \((\((a - b)\ \)\^2 + \((d - e)\)\^2)\)\ l\^2)\) + \[IndentingNewLine]6\ \((\((\((c - a)\)\ \((c - b)\) + \((f - d)\)\ \((f - e)\))\)\ j\ k + \((\((b - a)\)\ \((b - c)\) + \((e - d)\)\ \((e - f)\))\)\ j\ l + \[IndentingNewLine]\((\((a - b)\)\ \((a - c)\) + \((d - e)\)\ \((d - f)\))\)\ k\ l)\), \[IndentingNewLine] (*27*) L4\[Sigma] = \ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - \ a)\)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((f - d)\)\^2 - \((g - h)\)\^2 - \ \((h - i)\)\^2 - \((i - g)\)\^2 - \((j - k)\)\^2 - \((k - l)\)\^2 - \((l - j)\ \)\^2)\)^2/4 + 3\ \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((b\ \ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 - \((e\ g - f\ g - d\ h + f\ h + d\ \ i - e\ i)\)\^2 + \((b\ j - c\ j - a\ k + c\ k + a\ l - b\ l)\)\^2 - \((\(-e\ \)\ j + f\ j + d\ k - f\ k - d\ l + e\ l)\)\^2 - \((h\ j - i\ j - g\ k + i\ k \ + g\ l - h\ l)\)\^2)\), \[IndentingNewLine] (*28*) L4a3 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + fl)\)\^2)\ \)\^2/4 + agek, (*29*) L4b3 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - fl)\)\^2)\ \)\^2/4 + agek, (*30*) L4c3 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + fd)\)\^2)\ \)\^2/4 - abef, (*31*) L4d3 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - fd)\)\^2)\ \)\^2/4 - abef, (*32*) L4a6 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)\^2/4 + agek, (*33*) L4b6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/4 + abef, (*34*) L4c6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/4 - abef, \n\t (*35*) L4d6 = \ \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)\^2/ 4 - agek, \[IndentingNewLine] (*36*) L49 = \((\((ab\^2 - de\^2)\)\^2 + \((ab\^2 - ef\^2)\)\^2 + \ \((ab\^2 - fd\^2)\)\^2 + \((bc\^2 - de\^2)\)\^2 + \((bc\^2 - ef\^2)\)\^2 + \ \((bc\^2 - fd\^2)\)\^2 + \((ca\^2 - de\^2)\)\^2 + \((ca\^2 - ef\^2)\)\^2 + \ \((ca\^2 - fd\^2)\)\^2)\), \[IndentingNewLine] (*37*) L49a = \((\((ag\^2 - dj\^2)\)\^2 + \((ag\^2 - ek\^2)\)\^2 + \((ag\ \^2 - fl\^2)\)\^2 + \((bh\^2 - dj\^2)\)\^2 + \((bh\^2 - ek\^2)\)\^2 + \ \((bh\^2 - fl\^2)\)\^2 + \((ci\^2 - dj\^2)\)\^2 + \((ci\^2 - ek\^2)\)\^2 + \ \((ci\^2 - fl\^2)\)\^2)\), \[IndentingNewLine] (*38*) L49b = \((\((ab\^2 - dj\^2)\)\^2 + \((ab\^2 - ek\^2)\)\^2 + \((ab\ \^2 - fl\^2)\)\^2 + \((bc\^2 - dj\^2)\)\^2 + \((bc\^2 - ek\^2)\)\^2 + \ \((bc\^2 - fl\^2)\)\^2 + \((ca\^2 - dj\^2)\)\^2 + \((ca\^2 - ek\^2)\)\^2 + \ \((ca\^2 - fl\^2)\)\^2)\), \[IndentingNewLine] (*39*) L49c = \((\((ag\^2 - de\^2)\)\^2 + \((ag\^2 - ef\^2)\)\^2 + \((ag\ \^2 - fd\^2)\)\^2 + \((bh\^2 - de\^2)\)\^2 + \((bh\^2 - ef\^2)\)\^2 + \ \((bh\^2 - fd\^2)\)\^2 + \((ci\^2 - de\^2)\)\^2 + \((ci\^2 - ef\^2)\)\^2 + \ \((ci\^2 - fd\^2)\)\^2)\)}];\)\) dozalcons=sh12[{a,b,c,d,e,f,g,h,i,j,k,l}]; ds={"o1","o2","o3","o4","l22a","l22b","p22","q22","r22","s22","p23","q23", "r23","s23","q24","s24","p26","q26","r26","s26","l3","l4p","l4q","l4r", "l4s","L4Q","L4\[Sigma]","L4a3","L4b3","L4c3","L4d3","L4a6","L4b6", "L4c6","L4d6","L49","L49a","L49b","L49c"}; \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 2, 15, 4, 13, 6, 11, 8, 9, 10, 7, 12, 5, 14, 3, 16, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, \((a - e + i - m)\)^2 - \((b - f + j - n)\)^2 + \((c - g + k - o)\)^2 - \((d - h + l - p)\)^2, \(-2\)\ \((\((a - i)\)\ \((c - k)\) - \((b - j)\)\ \((d - l)\) + \((c - k)\)\ \((e - m)\) - \((d - l)\)\ \((f - n)\) - \((a - i)\)\ \((g - o)\) + \((e - m)\)\ \((g - o)\) + \((b - j)\)\ \((h - p)\) - \((f - n)\)\ \((h - p)\))\)^2 + \((\((a - i)\)\^2 - \((b - j)\)\^2 + \((c \ - k)\)\^2 - \((d - l)\)\^2 + \((e - m)\)\^2 - \((f - n)\)\^2 + \((g - o)\)\^2 \ - \((h - p)\)\^2)\)^2}]; sh["\"] := Module[{}, gp =. ; gi = {1, 10, 15, 12, 13, 14, 11, 16, 9, 2, 7, 4, 5, 6, 3, 8, 1, 1, 1, 1, 2, 2}; {a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, \((a - e + i - m)\)^2 - \((b - f + j - n)\)^2 + \((c - g + k - o)\)^2 - \((d - h + l - p)\)^2, \[IndentingNewLine]\((a - i)\)\^4 + 2\ \((a - i)\)\^2\ \((b - j)\)\^2 + \((b - j)\)\^4 + 2\ \((b - j)\)\^2\ \((c - k)\)\^2 + \((c - k)\)\^4 - 4\ \((a - i)\)\ \((b - j)\)\ \((c - k)\)\ \((d - l)\) + 2\ \((a - i)\)\^2\ \((d - l)\)\^2 + 2\ \((c - k)\)\^2\ \((d - l)\)\^2 + \((d - l)\)\^4 - 4\ \((a - i)\)\ \((c - k)\)\^2\ \((e - m)\) - 4\ \((b - j)\)\ \((c - k)\)\ \((d - l)\)\ \((e - m)\) + 2\ \((a - i)\)\^2\ \((e - m)\)\^2 + 2\ \((b - j)\)\^2\ \((e - m)\)\^2 + 2\ \((d - l)\)\^2\ \((e - m)\)\^2 + \((e - m)\)\^4 - 4\ \((a - i)\)\ \((c - k)\)\ \((d - l)\)\ \((f - n)\) - 4\ \((b - j)\)\ \((d - l)\)\^2\ \((f - n)\) - 4\ \((c - k)\)\ \((d - l)\)\ \((e - m)\)\ \((f - n)\) + 2\ \((a - i)\)\^2\ \((f - n)\)\^2 + 2\ \((b - j)\)\^2\ \((f - n)\)\^2 + 2\ \((c - k)\)\^2\ \((f - n)\)\^2 + 2\ \((e - m)\)\^2\ \((f - n)\)\^2 + \((f - n)\)\^4 + 4\ \((a - i)\)\^2\ \((c - k)\)\ \((g - o)\) + 4\ \((a - i)\)\ \((b - j)\)\ \((d - l)\)\ \((g - o)\) - 4\ \((b - j)\)\ \((d - l)\)\ \((e - m)\)\ \((g - o)\) - 4\ \((c - k)\)\ \((e - m)\)\^2\ \((g - o)\) + 4\ \((a - i)\)\ \((d - l)\)\ \((f - n)\)\ \((g - o)\) - 4\ \((d - l)\)\ \((e - m)\)\ \((f - n)\)\ \((g - o)\) + 2\ \((b - j)\)\^2\ \((g - o)\)\^2 + 2\ \((c - k)\)\^2\ \((g - o)\)\^2 + 2\ \((d - l)\)\^2\ \((g - o)\)\^2 + 4\ \((a - i)\)\ \((e - m)\)\ \((g - o)\)\^2 + 2\ \((f - n)\)\^2\ \((g - o)\)\^2 + \((g - o)\)\^4 + 4\ \((a - i)\)\ \((b - j)\)\ \((c - k)\)\ \((h - p)\) + 4\ \((b - j)\)\^2\ \((d - l)\)\ \((h - p)\) + 4\ \((b - j)\)\ \((c - k)\)\ \((e - m)\)\ \((h - p)\) - 4\ \((a - i)\)\ \((c - k)\)\ \((f - n)\)\ \((h - p)\) - 4\ \((c - k)\)\ \((e - m)\)\ \((f - n)\)\ \((h - p)\) - 4\ \((d - l)\)\ \((f - n)\)\^2\ \((h - p)\) - 4\ \((a - i)\)\ \((b - j)\)\ \((g - o)\)\ \((h - p)\) + 4\ \((b - j)\)\ \((e - m)\)\ \((g - o)\)\ \((h - p)\) + 4\ \((a - i)\)\ \((f - n)\)\ \((g - o)\)\ \((h - p)\) - 4\ \((e - m)\)\ \((f - n)\)\ \((g - o)\)\ \((h - p)\) + 2\ \((a - i)\)\^2\ \((h - p)\)\^2 + 2\ \((c - k)\)\^2\ \((h - p)\)\^2 + 2\ \((d - l)\)\^2\ \((h - p)\)\^2 + 2\ \((e - m)\)\^2\ \((h - p)\)\^2 + 4\ \((b - j)\)\ \((f - n)\)\ \((h - p)\)\^2 + 2\ \((g - o)\)\^2\ \((h - p)\)\^2 + \((h - p)\)\^4}]; sh["\"] := Module[{}, gp =. ; gi = {1, 2, 7, 4, 5, 6, 3, 8, 9, 10, 15, 12, 13, 14, 11, 16, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + o + p, a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, \((a - e + i - m)\)\^2 - \((b - f + j - n)\)\^2 + \((c - g + k - \ o)\)\^2 - \((d - h + l - p)\)\^2, \((a - e - i + m)\)\^2 - \((b - f - j + \ n)\)\^2 + \((c - g - k + o)\)\^2 - \((d - h - l + p)\)\^2}];\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 4, 3, 2, 13, 16, 15, 14, 9, 12, 11, 10, 5, 8, 7, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((e + f + g + h - m - \ n - o - p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((e - f + g - h - \ m + n - o + p)\)\^2, \((a - c + e - g + i - k + m - o)\)\^2 + \((b - d + f - \ h + j - l + n - p)\)\^2, \((a - c - e + g + i - k - m + o)\)\^2 + \((b - d - \ f + h + j - l - n + p)\)\^2, \((a - c + f - h - i + k - n + p)\)\^2 + \((b - \ d - e + g - j + l + m - o)\)\^2, \((a - c - f + h - i + k + n - p)\)\^2 + \ \((b - d + e - g - j + l - m + o)\)\^2}]; sh["\"] := Module[{}, gp =. ; gi = {1, 6, 7, 8, 5, 2, 3, 4, 9, 14, 15, 16, 13, 10, 11, 12, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + o + p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, \((a - e + i - m)\)\^2 + \((b - f + j - n)\)\^2 + \((c - g + k - \ o)\)\^2 + \((d - h + l - p)\)\^2, \((a - e - i + m)\)\^2 + \((b - f - j + \ n)\)\^2 + \((c - g - k + o)\)\^2 + \((d - h - l + p)\)\^2}];\) sh["KK"]:=Module[{}, gi={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1}; gp={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1};{a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p, a-b-c+d+e-f-g+h+i-j-k+l+m-n-o+p,a-b+c-d+e-f+g-h+i-j+k-l+m-n+o-p, a+b-c-d+e+f-g-h+i+j-k-l+m+n-o-p,a-b+c-d-e+f-g+h+i-j+k-l-m+n-o+p, a+b-c-d-e-f+g+h+i+j-k-l-m-n+o+p,a+b+c+d-e-f-g-h+i+j+k+l-m-n-o-p, a-b-c+d-e+f+g-h+i-j-k+l-m+n+o-p,a+b+c+d+e+f+g+h-i-j-k-l-m-n-o-p, a+b+c+d-e-f-g-h-i-j-k-l+m+n+o+p,a+b-c-d-e-f+g+h-i-j+k+l+m+n-o-p, a-b+c-d-e+f-g+h-i+j-k+l+m-n+o-p,a+b-c-d+e+f-g-h-i-j+k+l-m-n+o+p, a-b-c+d+e-f-g+h-i+j+k-l-m+n+o-p,a-b+c-d+e-f+g-h-i+j-k+l-m+n-o+p, a-b-c+d-e+f+g-h-i+j+k-l+m-n-o+p}]; toPol[{a_,b_,c_,d_,e_,f_,g_,h_,i_,j_,k_,l_,m_,n_,o_,p_}, "KK"]:={a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p,a-b-c+d+e-f-g+h+i-j-k+l+m-n-o+p, a-b+c-d+e-f+g-h+i-j+k-l+m-n+o-p,a+b-c-d+e+f-g-h+i+j-k-l+m+n-o-p, a-b+c-d-e+f-g+h+i-j+k-l-m+n-o+p,a+b-c-d-e-f+g+h+i+j-k-l-m-n+o+p, a+b+c+d-e-f-g-h+i+j+k+l-m-n-o-p,a-b-c+d-e+f+g-h+i-j-k+l-m+n+o-p, a+b+c+d+e+f+g+h-i-j-k-l-m-n-o-p,a+b+c+d-e-f-g-h-i-j-k-l+m+n+o+p, a+b-c-d-e-f+g+h-i-j+k+l+m+n-o-p,a-b+c-d-e+f-g+h-i+j-k+l+m-n+o-p, a+b-c-d+e+f-g-h-i-j+k+l-m-n+o+p,a-b-c+d+e-f-g+h-i+j+k-l-m+n+o-p, a-b+c-d+e-f+g-h-i+j-k+l-m+n-o+p,a-b-c+d-e+f+g-h-i+j+k-l+m-n-o+p}; toVec[{a_,b_,c_,d_,e_,f_,g_,h_,i_,j_,k_,l_,m_,n_,o_,p_}, "KK"]:={a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p,a-b-c+d-e+f+g-h+i+j+k-l+m-n-o-p, a-b+c-d+e-f+g-h+i+j-k+l-m-n+o-p,a+b-c-d-e-f+g+h+i+j-k-l-m+n-o+p, a+b+c+d-e-f-g-h+i-j-k-l+m+n+o-p,a-b-c+d+e-f-g+h+i-j-k+l+m-n-o+p, a-b+c-d-e+f-g+h+i-j+k-l-m-n+o+p,a+b-c-d+e+f-g-h+i-j+k+l-m+n-o-p, a+b+c+d+e+f+g+h-i-j-k-l-m-n-o-p,a-b-c+d-e+f+g-h-i-j-k+l-m+n+o+p, a-b+c-d+e-f+g-h-i-j+k-l+m+n-o+p,a+b-c-d-e-f+g+h-i-j+k+l+m-n+o-p, a+b+c+d-e-f-g-h-i+j+k+l-m-n-o+p,a-b-c+d+e-f-g+h-i+j+k-l-m+n+o-p, a-b+c-d-e+f-g+h-i+j-k+l+m+n-o-p,a+b-c-d+e+f-g-h-i+j-k-l+m-n+o+p}/16; \!\(sh["\"] := Module[{}, gi = {1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}; gp = {1, 1, 1, 1, 1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a + b + c + d - e - f - g - h - i - j - k - l + m + n + o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, a - b + c - d - e + f - g + h - i + j - k + l + m - n + o - p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p, \((a - c - e + g - i + k + m - o)\)\^2 + \((b - d - f + h - j + \ l + n - p)\)\^2, \((a - c + e - g - i + k - m + o)\)\^2 + \((b - d + f - h - \ j + l - n + p)\)\^2, \((a - c - e + g + i - k - m + o)\)\^2 + \((b - d - f + \ h + j - l - n + p)\)\^2, \((a - c + e - g + i - k + m - o)\)\^2 + \((b - d + \ f - h + j - l + n - p)\)\^2}]; toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, o_, p_}, "\"] := \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a + b + c + d - e - f - g - h - i - j - k - l + m + n + o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, \[IndentingNewLine]a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, \[IndentingNewLine]a - b + c - d - e + f - g + h - i + j - k + l + m - n + o - p, \[IndentingNewLine]a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, \[IndentingNewLine]a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, \[IndentingNewLine]a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p, \[IndentingNewLine] (*ij*) \((a - c - e + g - i + k + m - o)\)\^2 \ + \((b - d - f + h - j + l + n - p)\)\^2, If[b - d - f + h - j + l + n - p \[Equal] 0, 0, ArcTan[a - c - e + g - i + k + m - o, b - d - f + h - j + l + n - p]], \[IndentingNewLine] (*kl*) \((a - c + e - g - i + k - m + o)\ \)\^2 + \((b - d + f - h - j + l - n + p)\)\^2, If[b - d + f - h - j + l - n + p \[Equal] 0, 0, ArcTan[a - c + e - g - i + k - m + o, b - d + f - h - j + l - n + p]], \[IndentingNewLine] (*mn*) \((a - c - e + g + i - k - m + o)\ \)\^2 + \((b - d - f + h + j - l - n + p)\)\^2, If[b - d - f + h + j - l - n + p \[Equal] 0, 0, ArcTan[a - c - e + g + i - k - m + o, b - d - f + h + j - l - n + p]], \[IndentingNewLine] (*op*) \((a - c + e - g + i - k + m - o)\ \)\^2 + \((b - d + f - h + j - l + n - p)\)\^2, If[b - d + f - h + j - l + n - p \[Equal] 0, 0, ArcTan[a - c + e - g + i - k + m - o, b - d + f - h + j - l + n - p]]}; toVec[{a_, b_, c_, d_, e_, f_, g_, h_, ii_, j_, kk_, l_, mm_, n_, oo_, p_}, \[IndentingNewLine]"\"] := Module[{i = 2 \@ ii, k = 2 \@ kk, m = 2 \@ mm, o = 2 \@ oo}, \[IndentingNewLine]{\ \ (*a*) a + b + c + d + e + f + g + h + i\ Cos[j] + k\ Cos[l] + m\ Cos[n] + o\ Cos[p], \[IndentingNewLine] (*b*) a + b - c - d - e - f + g + h + i\ Sin[j] + k\ Sin[l] + m\ Sin[n] + o\ Sin[p], \[IndentingNewLine] (*c*) a + b + c + d + e + f + g + h - i\ Cos[j] - k\ Cos[l] - m\ Cos[n] - o\ Cos[p], \[IndentingNewLine] (*d*) a + b - c - d - e - f + g + h - i\ Sin[j] - k\ Sin[l] - m\ Sin[n] - o\ Sin[p], \[IndentingNewLine] (*e*) a - b + c - d - e + f - g + h - i\ Cos[j] + k\ Cos[l] - m\ Cos[n] + o\ Cos[p], \[IndentingNewLine] (*f*) a - b - c + d + e - f - g + h - i\ Sin[j] + k\ Sin[l] - m\ Sin[n] + o\ Sin[p], \[IndentingNewLine] (*g*) a - b + c - d - e + f - g + h + i\ Cos[j] - k\ Cos[l] + m\ Cos[n] - o\ Cos[p], \[IndentingNewLine] (*h*) a - b - c + d + e - f - g + h + i\ Sin[j] - k\ Sin[l] + m\ Sin[n] - o\ Sin[p], \[IndentingNewLine] (*i*) a - b - c + d - e + f + g - h - i\ Cos[j] - k\ Cos[l] + m\ Cos[n] + o\ Cos[p], \[IndentingNewLine] (*j*) a - b + c - d + e - f + g - h - i\ Sin[j] - k\ Sin[l] + m\ Sin[n] + o\ Sin[p], \[IndentingNewLine] (*k*) a - b - c + d - e + f + g - h + i\ Cos[j] + k\ Cos[l] - m\ Cos[n] - o\ Cos[p], \[IndentingNewLine] (*l*) a - b + c - d + e - f + g - h + i\ Sin[j] + k\ Sin[l] - m\ Sin[n] - o\ Sin[p], \[IndentingNewLine] (*m*) a + b - c - d + e + f - g - h + i\ Cos[j] - k\ Cos[l] - m\ Cos[n] + o\ Cos[p], \[IndentingNewLine] (*n*) a + b + c + d - e - f - g - h + i\ Sin[j] - k\ Sin[l] - m\ Sin[n] + o\ Sin[p], \[IndentingNewLine] (*o*) a + b - c - d + e + f - g - h - i\ Cos[j] + k\ Cos[l] + m\ Cos[n] - o\ Cos[p], \[IndentingNewLine] (*p*) a + b + c + d - e - f - g - h - i\ Sin[j] + k\ Sin[l] + m\ Sin[n] - o\ Sin[p]}/16];\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 2, 3, 4, 13, 16, 15, 14, 9, 10, 11, 12, 5, 8, 7, 6, 1, 1, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((\(-e\) - f - g - h + \ m + n + o + p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((\(-e\) + f - \ g + h + m - n + o - p)\)\^2, \((a - c + i - k)\)\^2 - \((b - d + j - l)\)\^2 \ - \((e - g + m - o)\)\^2 + \((f - h + n - p)\)\^2, \((a - c - i + k)\)\^2 - \ \((b - d - j + l)\)\^2 + \((e - g - m + o)\)\^2 - \((f - h - n + p)\)\^2}]; sh["\"] := Module[{}, gp =. ; gi = {1, 4, 3, 2, 13, 14, 15, 16, 9, 12, 11, 10, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 2, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((\(-e\) - f - g - h + \ m + n + o + p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((\(-e\) + f - \ g + h + m - n + o - p)\)\^2, \((a - c + i - k)\)\^2 + \((b - d + j - l)\)\^2 \ - \((e - g + m - o)\)\^2 - \((f - h + n - p)\)\^2, \((a - c - i + k)\)\^2 + \ \((b - d - j + l)\)\^2 + \((e - g - m + o)\)\^2 + \((f - h - n + p)\)\^2}];\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, a + b + c - d - e - f - g + h + i + j + k - l - m - n - o + p, a + b - c + d - e - f + g - h + i + j - k + l - m - n + o - p, a - b + c + d - e + f - g - h + i - j + k + l - m + n - o - p, a - b - c - d - e + f + g + h + i - j - k - l - m + n + o + p, 4\ \((\((a - i)\)\ \((e - m)\) - \((b - j)\)\ \((f - n)\) - \((c - k)\)\ \((g - o)\) - \((d - l)\)\ \((h - p)\))\)^2 + \((\((a - i)\)\^2 - \((b - j)\)\^2 - \((c \ - k)\)\^2 - \((d - l)\)\^2 - \((e - m)\)\^2 + \((f - n)\)\^2 + \((g - o)\)\^2 \ + \((h - p)\)\^2)\)^2}]; sh["\"] := Module[{}, gp =. ; gi = {1, 10, 11, 12, 13, 14, 15, 16, 9, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 4}; \[IndentingNewLine]{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, a + b - c - d - e - f + g + h + i + j - k - l - m - n + o + p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, a - b - c + d - e + f + g - h + i - j - k + l - m + n + o - p, \((a - i)\)\^2 + \((b - j)\)\^2 + \((c - k)\)\^2 + \((d - \ l)\)\^2 + \((e - m)\)\^2 + \((f - n)\)\^2 + \((g - o)\)\^2 + \((h - \ p)\)\^2}];\) \!\(\(sh["\"] := Module[{}, gp =. ; gi = {1, 6, 7, 8, 5, 2, 3, 4, 9, 10, 11, 12, 13, 14, \[IndentingNewLine]15, 16, 1, 1, 1, 1, 1, 1, 1, 1, 4}; \[IndentingNewLine]{a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + o + p, a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p, \((a - e)\)\^2 + \((b - f)\)\^2 + \((c - g)\)\^2 + \((d - h)\)\ \^2 - \((j - n)\)\^2 - \[IndentingNewLine]\((k - o)\)\^2 - \((l - p)\)\^2 - \ \((i - m)\)\^2}];\)\) \!\(sh["\"] := Module[{}, (*see\ Louenesto\ p86*) gp =. ; gi = {1, 2, 3, \(-4\), 5, \(-6\), \(-7\), \(-8\), 9, \(-10\), \(-11\), \(-12\), \(-13\), \(-14\), \(-15\), 16, 4}; {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)^2 - 4 \((\((a\ b + c\ d + e\ f + g\ h + i\ j + k\ l + m\ n + o\ p)\)^2 + \((a\ c - b\ d + e\ g - f\ h + i\ k - j\ l + m\ o - n\ p)\)^2 + \((a\ e - b\ f - c\ g + d\ h + i\ m - j\ n - k\ o + l\ p)\)^2 + \((a\ i - b\ j - c\ k + d\ l - e\ m + f\ n + g\ o - h\ p)\)^2 + \((h\ i - g\ j + f\ k - e\ l - d\ m + c\ n - b\ o + a\ p)\)^2)\)}]; sh["\"] := Module[{}, gp =. ; gi = {1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), 8, 9, 10, 11, \(-12\), 13, \(-14\), \(-15\), \(-16\), 4}; {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)^2 - 4 \((\((\(-d\)\ e + c\ f - b\ g + a\ h - l\ m + k\ n - j\ o + i\ p)\)^2 + \((a\ i - b\ j - c\ k + d\ l - e\ m + f\ n + g\ o - h\ p)\)^2 + \[IndentingNewLine]\((b\ i + a\ j - d\ k - c\ l - f\ m - e\ n + h\ o + g\ p)\)^2 + \((c\ i + d\ j + a\ k + b\ l - g\ m - h\ n - e\ o - f\ p)\)^2 + \((e\ i + f\ j + g\ k + h\ l + a\ m + b\ n + c\ o + d\ p)\)^2)\)}]; (*\ found\ from\ conjugation, \ with\ negated\ \(terms!\)\ *) \[IndentingNewLine] \(sh["\"] = Module[{}, gp =. ; gi = {1, \(-2\), \(-3\), \(-4\), 5, 6, 7, \(-8\), 9, 10, 11, \(-12\), \(-13\), 14, 15, 16, 4}; {\((a\^2 + b\^2 + c\^2 + d\^2 - e\^2 - f\^2 - g\^2 - h\^2 - i\^2 - j\^2 - k\^2 - l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)^2 - 4 \((\(-\((\(-d\)\ e + c\ f - b\ g + a\ h + l\ m - k\ n + j\ o - i\ p)\)^2\) - \((\(-d\)\ i + c\ j - b\ k + a\ l - h\ m + g\ n - f\ o + e\ p)\)^2\[IndentingNewLine] + \((\(-f\)\ i + e\ j - h\ k + g\ l - b\ m + a\ n - d\ o + c\ p)\)^2 + \((\(-g\)\ i + h\ j + e\ k - f\ l - c\ m + d\ n + a\ o - b\ p)\)^2\[IndentingNewLine] + \((\(-h\)\ i - g\ j + f\ k + e\ l - d\ m - c\ n + b\ o + a\ p)\)^2)\)}];\)\) \!\(\(\( (*\ found\ from\ conjugation, \ with\ negated\ \(terms!\)\ *) \)\(sh["\"] := Module[{}, gp =. ; gi = {1, \(-2\), 3, 4, 5, 6, \(-7\), 8, 9, 10, \(-11\), 12, \(-13\), 14, \(-15\), \(-16\), 4}; {\((a\^2 + b\^2 - c\^2 - d\^2 - e\^2 - f\^2 + g\^2 + h\^2 - i\^2 - j\^2 + k\^2 + l\^2 + m\^2 + n\^2 - o\^2 - p\^2)\)^2 - 4 \((\((\(-d\)\ e + c\ f - b\ g + a\ h + l\ m - k\ n + j\ o - i\ p)\)^2 + \((\(-d\)\ i + c\ j - b\ k + a\ l - h\ m + g\ n - f\ o + e\ p)\)^2 + \((\(-f\)\ i + e\ j + h\ k - g\ l - b\ m + a\ n + d\ o - c\ p)\)^2 - \((\(-g\)\ i + h\ j + e\ k - f\ l - c\ m + d\ n + a\ o - b\ p)\)^2 - \((\(-h\)\ i - g\ j + f\ k + e\ l - d\ m - c\ n + b\ o + a\ p)\)^2)\)}]; sh["\"] := Module[{}, gp =. ; plex = {a, b, c, \(-d\), e, \(-f\), \(-g\), \(-h\), i, \(-j\), \(-k\), \(-l\), \(-m\), \(-n\), \(-o\), p}; gi = {1, 2, 3, \(-4\), \(-5\), 6, 7, 8, 9, \(-10\), \(-11\), \(-12\), 13, 14, 15, \(-16\), 4}; {\((a\^2 + b\^2 + c\^2 + d\^2 - e\^2 - f\^2 - g\^2 - h\^2 + i\^2 + j\^2 + k\^2 + l\^2 - m\^2 - n\^2 - o\^2 - p\^2)\)^2 - 4 \((\((a\ b + \ c\ d - \ e\ f - \ g\ h + \ i\ j + \ k\ l - \ m\ n - \ o\ p)\)^2 + \((\ a\ c - \ b\ d - \ e\ g + \ f\ h + \ i\ k - \ j\ l - \ m\ o + \ n\ p)\)^2 - \[IndentingNewLine]\((\ a\ e - \ b\ f - \ c\ g + \ d\ h + \ i\ m - \ j\ n - \ k\ o + \ l\ p)\)^2 + \((\ a\ i - \ b\ j - \ c\ k + \ d\ l + \ e\ m - \ f\ n - \ g\ o + \ h\ p)\)^2 - \((\ h\ i - \ g\ j + \ f\ k - \ e\ l - \ d\ m + \ c\ n - \ b\ o + \ a\ p)\)^2)\)}]; sh["\"] := Module[{}, gp =. ; gi = {1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), 8, \(-9\), \(-10\), \(-11\), 12, \(-13\), 14, 15, 16, 4}; {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)^2 - 4 \((\((\(-d\)\ e + c\ f - b\ g + a\ h - l\ m + k\ n - j\ o + i\ p)\)^2 + \((\(-d\)\ i + c\ j - b\ k + a\ l + h\ m - g\ n + f\ o - e\ p)\)^2 + \((\(-f\)\ i + e\ j - h\ k + g\ l - b\ m + a\ n - d\ o + c\ p)\)^2 + \((\(-g\)\ i + h\ j + e\ k - f\ l - c\ m + d\ n + a\ o - b\ p)\)^2 + \((\(-h\)\ i - g\ j + f\ k + e\ l - d\ m - c\ n + b\ o + a\ p)\)^2)\)}];\)\)\) \!\(sh["\"] := Module[{}, gp = {1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }; gi = {1, 2, 3, 4, 5, 6, 7, 8, \(-9\), \(-10\), \(-11\), \(-12\), \(-13\), \(-14\), \(-15\), \ \(-16\), 1, 1, 1, 1, 1, 1, 1, 1}; {\((a + b + c + d + e + f + g + h)\)\^2 + \((i + j + k + l + m \ + n + o + p)\)\^2, \((a - b - c + d + e - f - g + h)\)\^2 + \((i - j - k + l \ + m - n - o + p)\)\^2, \((a - b + c - d + e - f + g - h)\)\^2 + \((i - j + k \ - l + m - n + o - p)\)\^2, \((a + b - c - d + e + f - g - h)\)\^2 + \((i + j \ - k - l + m + n - o - p)\)\^2, \((a + b + c + d - e - f - g - h)\)\^2 + \((i \ + j + k + l - m - n - o - p)\)\^2, \((a - b - c + d - e + f + g - h)\)\^2 + \ \((i - j - k + l - m + n + o - p)\)\^2, \((a - b + c - d - e + f - g + \ h)\)\^2 + \((i - j + k - l - m + n - o + p)\)\^2, \((a + b - c - d - e - f + \ g + h)\)\^2 + \((i + j - k - l - m - n + o + p)\)\^2}]; toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, o_, p_}, "\"] := \ {\ \ \ \ \ \ \((a + b + c + d + e + f + g + \ h)\)\^2 + \((i + j + k + l + m + n + o + p)\)\^2, ArcTan[a + b + c + d + e + f + g + h, i + j + k + l + m + n + o + p], \[IndentingNewLine]\ \ \ \ \ \ \((a - b - c + d + e - f - g + \ h)\)\^2 + \((i - j - k + l + m - n - o + p)\)\^2, ArcTan[a - b - c + d + e - f - g + h, i - j - k + l + m - n - o + p], \[IndentingNewLine]\ \ \ \ \ \ \((a - b + c - d + e - f + g - \ h)\)\^2 + \((i - j + k - l + m - n + o - p)\)\^2, ArcTan[a - b + c - d + e - f + g - h, i - j + k - l + m - n + o - p], \[IndentingNewLine]\ \ \ \ \ \ \((a + b - c - d + e + f - g - \ h)\)\^2 + \((i + j - k - l + m + n - o - p)\)\^2, ArcTan[a + b - c - d + e + f - g - h, i + j - k - l + m + n - o - p], \[IndentingNewLine]\ \ \ \ \ \ \((a + b + c + d - e - f - g - \ h)\)\^2 + \((i + j + k + l - m - n - o - p)\)\^2, ArcTan[a + b + c + d - e - f - g - h, i + j + k + l - m - n - o - p], \[IndentingNewLine]\ \ \ \ \ \ \((a - b - c + d - e + f + g - \ h)\)\^2 + \((i - j - k + l - m + n + o - p)\)\^2, ArcTan[a - b - c + d - e + f + g - h, i - j - k + l - m + n + o - p], \[IndentingNewLine]\ \ \ \ \ \ \ \((a - b + c - d - e + f - g + \ h)\)\^2 + \((i - j + k - l - m + n - o + p)\)\^2, ArcTan[a - b + c - d - e + f - g + h, i - j + k - l - m + n - o + p], \[IndentingNewLine]\ \ \ \ \ \ \ \((a + b - c - d - e - f + g + \ h)\)\^2 + \((i + j - k - l - m - n + o + p)\)\^2, ArcTan[a + b - c - d - e - f + g + h, i + j - k - l - m - n + o + p]}; toVec[{ii_, j_, kk_, l_, mm_, n_, oo_, p_, qq_, r_, ss_, t_, uu_, v_, ww_, x_}, "\"] := Module[{i = \@ii, k = \@kk, m = \@mm, o = \@oo, q = \@qq, s = \@ss, u = \@uu, w = \@ww}, 2 {i\ Cos[j] + k\ Cos[l] + m\ Cos[n] + o\ Cos[p] + q\ Cos[r] + s\ Cos[t] + u\ Cos[v] + w\ Cos[x], \[IndentingNewLine]i\ Cos[j] - k\ Cos[l] - m\ Cos[n] + o\ Cos[p] + q\ Cos[r] - s\ Cos[t] - u\ Cos[v] + w\ Cos[x], \[IndentingNewLine]i\ Cos[j] - k\ Cos[l] + m\ Cos[n] - o\ Cos[p] + q\ Cos[r] - s\ Cos[t] + u\ Cos[v] - w\ Cos[x], \[IndentingNewLine]i\ Cos[j] + k\ Cos[l] - m\ Cos[n] - o\ Cos[p] + q\ Cos[r] + s\ Cos[t] - u\ Cos[v] - w\ Cos[x], \[IndentingNewLine]i\ Cos[j] + k\ Cos[l] + m\ Cos[n] + o\ Cos[p] - q\ Cos[r] - s\ Cos[t] - u\ Cos[v] - w\ Cos[x], \[IndentingNewLine]i\ Cos[j] - k\ Cos[l] - m\ Cos[n] + o\ Cos[p] - q\ Cos[r] + s\ Cos[t] + u\ Cos[v] - w\ Cos[x], \[IndentingNewLine]i\ Cos[j] - k\ Cos[l] + m\ Cos[n] - o\ Cos[p] - q\ Cos[r] + s\ Cos[t] - u\ Cos[v] + w\ Cos[x], \[IndentingNewLine]i\ Cos[j] + k\ Cos[l] - m\ Cos[n] - o\ Cos[p] - q\ Cos[r] - s\ Cos[t] + u\ Cos[v] + w\ Cos[x], \[IndentingNewLine]i\ Sin[j] + k\ Sin[l] + m\ Sin[n] + o\ Sin[p] + q\ Sin[r] + s\ Sin[t] + u\ Sin[v] + w\ Sin[x], \[IndentingNewLine]i\ Sin[j] - k\ Sin[l] - m\ Sin[n] + o\ Sin[p] + q\ Sin[r] - s\ Sin[t] - u\ Sin[v] + w\ Sin[x], \[IndentingNewLine]i\ Sin[j] - k\ Sin[l] + m\ Sin[n] - o\ Sin[p] + q\ Sin[r] - s\ Sin[t] + u\ Sin[v] - w\ Sin[x], \[IndentingNewLine]i\ Sin[j] + k\ Sin[l] - m\ Sin[n] - o\ Sin[p] + q\ Sin[r] + s\ Sin[t] - u\ Sin[v] - w\ Sin[x], \[IndentingNewLine]i\ Sin[j] + k\ Sin[l] + m\ Sin[n] + o\ Sin[p] - q\ Sin[r] - s\ Sin[t] - u\ Sin[v] - w\ Sin[x], \[IndentingNewLine]i\ Sin[j] - k\ Sin[l] - m\ Sin[n] + o\ Sin[p] - q\ Sin[r] + s\ Sin[t] + u\ Sin[v] - w\ Sin[x], \[IndentingNewLine]i\ Sin[j] - k\ Sin[l] + m\ Sin[n] - o\ Sin[p] - q\ Sin[r] + s\ Sin[t] - u\ Sin[v] + w\ Sin[x], \[IndentingNewLine]i\ Sin[j] + k\ Sin[l] - m\ Sin[n] - o\ Sin[p] - q\ Sin[r] - s\ Sin[t] + u\ Sin[v] + w\ Sin[x]}/16];\) \!\(sh["\"] := Module[{fa = a - b - g + h + m - n - s + t, fb = b - c - h + i + n - o - t + u, fc = c - a - i + g + o - m - u + s, fd = d - e - j + k + p - q - v + w, fe = e - f - k + l + q - r - w + x, ff = f - d - l + j + r - p - x + v, ga = a - b - g + h - m + n + s - t, gb = b - c - h + i - n + o + t - u, gc = c - a - i + g - o + m + u - s, gd = d - e - j + k - p + q + v - w, ge = e - f - k + l - q + r + w - x, gf = f - d - l + j - r + p - v + x}, gp =. ; gi = {1, 3, 2, 10, 12, 11, 7, 9, 8, 4, 6, 5, 13, 15, 14, 22, 24, 23, 19, 21, 20, 16, 18, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}; {a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q + r + s + t + u + v + w + x, \[IndentingNewLine]a + b + c - d - e - f + g + h + i - j - k - l + m + n + o - p - q - r + s + t + u - v - w - x, \[IndentingNewLine]a + b + c + d + e + f + g + h + i + j + k + l - m - n - o - p - q - r - s - t - u - v - w - x, \[IndentingNewLine]a + b + c - d - e - f + g + h + i - j - k - l - m - n - o + p + q + r - s - t - u + v + w + x, \[IndentingNewLine]\((a + b + c - g - h - i - m - n - o + s + t \ + u)\)\^2 + \((d + e + f - j - k - l - p - q - r + v + w + x)\)\^2, \ \[IndentingNewLine]\((a + b + c - g - h - i + m + n + o - s - t - u)\)\^2 + \ \((d + e + f - j - k - l + p + q + r - v - w - x)\)\^2, \ \[IndentingNewLine]\((\ \ \ \ \((a - b + d - e + g - h + j - k + m - n + p - \ q + s - t + v - w)\)\^2 + \((b - c + e - f + h - i + k - l + n - o + q - r + \ t - u + w - x)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l - m + o - p + r - \ s + u - v + x)\)\^2)\)/ 2, \[IndentingNewLine]\((\ \ \ \ \((a - b + d - e + g - h + j - k - \ m + n - p + q - s + t - v + w)\)\^2 + \((b - c + e - f + h - i + k - l - n + \ o - q + r - t + u - w + x)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l + m - \ o + p - r + s - u + v - x)\)\^2)\)/ 2, \[IndentingNewLine]\((\ \ \ \ \((a - b - d + e + g - h - j + k + \ m - n - p + q + s - t - v + w)\)\^2 + \((b - c - e + f + h - i - k + l + n - \ o - q + r + t - u - w + x)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l - m + \ o + p - r - s + u + v - x)\)\^2)\)/ 2, \[IndentingNewLine]\((\ \ \ \ \((a - b - d + e + g - h - j + k - \ m + n + p - q - s + t + v - w)\)\^2 + \((b - c - e + f + h - i - k + l - n + \ o + q - r - t + u + w - x)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l + m - \ o - p + r + s - u - v + x)\)\^2)\)/ 2, \[IndentingNewLine]\((\((fa + fd)\)\^4 + \((fa - fd)\)\^4 + \ \((fb + fe)\)\^4 + \((fb - fe)\)\^4 + \((fc + ff)\)\^4 + \((fc - ff)\)\^4)\)/ 4 - \((fa\^2 + fb\^2 + fc\^2)\) \((fd\^2 + fe\^2 + ff\^2)\), \[IndentingNewLine]\((\((ga + gd)\)\^4 + \((ga - \ gd)\)\^4 + \((gb + ge)\)\^4 + \((gb - ge)\)\^4 + \((gc + gf)\)\^4 + \((gc - \ gf)\)\^4)\)/4 - \((ga\^2 + gb\^2 + gc\^2)\) \((gd\^2 + ge\^2 + gf\^2)\)}]\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 18, 3, 4, 5, 6, 23, 8, 9, 10, 27, 12, 29, 14, 31, 32, 17, 2, 19, 20, 21, 22, 7, 24, 25, 26, 11, 28, 13, 30, 15, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4}; {a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p + q + r + s + t + u + v + w + x - y - z - \[Alpha] - \[Beta] - \[Gamma] - \[Delta] - \ \[CurlyEpsilon] - \[Zeta], a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p + q + r + s + t - u - v - w - x + y + z + \[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p + q + r - s - t + u + v - w - x + y + z - \[Alpha] - \[Beta] + \[Gamma] + \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a + b - c - d - e - f + g + h - i - j + k + l + m + n - o - p + q + r - s - t - u - v + w + x - y - z + \[Alpha] + \[Beta] + \[Gamma] + \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p + q - r + s - t + u - v + w - x + y - z + \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b + c - d - e + f - g + h - i + j - k + l + m - n + o - p + q - r + s - t - u + v - w + x - y + z - \[Alpha] + \[Beta] + \[Gamma] - \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p + q - r - s + t + u - v - w + x - y + z + \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d - e + f + g - h + i - j - k + l - m + n + o - p + q - r - s + t - u + v + w - x + y - z - \[Alpha] + \[Beta] - \[Gamma] + \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d - e + f + g - h - i + j + k - l + m - n - o + p + q - r - s + t - u + v + w - x - y + z + \[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p + q - r - s + t + u - v - w + x + y - z - \[Alpha] + \[Beta] + \[Gamma] - \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p + q - r + s - t - u + v - w + x + y - z + \[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p + q - r + s - t + u - v + w - x - y + z - \[Alpha] + \[Beta] - \[Gamma] + \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a + b - c - d - e - f + g + h + i + j - k - l - m - n + o + p + q + r - s - t - u - v + w + x + y + z - \[Alpha] - \[Beta] - \[Gamma] - \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b - c - d + e + f - g - h - i - j + k + l - m - n + o + p + q + r - s - t + u + v - w - x - y - z + \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b + c + d - e - f - g - h - i - j - k - l + m + n + o + p + q + r + s + t - u - v - w - x - y - z - \[Alpha] - \[Beta] + \[Gamma] + \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q + r + s + t + u + v + w + x + y + z + \[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[CurlyEpsilon] + \ \[Zeta], \(-4\)\ \((\(-\((\(-\((h - x)\)\)\ \((i - y)\) - \((g - w)\)\ \((j - z)\) + \((f - v)\)\ \((k - \[Alpha])\) + \((e - u)\)\ \((l - \[Beta])\) - \((d - t)\)\ \((m - \[Gamma])\) - \((c - s)\)\ \((n - \[Delta])\) + \((b - r)\)\ \((o - \[CurlyEpsilon])\) + \((a - q)\)\ \((p - \[Zeta])\))\)^2\) - \((\(-\((g - w)\)\)\ \((i - y)\) + \((h - x)\)\ \((j - z)\) + \((e - u)\)\ \((k - \[Alpha])\) - \((f - v)\)\ \((l - \[Beta])\) - \((c - s)\)\ \((m - \[Gamma])\) + \((d - t)\)\ \((n - \[Delta])\) + \((a - q)\)\ \((o - \[CurlyEpsilon])\) - \((b - r)\)\ \((p - \[Zeta])\))\)^2 + \((\(-\((f - v)\)\)\ \((i - y)\) + \((e - u)\)\ \((j - z)\) + \((h - x)\)\ \((k - \[Alpha])\) - \((g - w)\)\ \((l - \[Beta])\) - \((b - r)\)\ \((m - \[Gamma])\) + \((a - q)\)\ \((n - \[Delta])\) + \((d - t)\)\ \((o - \[CurlyEpsilon])\) - \((c - s)\)\ \((p - \[Zeta])\))\)^2 + \((\(-\((d - t)\)\)\ \((i - y)\) + \((c - s)\)\ \((j - z)\) - \((b - r)\)\ \((k - \[Alpha])\) + \((a - q)\)\ \((l - \[Beta])\) - \((h - x)\)\ \((m - \[Gamma])\) + \((g - w)\)\ \((n - \[Delta])\) - \((f - v)\)\ \((o - \[CurlyEpsilon])\) + \((e - u)\)\ \((p - \[Zeta])\))\)^2 + \((\(-\((d - t)\)\)\ \((e - u)\) + \((c - s)\)\ \((f - v)\) - \((b - r)\)\ \((g - w)\) + \((a - q)\)\ \((h - x)\) + \((l - \[Beta])\)\ \((m - \[Gamma])\) - \ \((k - \[Alpha])\)\ \((n - \[Delta])\) + \((j - z)\)\ \((o - \[CurlyEpsilon])\) - \((i - y)\)\ \((p - \[Zeta])\))\)^2)\) + \((\((a - \ q)\)\^2 + \((b - r)\)\^2 - \((c - s)\)\^2 - \((d - t)\)\^2 - \((e - u)\)\^2 - \ \((f - v)\)\^2 + \((g - w)\)\^2 + \((h - x)\)\^2 - \((i - y)\)\^2 - \((j - z)\ \)\^2 + \((k - \[Alpha])\)\^2 + \((l - \[Beta])\)\^2 + \((m - \[Gamma])\)\^2 \ + \((n - \[Delta])\)\^2 - \((o - \[CurlyEpsilon])\)\^2 - \((p - \ \[Zeta])\)\^2)\)^2}]\) \!\(sh["\"] := Module[{}, gp =. ; gi = {1, 2, 3, 20, 5, 22, 23, 24, 9, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 4, 21, 6, 7, 8, 25, 10, 11, 12, 13, 14, 15, 32, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4}; {a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p + q + r + s + t + u + v + w + x - y - z - \[Alpha] - \[Beta] - \[Gamma] - \[Delta] - \ \[CurlyEpsilon] - \[Zeta], a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p + q + r + s + t - u - v - w - x + y + z + \[Alpha] + \[Beta] - \[Gamma] - \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a + b - c - d + e + f - g - h + i + j - k - l + m + n - o - p + q + r - s - t + u + v - w - x + y + z - \[Alpha] - \[Beta] + \[Gamma] + \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a + b - c - d - e - f + g + h - i - j + k + l + m + n - o - p + q + r - s - t - u - v + w + x - y - z + \[Alpha] + \[Beta] + \[Gamma] + \[Delta] - \[CurlyEpsilon] - \ \[Zeta], a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p + q - r + s - t + u - v + w - x + y - z + \[Alpha] - \[Beta] + \[Gamma] - \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b + c - d - e + f - g + h - i + j - k + l + m - n + o - p + q - r + s - t - u + v - w + x - y + z - \[Alpha] + \[Beta] + \[Gamma] - \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p + q - r - s + t + u - v - w + x - y + z + \[Alpha] - \[Beta] - \[Gamma] + \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d - e + f + g - h + i - j - k + l - m + n + o - p + q - r - s + t - u + v + w - x + y - z - \[Alpha] + \[Beta] - \[Gamma] + \[Delta] + \[CurlyEpsilon] - \ \[Zeta], a - b - c + d - e + f + g - h - i + j + k - l + m - n - o + p + q - r - s + t - u + v + w - x - y + z + \[Alpha] - \[Beta] + \[Gamma] - \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p + q - r - s + t + u - v - w + x + y - z - \[Alpha] + \[Beta] + \[Gamma] - \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p + q - r + s - t - u + v - w + x + y - z + \[Alpha] - \[Beta] - \[Gamma] + \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p + q - r + s - t + u - v + w - x - y + z - \[Alpha] + \[Beta] - \[Gamma] + \[Delta] - \[CurlyEpsilon] + \ \[Zeta], a + b - c - d - e - f + g + h + i + j - k - l - m - n + o + p + q + r - s - t - u - v + w + x + y + z - \[Alpha] - \[Beta] - \[Gamma] - \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b - c - d + e + f - g - h - i - j + k + l - m - n + o + p + q + r - s - t + u + v - w - x - y - z + \[Alpha] + \[Beta] - \[Gamma] - \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b + c + d - e - f - g - h - i - j - k - l + m + n + o + p + q + r + s + t - u - v - w - x - y - z - \[Alpha] - \[Beta] + \[Gamma] + \[Delta] + \[CurlyEpsilon] + \ \[Zeta], a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q + r + s + t + u + v + w + x + y + z + \[Alpha] + \[Beta] + \[Gamma] + \[Delta] + \[CurlyEpsilon] + \ \[Zeta], \(-\((2 \((h - x)\) \((i - y)\) - 2 \((g - w)\) \((j - z)\) + 2 \((f - v)\) \((k - \[Alpha])\) - 2 \((e - u)\) \((l - \[Beta])\) - 2 \((d - t)\) \((m - \[Gamma])\) + 2 \((c - s)\) \((n - \[Delta])\) - 2 \((b - r)\) \((o - \[CurlyEpsilon])\) + 2 \((a - q)\) \((p - \[Zeta])\))\)^2\) - \((2 \((a - q)\) \((i - y)\) - 2 \((b - r)\) \((j - z)\) - 2 \((c - s)\) \((k - \[Alpha])\) + 2 \((d - t)\) \((l - \[Beta])\) - 2 \((e - u)\) \((m - \[Gamma])\) + 2 \((f - v)\) \((n - \[Delta])\) + 2 \((g - w)\) \((o - \[CurlyEpsilon])\) - 2 \((h - x)\) \((p - \[Zeta])\))\)^2 - \((2 \((a - q)\) \((e - u)\) - 2 \((b - r)\) \((f - v)\) - 2 \((c - s)\) \((g - w)\) + 2 \((d - t)\) \((h - x)\) + 2 \((i - y)\) \((m - \[Gamma])\) - 2 \((j - z)\) \((n - \[Delta])\) - 2 \((k - \[Alpha])\) \((o - \[CurlyEpsilon])\) + 2 \((l - \[Beta])\) \((p - \[Zeta])\))\)^2 - \((2 \((a - q)\) \((c - s)\) - 2 \((b - r)\) \((d - t)\) + 2 \((e - u)\) \((g - w)\) - 2 \((f - v)\) \((h - x)\) + 2 \((i - y)\) \((k - \[Alpha])\) - 2 \((j - z)\) \((l - \[Beta])\) + 2 \((m - \[Gamma])\) \((o - \[CurlyEpsilon])\) - 2 \((n - \[Delta])\) \((p - \[Zeta])\))\)^2 - \((2 \((a - q)\) \((b - r)\) + 2 \((c - s)\) \((d - t)\) + 2 \((e - u)\) \((f - v)\) + 2 \((g - w)\) \((h - x)\) + 2 \((i - y)\) \((j - z)\) + 2 \((k - \[Alpha])\) \((l - \[Beta])\) + 2 \((m - \[Gamma])\) \((n - \[Delta])\) + 2 \((o - \[CurlyEpsilon])\) \((p - \[Zeta])\))\)^2 + \((\((a \ - q)\)\^2 + \((b - r)\)\^2 + \((c - s)\)\^2 + \((d - t)\)\^2 + \((e - u)\)\^2 \ + \((f - v)\)\^2 + \((g - w)\)\^2 + \((h - x)\)\^2 + \((i - y)\)\^2 + \((j - \ z)\)\^2 + \((k - \[Alpha])\)\^2 + \((l - \[Beta])\)\^2 + \((m - \ \[Gamma])\)\^2 + \((n - \[Delta])\)\^2 + \((o - \[CurlyEpsilon])\)\^2 + \((p \ - \[Zeta])\)\^2)\)^2}]\) \!\(\(abbr[t_, o_: 1, n_: {}] := Module[{a = {}, l = Length[t] - 1, f = First[t], r = Drop[t, 1], s, u = Union[Drop[t, 1]], ui}, Do[ui = u[\([i]\)]; s = Length[Position[r, ui]]\ ; If[s > 1, u[\([i]\)] = ui\_s], {i, Length[u]}]; Flatten[Join[{o\ f}, u, n]]];\)\) gpd[gg_]:=Module[{g= If[Length[gg]\[Equal]1&&Head[gg]\[Equal]TraditionalForm,gg[[1]],gg]}, If[Length[g]\[Equal] Length[g[[1]]]&&(MatrixQ[g[[1,1]]]|| Union[Flatten[Abs[g],1]]\[Equal]Sort[Abs[g[[1]]]]),g, Print[Short[g]," is not a groupoid"];Abort[]]]; shape[a_?VectorQ,g_:gmn]:=Chop[sh[g]/.fromAlph[a],hmin]; hoopFactor[f_Symbol:fp]:= Module[{i,gf,gl=Length[glo],n,res={}},gi=Position[Abs[glo],1]; gi=Drop[Transpose[gi],1][[1]]* Table[i=ii;Sign[glo[[gi[[i,1]],gi[[i,2]]]]],{ii,gl}];gf=f[glo]; gf=If[Head[gf]===Times,List@@gf,{gf}];(* that deals with single and repeated single factors. Remove any -1 factor *) If[gf[[1]]===-1,gf=Drop[gf,1]]; Do[n=gf[[i]]; If[Head[n]===Power,AppendTo[res,n[[1]]];AppendTo[gi,n[[-1]]], AppendTo[res,n];AppendTo[gi,1]],{i,Length[gf]}];res]; hoopShape[]:=Which[Head[glo]===List,hoopShape[glo,-1], Head[gmn]===Symbol,Print["Neither glo nor gmn set"], Head[sh[gmn]]===sh,frag[hoopFactor[fa]], True,sh[gmn]]; hoopShape[m_?IntegerQ,n_]:=hoopShape[mp[m,n]]; hoopShape[X_,o_:0]/;(Head[X]===String)||Head[X]===Symbol:= hoopShape[makeProtoloop[X],o]; hoopShape[X_,o_:0]/;Head[X]\[Equal]TraditionalForm:=hoopShape[X[[1]],o]; hoopShape[X_,o_:0]/;MatrixQ[X]:= Module[{g=X,gf,gm,res={},r={}}, If[X===glo,Null,g=If[o\[Equal]-1,X,caylindex[gpd[X]]]]; If[o>0&&!(conservativeQ[]),Print["Table not conservative"];Abort[]]; id[g];hoopFactor[]; Do[If[Head[X\[LeftDoubleBracket]in,1\[RightDoubleBracket]]===Symbol, AppendTo[r, alph\[LeftDoubleBracket]in\[RightDoubleBracket]\[Rule] X\[LeftDoubleBracket]in,1\[RightDoubleBracket]]],{in,mm}]; gm=fp[g]/.r;res=sh[gmn]; Which[ gmn\[Equal]"Unrecognised"||Head[res]\[Equal]sh,res=frag[fa[g]]; sh[gmn]=frag[res,o], (gmn\[NotEqual]Unrecognized)&&fp[makeProtoloop[gmn]]===gm, If[o>0,Print[gmn<>Protoloop]]; If[Head[res]\[Equal]sh,res=hoopFactor[fa];sh[gmn]=res], gmn\[NotEqual]Unrecognized&&Length[res]===Length[gm], If[o>0,Print[gmn<>Isomorph]], True,If[o\[NotEqual]0,Print[gmn<>"Shape found"]]]; If[Length[r]\[Equal]mm,res/.r,res]]; hoopShape[Unrecognized,o_:0]:=sh[Unrecognized]; hoopShape[X_,o_:0]/;Head[X]\[Equal]List:= hoopShape[makeProtoloop[{X[[1]],X[[2]]}],0]; hoopShape[gg_,o_:0]:= Module[{g=makeProtoloop[gg],res}, res=If[g==="No Protoloop",hoopFactor[fa];sh[Unrecognized], sh[gd[]\[LeftDoubleBracket]3,1\[RightDoubleBracket]]]; frag[If[Head[res]\[Equal]sh,hoopFactor[fa],res,res],o]]; polsort[P_]:= Module[{i,j,l,p=Expand[P],t,v},v=Variables[p]; l=Length[v]; t=Table[Position[p,v[[i]]],{i,l}]; Table[Table[p[[t[[v,j,1]]]],{j,Length[t[[v]]]}],{v,l}]]; frag[p_,o_:0]:= Module[{P,res},P=If[MatrixQ[p],fa[p],p];If[Head[P]===Plus,P={P}]; If[Head[res=sh[P]]===List,res,res={}; Do[AppendTo[res, If[Head[P[[ii]]]===Power,frag2[P[[ii,1]],o], frag2[P[[ii]],o]]],{ii,Length[P]}];res]]; \!\(\* RowBox[{ RowBox[{ RowBox[{\(frag2[\((P_)\)?PolynomialQ, o_: 0]\), ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{\({ai, aj, d = 1, err, f, fn = 0, f2 = {{}, {}, {}, {}}, i, kf = {}, l, np = 1, nw, p = polsort[Expand[P]], pi, pip, p1, p2, res = {}, sgn, t, v, vf = {}, addnew}\), ",", RowBox[{ RowBox[{"If", "[", RowBox[{\(P === \(-a\^2\) - b\^2 || \((p = polsort[Expand[P]]; MatrixQ[p] === False)\) || Max[Exponent[Plus @@ p, a]] \[NotEqual] 2\), ",", \(Return[P]\), ",", "\[IndentingNewLine]", RowBox[{\(addnew[t_] := If[MemberQ[res, t] === False, AppendTo[res, t]]\), ";", " ", \(l = Length[p]\), ";", " ", \(f = FactorInteger[l]\), ";", " ", \(v = Variables[p]\), ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{\(pi = p[\([i]\)]\), ";", " ", \(ai = v[\([i]\)]\), ";", " ", \(sgn = If[Position[pi, \(-ai^2\)] \[Equal] {}, 1, \(-1\)]\), ";", "\[IndentingNewLine]", RowBox[{"Which", "[", RowBox[{\(Length[pi] \[Equal] 1\), ",", \(addnew[pi]\), ",", "\[IndentingNewLine]", \(Length[pi] \[Equal] 2\), ",", \(addnew[ sgn*pos[ ai + \((sgn*\((Plus @@ pi - sgn*ai^2)\))\)/\((2*ai)\)]^2]\), ",", \(Length[ Union[Position[pi, sgn*2], Position[ pi, \(-sgn\)*2]]] \[GreaterEqual] Length[pi] - 1\), ",", "\[IndentingNewLine]", " ", \(addnew[ sgn*pos[ Simplify[\((ai/ 2 + \((sgn* Plus @@ pi)\)/\((ai*2)\))\)^2]]]\), ",", \(Mod[Length[pi], 3] \[Equal] 0\), ",", RowBox[{\(d = 2\), ";", " ", \(If[\((p2 = Position[vf, ai])\) === {}, \(fn++\); \ np = fn; \ AppendTo[vf, Variables[pi]]; \ AppendTo[kf, sgn], aj = ai; \ If[True, aj = \(-ai\)]; \ np = First[Flatten[p2]]]\), ";", "\[IndentingNewLine]", \(pip = Expand[Plus @@ pi/ai - sgn*ai]\), ";", \(AppendTo[f2[\([np]\)], pos[Plus @@ Flatten[{ai, 1/2*Union[ sgn*Extract[pip, Position[pip, 2*_]], sgn*Extract[pip, Position[pip, \(-2\)*_]]]}]]]\), ";", RowBox[{"If", "[", RowBox[{\(o > 2\), ",", StyleBox[\(Print[{"\", pi, pip, f2}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}]}], ",", "\[IndentingNewLine]", StyleBox["_", FontColor->RGBColor[1, 0, 1]], StyleBox[",", FontColor->RGBColor[1, 0, 1]], StyleBox[\(Abort[]\), FontColor->RGBColor[1, 0, 1]]}], "]"}]}], ",", \({i, l}\)}], "]"}], ";", " ", RowBox[{"If", "[", RowBox[{\(d \[Equal] 2\), ",", RowBox[{"Do", "[", RowBox[{ RowBox[{\(p1 = Union[f2[\([i]\)]]\), ";", " ", \(If[p1 \[NotEqual] {}, sgn = kf[\([i]\)]; \ AppendTo[res, sgn*\((p1[\([1]\)] - p1[\([2]\)])\)^2]; \ AppendTo[res, sgn*\((p1[\([2]\)] - p1[\([3]\)])\)^2]; \ AppendTo[res, sgn*\((p1[\([1]\)] - p1[\([3]\)])\)^2]]\), ";", " ", RowBox[{"If", "[", RowBox[{\(o > 0\), ",", StyleBox[\(Print[{"\", i, p1, res}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}]}], ",", \({i, Length[f2]}\)}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", \(res = Plus @@ res/d\), ";", " ", \(If[Head[res] \[Equal] List, res = res[\([1]\)]]\), ";", "\[IndentingNewLine]", \(err = Flatten[{Expand[ P - res]} /. {Plus[a_, Times[c_, b_]] \[Rule] {a, b\ c}, Plus[a_, b_] \[Rule] {a, b}}]\), ";", "\[IndentingNewLine]", \(If[ err \[NotEqual] {0}, \[IndentingNewLine]Do[ v = Variables[err[\([i]\)]]; \ pip = Position[res, v[\([1]\)] + v[\([2]\)] \((sgn = If[err[\([1]\)] < 0, 1, \(-1\), \(-1\)])\)]; \ \[IndentingNewLine]res = ReplacePart[res, v[\([1]\)] - sgn*v[\([2]\)], pip], \[IndentingNewLine]{i, Length[ err]}] (*End\ Do*) \[IndentingNewLine]]\), \( \ (*End\ \(\(If\)\([\)\(err \[NotEqual] {0}\)\)*) \), ";"}]}], "\[IndentingNewLine]", "]"}], \( (*End\ \(\(If\)\([\)\(P\)\(\ \)\)*) \), ";", \(err = Flatten[{Expand[ P - res]} /. {Plus[a_, Times[c_, b_]] \[Rule] {a, b\ c}, Plus[a_, b_] \[Rule] {a, b}}]\), ";", "\[IndentingNewLine]", \(If[err \[NotEqual] {0}, Print["\"]; Abort[]]\), ";", "res"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", \(SetAttributes[frag2, Listable]\)}]\) tidy[gg_?indexTableQ,n_:-1,o_:1,p_:{}]:= Module[{a,b,c,d,g=If[Length[gg]\[Equal]1,gg[[1]],gg],gl,m,nb,r,q}, gl=Length[g]; If[Length[n]==0,b=subgroups[g][[-n]],b=n,b=n]; (* Find subgroup if list not supplied *) nb=Length[b];m=gl/nb; If[!IntegerQ[m],Abort[]]; (* nb must divide gl *) r=Complement[Range[gl],b];If[MemberQ[r,o],b=Join[b,{o}]]; r=Complement[r,b];c=Join[b,r];b=c; d=Transpose[reorder[g,c]][[nb+1]];(* Run to find column for sorting *) Do[b[[j+nb]]=c[[d[[j]]]],{j,2,nb}]; (* Rearrange col nb+1 *) If[m>2,b=Take[b,2nb];r=Complement[Range[gl],b]; If[Length[p]>1,b=Join[b,p];b=Join[b,Complement[Range[gl],b]], If[MemberQ[r,p],b=Join[b,{p}]]; b=Join[b,Complement[r,b]]]];(* Add other elements *) Print[b];reorder[g,b]] \!\(\* RowBox[{ RowBox[{\(ts[a___] := toSignedTable[a]\), ";", RowBox[{\(toSignedTable[g_?indexTableQ, r_Integer : 2, reo_ : {}, o_ : 0]\), ":=", RowBox[{"Module", "[", RowBox[{\({diags, diagelements, dis, gg = g, i, j, l = Length[g], nex = 1, n = Abs[r], ps, rr = {}, res, s}\), ",", RowBox[{ RowBox[{"If", "[", RowBox[{\(reo \[NotEqual] {}\), ",", RowBox[{\(gg = reorder[gg, reo]\), ";", RowBox[{"If", "[", RowBox[{\(o > 1\), ",", StyleBox[\(Print[{"\", tf[gg]}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}]}]}], "]"}], ";", RowBox[{"If", "[", RowBox[{\(Length[gg] === Length[g]\), ",", RowBox[{"If", "[", RowBox[{\(negatableQ[gg, n] \[Equal] True\), ",", RowBox[{ RowBox[{"If", "[", RowBox[{\(o > 2\), ",", StyleBox[\(Print[{"\", n, "\"}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}], ";", \(gg = Take[Transpose[Take[gg, l\/n]], l\/n]\)}], ",", RowBox[{\(diags = Table[gg\[LeftDoubleBracket]ii, ii\[RightDoubleBracket], {ii, l}]\), ";", \(diagelements = Union[diags]\), ";", RowBox[{"If", "[", RowBox[{\(o \[NotEqual] 0\), ",", StyleBox[\(Print[{diags, "\", diagelements}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}], ";", \(res = Table[0, {l\/n}]\), ";", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"If", "[", RowBox[{\(Mod[ Length[ diagelements\[LeftDoubleBracket] jj\[RightDoubleBracket]], n] \[NotEqual] 0\), ",", RowBox[{ StyleBox[\(Print[{"\", n, "\"}]\), FontColor->RGBColor[1, 0, 1]], ";", \(Abort[]\)}], ",", RowBox[{\(ps = Position[ diags, \(Union[ diagelements]\)\[LeftDoubleBracket]jj\ \[RightDoubleBracket]]\), ";", RowBox[{"If", "[", RowBox[{\(o > 1\), ",", StyleBox[\(Print[{"\", ps, jj, nex}]\), FontColor->RGBColor[0, 1, 0]]}], "]"}], ";", RowBox[{"If", "[", RowBox[{\(Mod[Length[ps], n] \[NotEqual] 0\), ",", RowBox[{ StyleBox[\(Print[{"\", n, "\"}]\), FontColor->RGBColor[1, 0, 1]], ";", \(Abort[]\)}]}], "]"}], ";", \(Do[ res\[LeftDoubleBracket]nex\ \[RightDoubleBracket] = ps\[LeftDoubleBracket]nx\ \[RightDoubleBracket]; \(nex++\), {nx, Length[ps]\/n}]\)}]}], "]"}], ";", RowBox[{"If", "[", RowBox[{\(o \[NotEqual] 0\), ",", RowBox[{ StyleBox["Print", FontColor->RGBColor[0, 1, 0]], StyleBox["[", FontColor->RGBColor[0, 1, 0]], StyleBox[\(Flatten[{"\", res}]\), FontColor->RGBColor[0, 1, 0]], "]"}]}], "]"}]}], ",", \({jj, Length[diagelements\/n]}\)}], "]"}], ";", \(gg = reorder[g, Flatten[res]]\)}]}], "]"}]}], "]"}], ";", \(s = Length[gg]\), ";", \(i =. \), ";", \(rr = {}\), ";", \(Do[ Which[n \[Equal] 2, AppendTo[rr, i + s \[Rule] \(-i\)], n \[Equal] 3, AppendTo[rr, i + s \[Rule] \[DoubleStruckCapitalJ]\ i]; AppendTo[rr, i + 2\ s \[Rule] \[DoubleStruckCapitalJ]\ \ \[DoubleStruckCapitalJ]\ i], n \[Equal] 4, AppendTo[rr, i + s \[Rule] \[ImaginaryI]\ i]; AppendTo[rr, i + 2\ s \[Rule] \(-i\)]; AppendTo[rr, i + 3\ s \[Rule] \(-\[ImaginaryI]\)\ i]], {i, s}]\), ";", \(If[o > 1, Print[{rr, s}]]\), ";", \(glo = Table[gg\[LeftDoubleBracket]i, j\[RightDoubleBracket] /. \[InvisibleSpace]rr, {i, s}, {j, s}]\), ";", \(mm = Length[glo]\), ";", RowBox[{"If", "[", RowBox[{\(indexTableQ[glo]\), ",", RowBox[{"If", "[", RowBox[{\(r > 0 && conservativeQ[glo] \[NotEqual] True\), ",", RowBox[{ StyleBox[\(Print["\"]\), FontColor->RGBColor[1, 0, 1]], ";", \("\"[glo]\)}], ",", "glo"}], "]"}], ",", StyleBox[\(Print["\"]\), FontColor->RGBColor[1, 0, 1]]}], "]"}], ";", \(id[glo]\), ";", "glo"}]}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox[\(collapse[g_, r_]\), FormatType->StandardForm, FontFamily->"Courier New"], StyleBox[":=", FormatType->StandardForm, FontFamily->"Courier New"], StyleBox[\( (*Skeleton\ toSignedTable*) \), FormatType->StandardForm, FontFamily->"Courier New", FontColor->RGBColor[1, 0, 0]], StyleBox["\[IndentingNewLine]", FormatType->StandardForm, FontFamily->"Courier New"], StyleBox[\(Module[{l = Length[g]/r, sr = {{}, {1, \(-1\)}, {1, J, JJ}, \ {1, \(-1\), i, \(-1\), \(-i\)}}}, \ Table[\((Mod[g[\([k, j]\)] - 1, l] + 1\ )\)\ sr[\([r, Floor[\((g[\([k, j]\)] - 1)\)/l] + 1]\)], {k, l}, {j, l}]]\), FormatType->StandardForm, FontFamily->"Courier New"]}], StyleBox[";", FormatType->StandardForm, FontFamily->"Courier New"]}]}]\) \!\(\* RowBox[{\(abbrFactors[g_, p_: power2]\), ":=", StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], \(Module[{ap, f, ff = fp[g, p], fi, fj, i = 1, l, res = {}, temp}, \[IndentingNewLine]f = If[NumberQ[First[ff]], Drop[ff, 1], ff]; f = If[NumberQ[First[f]], {Drop[f, 1]}, f]; If[First[f] < 0, f = \(-f\)]; \[IndentingNewLine]While[ i \[LessEqual] Length[f], fi = f[\([i]\)]; If[Length[f] === 1, \(i++\); fi = \(Exponent[f, "\"]\)[\([1]\)]; fj = 1; Goto[done]]; \[IndentingNewLine]If[ Head[f] === Power, \(i++\); fj = Last[f]; fi = If[Head[f[\([\(-2\), \(-1\)]\)]] === Power, f[\([\(-2\), \(-1\), \(-1\)]\)], 1]; Goto[done]]; \[IndentingNewLine]If[ Head[f] === Plus && \((Head[f[\([\(-1\)]\)]] === Power || Head[f[\([\(-1\)]\)]] === Times)\), i = Length[f]; fi = Exponent[f, "\"]; fj = 1; Goto[done]]; \[IndentingNewLine]If[NumberQ[\((fj = Last[fi])\)], fi = Last[Drop[fi, \(-1\)]], fj = 1]; \[IndentingNewLine]fi = Switch[Length[fi], 0, 1, 1, 1, 2, Last[fi], _, Last[Last[fi]]]; \[IndentingNewLine]Label[done]; fi = fi /. {"\" \[Rule] 1, \(-"\"\) \[Rule] 1, a -> 1}; AppendTo[res, If[fj \[Equal] 1, ToString[ fi], \((ToString[fi])\)\^ToString[fj]]]; \[IndentingNewLine]\(i\ ++\)]; \[IndentingNewLine]temp = Union[res = Sort[Flatten[res]]]; \[IndentingNewLine]Table[ ap = temp[\([i]\)]; l = Length[Position[res, ap, 1]]; Which[Length[ap] > 1 && l \[Equal] 1, ap[\([1]\)]\^ap[\([2]\)], Length[ap] > 1, \(ap[\([1]\)]\)\_l\%\(ap[\([2]\)]\), l \[Equal] 1, ap, True, ap\_l], {i, Length[temp]}]]\)}]\) cosorbit[\[Theta]_,phaselist_:{0,1,2,3,4,5},scale_:1,meanlist_:Table[0,{24}]]:= Module[{m=Length[phaselist],i}, Table[i=phaselist[[j]]; meanlist[[j]]+ If[NumberQ[i],scale*Cos[\[Theta]+2*Mod[i,m] \[Pi]/m],0],{j,m}]] \[Omega]=2\[Pi]/3;(* Used widely in orbits. Placed here to avoid evaluation in their definitions.*) ca[{m_,a_:1}]:=ca[m,a]; ca[m_Integer?Positive]:=Table[Mod[i+j-2,m]+1,{i,m},{j,m}]; ca[m_Integer?Positive,a_Integer]:= Module[{k=Abs[a],test}, test=If[GCD[m,k]\[Equal]1, Table[Mod[i+If[EvenQ[i],j k-k+1,j]-2,m]+1,{i,m},{j,m}], Print[m," ",k," ","GCD not 1"];{},{}]; If[a<0&&associativeQ[test,1]\[Equal]False,Print["Not a group"];test,test, test]];ca[m_,k_:0,c_]:=(*skeleton*) Table[Mod[i+If[EvenQ[i],j k-k+1,j]-2,m]+1, {i,m},{j,m}]; co[g_?indexTableQ,gg_?indexTableQ,hh_?indexTableQ,a_Integer,o_:0]:= co[g,co[gg,hh,a],o] co[g_?indexTableQ,h_?indexTableQ,gg_?indexTableQ,hh_?indexTableQ,a_Integer, o_Integer:0,p_:0]:=co[g,co[h,co[gg,hh,a],o],p] co[gg_?indexTableQ,hh_?indexTableQ,a_:0]:= Module[{any,as=Abs[a], g1=Dimensions[gg]\[LeftDoubleBracket]1\[RightDoubleBracket], h1=Dimensions[hh]\[LeftDoubleBracket]1\[RightDoubleBracket], g=unsign[gg],h=unsign[hh],im,lm,test={},t,m},m=g1 h1; If[Position[gg,\[ImaginaryI]]\[NotEqual]{}, g=gg/.{any_/;Re[any]<0->-any+2g1, Complex[r_,i_]\[Rule]If[i<0,-i+3g1,i+g1]}]; If[Position[gg,\[DoubleStruckCapitalJ]]\[NotEqual]{}, g=Mod[(gg/.{Times[\[DoubleStruckCapitalJ] \[DoubleStruckCapitalJ], any_]\[Rule]any+2g1, Times[\[DoubleStruckCapitalJ],any_]\[Rule] any+g1,\[DoubleStruckCapitalJ] \[DoubleStruckCapitalJ]\ \[Rule]2g1+1,\[DoubleStruckCapitalJ]\[Rule]g1+1})-1,3g1]+1]; If[h\[NotEqual]hh,Print["Second matrix must not be signed"], If[as\[Equal]7&&h\[NotEqual]C3,Print["Second matrix must be C3"], Do[lm=l-1;Do[t={};Do[ Do[Which[ as\[Equal]0||as\[Equal]1, AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]], as<3, If[EvenQ[l]&&i>1, AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]g1+2-i,j\[RightDoubleBracket]], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], as\[Equal]3, If[EvenQ[j]&&k>1, AppendTo[t, Mod[(k+h\[LeftDoubleBracket]k, l\[RightDoubleBracket]-2) g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]-1, m]+1],AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], as\[Equal]4,im=4 lm (i-1); If[EvenQ[l]&&EvenQ[i], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]Mod[i+im-1,g1]+1, j\[RightDoubleBracket]], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], as\[Equal]7,If[lm\[Equal]2,lm=3];im=lm (i-1); AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]Mod[i+im-1,g1]+1, j\[RightDoubleBracket]], True,Print[Unknown a];Abort[]],{i,g1}],{k,h1}]; AppendTo[test,Mod[t-1,m]+1 ],{j,g1}],{l,h1}]]]; If[a<0||as\[Equal]1, If[loopQ[test], If[associativeQ[test],test,Print["Not associative"];test], Print["Not a quasigroup"]],test,test]]; co[g_,h_,a_:0,c_]:=(*skeleton*) Module[{g1=Dimensions[g]\[LeftDoubleBracket]1\[RightDoubleBracket], h1=Dimensions[h]\[LeftDoubleBracket]1\[RightDoubleBracket],im,lm, test={},t,m},m=g1 h1; Do[lm=l-1;Do[t={};Do[Do[ Which[a\[Equal]0||a\[Equal]1, AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1)g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]], a\[Equal]2, If[EvenQ[l]&&i>1, AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1)g1+ g\[LeftDoubleBracket]g1+2-i,j\[RightDoubleBracket]], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1)g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], a\[Equal]3, If[EvenQ[j]&&k>1, AppendTo[t, Mod[(k+h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-2)g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]-1,m]+1], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1)g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], a\[Equal]4,im=4lm (i-1); If[EvenQ[l]&&EvenQ[i], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1) g1+ g\[LeftDoubleBracket]Mod[i+im-1,g1]+1, j\[RightDoubleBracket]], AppendTo[ t,(h\[LeftDoubleBracket]k,l\[RightDoubleBracket]-1)g1+ g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]], True,Print["Unknown a ",a];Abort[]],{i,g1}],{k,h1}]; AppendTo[test,Mod[t-1,m]+1],{j,g1}],{l,h1}];test]; unsign[g_?MatrixQ]:= Module[{l=Length[g],r,e,i,ar,ai,m,p}, m=Switch[Length[Union[Flatten[Sign[g]]]],1,p=0;l,2,p=l;2l,_,p=2l;4l]; Table[e=g[[j,k]];r=Re[e];ar=Abs[r];i=Im[e];ai=Abs[i];r=ar+If[r<0,p,0]; i=ai+Which[i\[Equal]0,0,i<0,3l,i>0,l];Mod[r+i-1,m]+1,{j,l},{k,l}]]; sgroup[m_?IntegerQ]:=Module[{e=Permutations[Range[m]],n,p},n=Length[e]; Table[p=Position[e,Apply[permute,{e[[i]],e[[j]]}]]; If[p==={},0,p[[1,1]]],{i,n},{j,n}]] permfromQ[p_List,n_Integer]:= Length[Complement[p,Range[n]]]\[Equal]0&&Sort[p]\[Equal]Union[p] permute[P_List,p_List]:=If[permfromQ[p,Length[P]],P[[p]],Print[{P,p," error"}]]; centralizer[a_?IntegerQ,g_:glo]:= Module[{gg=If[Length[g]\[Equal]1,g[[1]],g],b,c={}}, Do[b=gg[[i,1]];If[gg[[a,i]]==gg[[i,a]],AppendTo[c,b]],{i,Length[gg]}];c] period[g_:glo]:= Module[{gg=If[Length[g]\[Equal]1,g[[1]],g],lg,a,aa,b,j,P={1}},lg=Length[gg]; Do[a=gg[[i,1]];aa=a;j=2;While[(b=gg[[a,aa]])!=1&&jl] jordanQ[gg_:glo,o_:0]:= Catch[Module[{a,i=1,j,k,g=gpd[gg],m,xx,axx,xy,axy},m=Length[g]; a=abelianQ[g]; While[i\[LessEqual]m,xx=g[[i,i]];axx=Abs[xx]; While[j\[LessEqual]m,xy=g\[LeftDoubleBracket]i,j\[RightDoubleBracket]; axy=Abs[xy]; If[Abs[g\[LeftDoubleBracket]axy,axx\[RightDoubleBracket]]\[NotEqual] Abs[g\[LeftDoubleBracket]i, Abs[g[[j,axx]]]\[RightDoubleBracket]], If[o\[NotEqual]0, Print[{g\[LeftDoubleBracket]axy,axx\[RightDoubleBracket], g\[LeftDoubleBracket]i, Abs[g[[j,axx]]]\[RightDoubleBracket]}];Throw[{False,i,j}], Throw[False],Throw[False]]]; j++;k=If[a,j,1]];i++;j=If[a,i,1]]; True]] jacobiQ[gg_:glo,o_:0]:= Catch[Module[{g=gpd[gg],i=1,j,k,m=Length[g]},If[abelianQ[g],Throw[False]]; While[i\[LessEqual]m, While[j\[LessEqual]m, If[g\[LeftDoubleBracket]i, j\[RightDoubleBracket]\[NotEqual]-g[[j,i]],Throw[False]]; While[k\[LessEqual]m, If[g[[i,g\[LeftDoubleBracket]j,k\[RightDoubleBracket]]]+ g[[j,g\[LeftDoubleBracket]k,i\[RightDoubleBracket]]]+ g[[k,g\[LeftDoubleBracket]i, j\[RightDoubleBracket]]]\[NotEqual]0, If[o\[NotEqual]0, Print[{g[[i,g\[LeftDoubleBracket]j,k\[RightDoubleBracket]]], g[[j,g\[LeftDoubleBracket]k,i\[RightDoubleBracket]]], g[[k,g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]]}]; Throw[{False,i,j,k}],Throw[False],Throw[False]]]; k++];j++;k=1];i++;j=1]; True]] MoufangQ[gg_:glo,o_:0]:= Catch[Module[{a,i=1,j=1,k=1,g=gpd[gg],m,axy,zx,azx,yz,ayz},m=Length[g]; a=abelianQ[g]; While[i\[LessEqual]m, While[j\[LessEqual]m, axy=Abs[g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]; While[k\[LessEqual]m, zx=g\[LeftDoubleBracket]k,i\[RightDoubleBracket];azx=Abs[zx]; yz=g\[LeftDoubleBracket]j,k\[RightDoubleBracket];ayz=Abs[yz]; If[Sign[zx]Sign[yz]g\[LeftDoubleBracket]azx, ayz\[RightDoubleBracket]\[NotEqual] g\[LeftDoubleBracket]Abs[g[[k,axy]]],k\[RightDoubleBracket], If[o\[NotEqual]0, Print[{Sign[zx]Sign[yz]g\[LeftDoubleBracket]azx, ayz\[RightDoubleBracket], g\[LeftDoubleBracket]Abs[g[[k,axy]]], k\[RightDoubleBracket]}];Throw[{False,i,j,k}], Throw[False],Throw[False]]]; k++];j++;k=If[a,j,1]];i++;j=If[a,i,1]]; True]] moufangQ[gg_:glo,o_:0]:= Catch[Module[{a,i=1,j=1,k=1,g=gpd[gg],m,axy,zx,azx,yz,ayz},m=Length[g]; a=abelianQ[g]; While[i\[LessEqual]m, While[j\[LessEqual]m, axy=Abs[g\[LeftDoubleBracket]i,j\[RightDoubleBracket]]; While[k\[LessEqual]m, zx=g\[LeftDoubleBracket]k,i\[RightDoubleBracket];azx=Abs[zx]; yz=g\[LeftDoubleBracket]j,k\[RightDoubleBracket];ayz=Abs[yz]; If[Abs[g\[LeftDoubleBracket]azx, ayz\[RightDoubleBracket]]\[NotEqual] Abs[g\[LeftDoubleBracket]Abs[g[[k,axy]]], k\[RightDoubleBracket]], If[o\[NotEqual]0, Print[{g\[LeftDoubleBracket]azx,ayz\[RightDoubleBracket], g\[LeftDoubleBracket]Abs[g[[k,axy]]], k\[RightDoubleBracket]}];Throw[{False,i,j,k}], Throw[False],Throw[False]]]; k++];j++;k=If[a,j,1]];i++;j=If[a,i,1]]; True]] moufangLQ[gg_:glo,o_:0]:= Catch[Module[{a,i=1,j=1,k=1,g=gpd[gg],m},m=Length[g];a=abelianQ[g]; While[i\[LessEqual]m, While[j\[LessEqual]m, While[k\[LessEqual]m, If[Abs[g[[Abs[ g\[LeftDoubleBracket]Abs[ g\[LeftDoubleBracket]k,j\[RightDoubleBracket]], k\[RightDoubleBracket]],i]]]\[NotEqual] Abs[g\[LeftDoubleBracket]k, Abs[g[[j, Abs[g\[LeftDoubleBracket]k, i\[RightDoubleBracket]]]]]\[RightDoubleBracket]]\ ,If[o\[NotEqual]0,Throw[{False,i,j,k}],Throw[False],Throw[False]]]; k++];j++;k=If[a,j,1]];i++;j=If[a,i,1]]; True]] moufangRQ[gg_:glo,o_:0]:= Catch[Module[{a,i=1,j=1,k=1,g=gpd[gg],m},m=Length[g];a=abelianQ[g]; While[i\[LessEqual]m, While[j\[LessEqual]m, While[k\[LessEqual]m, If[Abs[g[[i, Abs[g\[LeftDoubleBracket]k, Abs[g\[LeftDoubleBracket]j, k\[RightDoubleBracket]]\[RightDoubleBracket]]]]]\ \[NotEqual]Abs[g[[Abs[g\[LeftDoubleBracket]Abs[ g\[LeftDoubleBracket]i,k\[RightDoubleBracket]], j\[RightDoubleBracket]],k]]], If[o\[NotEqual]0,Throw[{False,i,j,k}],Throw[False], Throw[False]]]; k++];j++;k=If[a,j,1]];i++;j=If[a,i,1]]; True]] negatableQ[gg_:glo,f_:2]:=Module[{g=gpd[gg],g1,s1,i,j,g2,l,m,t}, l=Length[g];m=Length[g]/f; If[!IntegerQ[m],False,g1=pick[g,{1,m},{1,m}]; s1=Table[Sign[g1[[i,j]]],{i,m},{j,m}]; g1=g1/s1;g2=Table[Mod[g1+e m-1,l]+1*s1,{e,f}];j=0;Catch[While[i=0; While[If[MemberQ[g2,pick[g,{m i+1,m i+m},{m j+1,m j+m}]],Null, Throw[False]];i++;in[[j]],{j,l,1,-1}];gg/.s]; toIndexTable[g_:glo]:=unshuffle[Transpose[unshuffle[gpd[g]]]]; unshuffle[g_]:= Module[{p,l=Length[g]},p=Table[g[[i,1]],{i,l}]; Table[g[[Position[p,i][[1]]]][[1]],{i,l}]]; lmatrix[]:= Module[{s,t},t=Table[0,{k,mm},{l,mm}]; Do[Do[s=glo\[LeftDoubleBracket]l,k\[RightDoubleBracket]; t\[LeftDoubleBracket]Abs[s],k\[RightDoubleBracket]=l Sign[s],{k, mm}],{l,mm}];t]; lmatrix[g_?indexTableQ]:= Module[{gg=gpd[g],m,s,t},m=Length[gg];t=Table[0,{k,m},{l,m}]; Do[Do[s=gg\[LeftDoubleBracket]l,k\[RightDoubleBracket]; t\[LeftDoubleBracket]Abs[s],k\[RightDoubleBracket]=l Sign[s],{k, m}],{l,m}];t]; rmatrix[]:= Module[{s,t},t=Table[0,{k,mm},{l,mm}]; Do[Do[s=glo\[LeftDoubleBracket]l,k\[RightDoubleBracket]; t\[LeftDoubleBracket]l,Abs[s]\[RightDoubleBracket]=l Sign[s],{k, mm}],{l,mm}];t]; rmatrix[g_?indexTableQ]:= Module[{gg=gpd[g],m,s,t},m=Length[gg];t=Table[0,{k,m},{l,m}]; Do[Do[s=gg\[LeftDoubleBracket]l,k\[RightDoubleBracket]; t\[LeftDoubleBracket]l,Abs[s]\[RightDoubleBracket]=k Sign[s],{k, m}],{l,m}];t]; smax=100;streprpt[str_,rul_,imax_:smax]:= Module[{s1=str,s2,ii=1}, While[(s2=StringReplace[s1,rul])\[NotEqual]s1,ii++; If[ii>imax, If[imax\[Equal]99,OK="Fail", Print[s1," ",s2," ","Failed. Increase smax?"];Abort[]]];s1=s2];s2] matstrep[l_?MatrixQ,rul_,sm_:smax]:= Table[streprpt[l[[i,j]],rul,sm],{i,Dimensions[l,1][[1]]},{j, Dimensions[l,2][[1]]}] ruleExpand[a___]:= Table[Rule[stringExpand[ToString[a[[i,1]]]], stringExpand[ToString[a[[i,2]]]]],{i,Length[a]}]; stringExpand[t_]:= Module[{e,f,g,l,n=1,res="",s=ToString[t]}, If[s\[Equal]"",res,l=StringLength[s];g=StringTake[s,1];res=res<>g; While[g=StringTake[s,{n}];n++;n\[LessEqual]l,f=StringTake[s,{n}]; e=ToExpression[f]; If[IntegerQ[e], If[ng,{i,e-1}],res=res<>f]]]; res] ge[gp_?VectorQ,pre_:{},o_:0]:= Module[{g={},cr,j,l=IntegerPart[Plus@@gp]+2,le=Length[gp],ln=1,lh,n={""}, new,p,st,rhs,lhs,t=ruleExpand[pre],si,sij}, cr=Take[CharacterRange["a","g"],le]; Catch[(*Reject bad rules.*)If[o\[NotEqual]0, Do[(*i loop,Relators*) Do[(*r loop,each rule*) Do[(*k loop,lhs & rhs*) If[(rhs=Characters[t[[r,2]]])\[NotEqual] Sort[rhs]||((lhs=Characters[t[[r,1]]])\[Equal]Sort[lhs]&& Length[Union[lhs]]>1),Print[{"Bad order",t[[r]]}]; Throw[Null]]; If[Length[Complement[Union[lhs,rhs],cr]]>0, Print[{"Unknown Relator ",t[[r]],cr}];Throw[Null]]; j=Length[Position[Characters[t[[r,k]]],cr[[i]]]]; If[GCD[j,IntegerPart[gp[[i]]]]>1&&j\[NotEqual]0&& j\[NotEqual]IntegerPart[gp[[i]]], Print[{"Bad GCD ",j,gp[[i]],t[[r]]}];Throw[Null]],{k,2}],{r, Length[t]}],{i,Length[gp]}]];strep={}; Do[(*i loop. Add the rules to reduce repetitions*)st=cr[[i]]; AppendTo[t,Do[st=st<>cr[[i]],{IntegerPart[gp[[i]]]-1}]; Rule[st,""]],{i,le}]; Do[(*i,j loops. Add abelianQ rules*)new= Rule[lh=cr[[j]]<>cr[[i]],cr[[i]]<>cr[[j]]]; Do[(*k loop, redundant rules*)If[lh\[Equal]t[[k,1]],new=""],{k, Length[t]-1}]; If[Length[new]\[Equal]2,AppendTo[t,new]],{i,1,le},{j,i+1,le}]; Do[(*i loop,symbols. Build the element list*)si=cr[[i]];sij=si; p=IntegerPart[gp[[i]]]-1; Do[(*j loop.Repetitions*) Do[(*k loop,new element=j repeats of k'th existing element*) AppendTo[n,If[k\[Equal]1,sij,n[[k]]<>sij]],{k,ln}]; sij=sij<>si,{j,p}];ln=Length[n],{i,le}];(*element list completed*) Do[AppendTo[ g,(*add successive n[[i]]<>n rows to g*)MapThread[ StringJoin,{Table[n[[i]],{j,ln}],n}]],{i,ln}]; g=matstrep[g,t]; If[o==0,genel=.;matel=.;strep=., genel=n/.""\[Rule]1;matel=g/.""\[Rule]1; If[o<0,(*Noncommutative mods*)ClearAllOperators[]; SetOperators[Drop[genel,1]]]; Do[Do[st= Rule[If[o<0,matel[[i,1]]** matel[[1,j]], Diamond[matel[[i,1]], matel[[1,j]]]], matel[[i,j]]]; If[ MemberQ[strep,st],,AppendTo[strep,st]],{j,2,ln}], {i,2,ln}]]; g=caylindex[g];(*Convert to a signed table?*)If[!IntegerQ[ln=Last[gp]], le=Length[g]/ln;g=ts[g,IntegerPart[ln]]; If[o\[NotEqual]0,genel=Take[genel,le]; matel=Transpose[Take[Transpose[Take[matel,le]],le]]]]; If[Abs[o]\[NotEqual]0,(*Report properties*)If[loopQ[g], If[associativeQ[g,1],g,Print["Non-associative"];g], Print["Not a quasigroup"]]]; glo=g]]; cl[p_Integer,q_:0(*,gr_:{}*)]:= Module[{g,l,n=q,ng=Length[gr],o,result,z}, If[IntegerQ[n],o=Range[p],(* no, extract negative squares from list {n} *) o=Complement[Range[p],q];n=0]; l=2^(p+n); g=Table[IntegerDigits[i,2,p+n],{i,0,l-1}];(* binary form for indices e.g. if l=8, index 6 becomes {1,1,0} *) result=Table[walsh[g[[i]]*g[[j]],o]*walsh[g[[i]]*grayinv[g[[j]]]]* (FromDigits[Mod[g[[i]]+g[[j]],2],2] +1), {j,l},{i,l}];(* If[ng>0,(* Apply Grassmann negations *) Do[z=2^(gr[[i]]-1)+1;If[z0,result[[z,z]]=0],{i,ng}]];*) result]; cd[{g_String}]:=ToExpression[g];cd[{}]:={{1}};cd[{2}]:={{1,2},{2,3}}; cd[{-2}]:={{1,2},{2,-3}};cd[{2.}]:={{1,2},{2,-1}}; cd[{3}]:={{1,2,3},{2,3,1},{3,1,2}}(*C3*); cd[{3.}]:={{1,2,3},{2,3,-1},{3,-1,-2}};cd[{-3}]:={{1,2,3},{2,-3,1},{3,1,-2}}; cd[{4}]:={{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}}(*C4*); cd[{4.}]:={{1,2,3,4},{2,1,-4,-3},{3,-4,1,-2},{4,-3,-2,1}}; cd[{-4}]:={{1,2,3,4},{2,3,-4,-1},{3,-4,1,-2},{4,-1,-2,3}}; cd[{-4.}]:={{1,2,3,4},{2,3,4,-1},{3,4,-1,-2},{4,-1,-2,-3}}; cd[{gamma__}]:= Module[{l=Last[{gamma}],n,q,r,(*Recursion*)m=cd[Drop[{gamma},-1]]}, q=Length[m];n=IntegerPart[l]; Which[ListQ[l],co[m,ca[l]](*compose with group*), StringQ[l], co[m,ToExpression[l]] (*compose with hoop*), True (* new relator*), Table[Which[i\[LessEqual]q&&j\[LessEqual]q,m[[i,j]](* Top-left *), i\[LessEqual]q,modqr[q,m[[j-q,i]]] (* Top-right *), j\[LessEqual]q,r=m[[i-q,j]]; (* Bottom left*) Which[ !IntegerQ[l],modqr[q,r](* simple, real l *), j\[NotEqual]1,n modqr[q,r] (* negate all but col 1 *), True,q+Abs[r]] (* shift by default*) , True,r=m[[j-q,i-q]] (* Bottom-right i,j>q*); If[IntegerQ[l]&&j\[NotEqual]q+1&&q>1, r, n r(* negate all but col 1 if l negative real*)]], {i,2q},{j,2q}]]] modqr[q_,r_]:=(Mod[(q+Abs[r])-1,2q]+1)Sign[r] \!\(\* RowBox[{\(Clear[md]\), ";", RowBox[{\(md[g_?MatrixQ, l_: {}]\), StyleBox[\( (*From\ Loop\ in\ GAP*) \), FontColor->RGBColor[0, 0, 1]], ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{\({m = Length[g], gin = If[l \[Equal] {}, gInverse[g], l]}\), ",", "\[IndentingNewLine]", RowBox[{\(res = Table[0, {i, 2 m}, {j, 2 m}]\), ";", RowBox[{"Do", "[", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{\(res[\([i, j]\)] = g[\([i, j]\)]\), StyleBox[\( (*\ Top - left\ g\ *) \), FontColor->RGBColor[0, 0, 1]], ";", "\[IndentingNewLine]", \(res[\([i, j + m]\)] = g[\([j, i]\)] + m\), StyleBox[\( (*\ Top - right\ g\^T + m*) \), FontColor->RGBColor[0, 0, 1]], ";", "\[IndentingNewLine]", \(res[\([i + m, j]\)] = g[\([i, gin[\([j]\)]]\)] + m\), StyleBox[\( (*\ Bottom\ left\ g + m\ reordered*) \), FontColor->RGBColor[0, 0, 1]], ";", "\[IndentingNewLine]", \(res[\([i + m, j + m]\)] = g[\([gin[\([j]\)], i]\)]\)}], StyleBox[\( (*\ Bottom\ right\ g\^T\ reordered*) \), FontColor->RGBColor[0, 0, 1]], ",", "\[IndentingNewLine]", \({i, m}\)}], "]"}], ",", \({j, m}\)}], "]"}], ";", "res"}]}], "]"}]}], ";"}]\) groupFromRelators[gp_,post_:{}]:=(* skeleton version of ge, for small groups only *) Module[{g={},cr,j,l=Plus@@gp+2,le=Length[gp],ln=1,lh,n={""},new,p,st,rhs, si,sij,t=post},cr=Take[{"a","b","c","d","e","f","g"},le]; Do[(*i loop. Rules to reduce repetitions*)st=cr[[i]]; AppendTo[t,Do[st=st<>cr[[i]],{gp[[i]]-1}];Rule[st,""]],{i,le}]; Do[(*i,j loops.Add abelian rules*)new= Rule[lh=cr[[j]]<>cr[[i]],cr[[i]]<>cr[[j]]]; Do[(*k loop.Omit redundant rules*)If[lh\[Equal]t[[k,1]],new=""],{k, Length[t]-1}]; If[Length[new]\[Equal]2,AppendTo[t,new]],{i,1,le-1},{j,i+1,le}]; Do[(*i loop.Build the element list*)si=cr[[i]];sij=si;p=gp[[i]]-1; Do[(*j loop.Repetitions*) Do[(*k loop.New element=j repeats of k'th existing element*) AppendTo[n,If[k\[Equal]1,sij,n[[k]]<>sij]],{k,ln}]; sij=sij<>si,{j,p}];ln=Length[n],{i,le}];(*element list completed.*) Do[(*i loop.Build table by concatenation.*)AppendTo[ g,(*add successive n[[i]]<>n rows to g*)MapThread[ StringJoin,{Table[n[[i]],{j,ln}],n}]],{i,ln}]; g=Table[l=0;s1=g\[LeftDoubleBracket]i,j\[RightDoubleBracket]; s1=While[(s2=StringReplace[s1,t])\[NotEqual]s1&&l<99,l++;s1=s2]; s2,{i,ln},{j,ln}]; (*Conversion to indices.*)st= Table[g\[LeftDoubleBracket]j,1\[RightDoubleBracket]\[Rule]j,{j,ln, 1,-1}]; g/.st]; pe[mm_,maxel_:72]:= Module[{m,g={Range[Max[Flatten[mm]]]},i,j,k,n,o={}},permel={}; If[Depth[mm]<3,Print[mm,"is not a list of permutations or cycles"]; Goto[tag]]; If[MemberQ[mm,g[[1]]],Print[g[[1]]," unacceptable"];Goto[tag]]; Do[j=mm[[i]];k=If[VectorQ[j],j,fromCycles[j]]; AppendTo[g,k],{i,Length[mm]}]; j=2;k=1; While[j\[LessEqual]Length[g],k++; n=permute[g\[LeftDoubleBracket]j\[RightDoubleBracket], g\[LeftDoubleBracket]k\[RightDoubleBracket]]; If[!(MemberQ[g,n]),AppendTo[g,n]]; If[k\[GreaterEqual]Length[g],k=1;j++; If[j>maxel+1,Print["Large"];i=l+1;j=l+1;i=0;mm=0;nn=0;Goto[tag]]]]; permel=g;o= caylindex[ Table[permute[g\[LeftDoubleBracket]i\[RightDoubleBracket], g\[LeftDoubleBracket]j\[RightDoubleBracket]],{i,Length[g]},{j, Length[g]}]];Label[tag];glo=o] ma[{mm__?MatrixQ},maxel_:73,o_:0]:= Module[{m={mm},mi,g,j=2,k=1,mx=2,n},mi=minsign[m]; g=union[{IdentityMatrix[ Length[m\[LeftDoubleBracket]1\[RightDoubleBracket]]],mm}]; While[j\[LessEqual]Length[g],k++; n=g\[LeftDoubleBracket]j\[RightDoubleBracket].g\[LeftDoubleBracket] k\[RightDoubleBracket]/.mi; If[\[InvisibleSpace]!(MemberQ[g,n]),AppendTo[g,n];mx++; If[mx>maxel,k=9999;j=mx]]; If[k\[GreaterEqual]Length[g],k=1;j++; If[j>maxel,matel={};g={{}};Goto[tag]]]];matel=g; If[o\[Equal]0, g=caylindex[ Table[g\[LeftDoubleBracket] i\[RightDoubleBracket].g\[LeftDoubleBracket] j\[RightDoubleBracket]/.\[InvisibleSpace]mi,{i, Length[g]},{j,Length[g]}]];glo=g];Label[tag];g] \!\(\* RowBox[{ RowBox[{"ma", "[", RowBox[{\({mm__}\), ",", \(maxel_: 73\), ",", \(o_: 0\), ",", StyleBox[ RowBox[{ StyleBox["skeleton", FontColor->RGBColor[1, 0, 0]], "_"}]]}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{\(m = {mm}\), ",", "g", ",", \(j = 2\), ",", \(k = 1\), ",", "lg", ",", \(mx = 2\), ",", "n", ",", RowBox[{"rs", "=", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], "any_"], "\[Rule]", SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], \(Mod[any, 3]\)]}], ",", \(\[DoubleStruckK]\^any_ \[Rule] \ \[DoubleStruckK]\^Mod[any, 2]\)}], "}"}]}], ",", "st"}], "}"}], ",", \(g = Insert[m, IdentityMatrix[Length[m[\([1, 1]\)]]], 1]; While[lg = Length[g]; j \[LessEqual] lg, \(k++\); n = g\[LeftDoubleBracket]j\[RightDoubleBracket] . g\[LeftDoubleBracket] k\[RightDoubleBracket] /. \[InvisibleSpace]rs; If[\(! MemberQ[g, n]\), AppendTo[g, n]; \(mx++\); If[mx > maxel, k = 9999; j = mx]]; If[k \[GreaterEqual] lg, k = 1; \(j++\); If[j > maxel, g = {{}}; Goto[tag]]]]; g = Table[ g\[LeftDoubleBracket]i\[RightDoubleBracket] . g\[LeftDoubleBracket]j\[RightDoubleBracket] /. rs, {i, lg}, {j, lg}]; Label[tag]; st = Table[ g\[LeftDoubleBracket]j, 1\[RightDoubleBracket] -> j, {j, lg, 1, \(-1\)}]; \ng /. st\)}], "]"}]}]\) mg[v_?VectorQ,u_]:= Module[{},ma[Table[u[[Abs[v[[i]]]]]Sign[v[[i]]],{i,Length[v]}],mm+1]] ms[v__?ListQ,u_:u2,maxel_:64]:=(*Version 2s*) Module[{gg,gij,i=2,j=2,l,mi,new,n,p,s,tot=0, m=If[VectorQ[v],Table[Sign[v[[i]]]u[[Abs[v[[i]]]]],{i,Length[v]}],v]}, s=Union[{1,-1},Variables[m]\[Intersection]signlist]; If[MatrixQ[m[[1]]]===True,,Print[ "Not a list or matrix"];Abort[]]; l=Length[m[[1]]];n=Length[m]+1;matel=Prepend[m,IdentityMatrix[l]]; mi=minsign[matel]; While[j=2; While[new=matel[[i]].matel[[j]]/.mi; If[mstst[new,s]\[Equal]0,AppendTo[matel,new]];tot++; If[Length[matel]>maxel||tot>4 maxel,Print[matel];Abort[]]; jmaxel||tot>4 maxel,Print[matel];Abort[]]; j6&&l[[7]]\[NotEqual]{},mx[l[[7]],u2], (o==0||o\[Equal]3)&&Length[l]>7&&l[[8]]\[NotEqual]{},mx[l[[8]],u3], (o==0||o\[Equal]4)&&Length[l]>8&&l[[9]]\[NotEqual]{},mx[l[[9]],u4], (o==0||o\[Equal]5)&&Length[l]>9,ToExpression[l[[10]]], (o==0||o\[Equal]6)&&Length[l]>10,ToExpression[l[[11]]], True,Print[{G,"No Protoloop",o}];{}];id[res];glo=res]; gd[]:=gd[mm,nn];gd[{m_Integer,n_Integer}]:=gd[m,n]; gd[m_Integer:mm,n_Integer:nn]:= If[m>0&&loop[[m]]\[NotEqual]{}&&n>0&& Length[loop[[m]]]\[GreaterEqual]n,{mm=m,nn=n,loop[[m,n]]}, "Not a loop index"]; gd[g_]:=Module[{i},Which[ Head[g]===String&&(i=Position[loop,g,3,1])\[NotEqual]{}, gd[mm=i[[1,1]],nn=i[[1,2]]], Head[g]===String&&(i=Position[loop,g<>"?",3,1])\[NotEqual]{}, gd[mm=i[[1,1]],nn=i[[1,2]]], True,Print["Not a loop index or known mnemonic. Try as a string?"]]] \!\(\(idc[G_] := If[conservativeQ[G], id[G], "\"];\)\[IndentingNewLine] \(idc[G_, L_?ListQ] := If[conservativeQ[G], id[G, L], "\"];\)\[IndentingNewLine] \(id[] := id[{mm, nn}, {}];\)\[IndentingNewLine] id[G_: {mm, nn}] := id[G, {}]; id[G_: {mm, nn}, L_?ListQ] := Module[{ln, lg, j, jj, M}, nn = 1; If[Head[G] === TraditionalForm, M = G[\([1]\)], M = G]; \[IndentingNewLine]If[Head[G] === String, M = mp[G]]; Which[Dimensions[M] \[Equal] {2}, mm = M[\([1]\)]; nn = M[\([2]\)]; M = {}, \[IndentingNewLine]MatrixQ[M], M = If[Length[L] > 0, reorder[M, L], M]; mm = Length[M], \[IndentingNewLine]True, mm = Length[loop]; glo =. ; M = {}]; \[IndentingNewLine]lg = Length[M]; \[IndentingNewLine]If[ lg > 1 && \((j = Position[\(loop[\([lg]\)]\)[\([All, {2}]\)], dico[M], 3, 1])\) \[NotEqual] {}, nn = j[\([1, 1]\)], mm = Length[loop]]; \[IndentingNewLine]jj = loop[\([mm, nn, 3]\)]; ln = nn; \[IndentingNewLine]Switch[jj, 0, , \[IndentingNewLine]1, Switch[trico[M], {3, 3\_2}, "\", {9}, \(nn++\); "\", {6, 3}, nn += 4; "\"], \[IndentingNewLine]2, If[Length[fa[M]] \[Equal] 2, "\", nn += 3; "\"], \[IndentingNewLine]3, If[subgroups[M, 0] \[Equal] 9, nn += 3; "\", "\"], \[IndentingNewLine]4, If[subgroups[M, 0, 5, 5] \[NotEqual] 1, "\", \(nn++\); "\"], \[IndentingNewLine]5, If[trico[M] === {3, 3\_5}, "\", nn += 3; "\"], \[IndentingNewLine]6, If[trico[M] === {9, 1\_9}, nn += 3; "\", "\"], \[IndentingNewLine]7, If[Length[fa[M]] \[Equal] 2, "\", nn += 5; "\"], \[IndentingNewLine]8, Switch[trico[M], {3, 3\_8}, "\", {9, 9\_2}, \(nn++\); "\", {27}, nn += 4; "\"], \[IndentingNewLine]9, If[subgroups[M, 0] \[Equal] 1, "\", \(nn++\); "\"], \[IndentingNewLine]10, Switch[trico[M], {9, 9\_2}, \(nn++\); "\", {27}, "\"], \ \[IndentingNewLine]11, Switch[subgroups[M, 0, 0, 8], 33, nn += 31; "\", 37, nn += 22; "\", any_, "\"], \[IndentingNewLine]12, If[conservativeQ[M], "\", nn += 2; "\"], \[IndentingNewLine]13, If[trico[M] === {1, 1\_13, 2}, "\", \(nn++\); "\"], \[IndentingNewLine]14, If[Length[centre[M]] \[Equal] 8, "\", nn += 6; "\"], \[IndentingNewLine]15, If[trico[M] === {1, 1\_11, 2\_2}, "\", nn += 3; "\"], \[IndentingNewLine]16, Switch[trico[M], {1, 1\_13, 2}, "\", {1, 1\_11, 2\_2}, \(nn++\); "\", {1, 1\_15}, nn += 2; "\", {1, 1\_7, 2\_4}, nn += 3; "\"], \[IndentingNewLine]17, If[Length[centre[M]] \[Equal] 8, "\", nn += 6; "\"], \[IndentingNewLine]18, Switch[Max[ First[Transpose[ collect[subgroupTypes[subgroups[M, 1, 16, 16]]]]]], 2, mm = 32; nn = 25; "\", 3, mm = 32; nn = 30; "\", 4, mm = 32; nn = 31; "\"], \[IndentingNewLine]19, If[Length[centre[M]] == 4, nn += 6; "\", "\"], \[IndentingNewLine]20, If[conservativeQ[M], "\", nn += 19; "\"], 21, If[Length[centre[M]] \[Equal] 4, "\"; mm = 64; nn = ln + 13, "\"; mm = 64; nn = ln], \[IndentingNewLine]22, If[Position[ M, \(-\[ImaginaryI]\)] === {}, \(nn++\); "\", \ "\"], \[IndentingNewLine]23, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", nn += 6; "\"], \[IndentingNewLine]24, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", nn += 9; "\"], \[IndentingNewLine]25, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", nn += 9; "\"], \[IndentingNewLine]26, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", nn += 6; "\"], \[IndentingNewLine]27, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", nn += 8; "\"], \[IndentingNewLine]28, Switch[allFactorGroups[M, 0, 2, 2, {37, 38}], 37, "\"; mm = 64; nn = ln, 38, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]29, Switch[trico[M], {3, 3\_20}, "\", {9, 9\_6}, \(nn++\); "\"], \[IndentingNewLine]30, If[subgroups[M, 0, 3, 3] \[Equal] 1, "\", \(nn++\); "\"], \[IndentingNewLine]31, Switch[trico[M], {3, 3\_6, 21\_2}, \(nn++\); "\", {45, 3\_6}, "\"], \[IndentingNewLine]32, If[subgroups[M, 0, 4, 4] \[Equal] 7, "\", nn += 3; "\"], \[IndentingNewLine]33, If[Length[centre[M]] \[Equal] 4, \(nn++\); "\", "\"], \[IndentingNewLine]34, Switch[subgroups[M, 0, 32, 0, {3, 7, 11, 15, 36, 38, 42}], \[IndentingNewLine]36, "\"; mm = 64; nn = 116, \[IndentingNewLine]3, "\"; mm = 64; nn = 117, \[IndentingNewLine]7, "\"; mm = 64; nn = 117, \[IndentingNewLine]_, "\"; mm = 64; nn = 124], \[IndentingNewLine]35, If[subgroups[M, 0, 4, 4] \[Equal] 15, nn += 2; "\", "\"], \[IndentingNewLine]36, If[subgroups[M, 0, 4, 4] \[Equal] 3, If[Length[centre[M]] \[NotEqual] 4, "\", \(nn++\); "\"], nn += 2; "\"], \[IndentingNewLine]37, Switch[subgroups[M, 0, 4, 4], 11, "\", 7, nn += 2; "\"], \[IndentingNewLine]38, If[subgrpPerm[M, 0, 7, 7] \[Equal] 1, "\", \(nn++\); "\"], \[IndentingNewLine]39, If[subgrpPerm[M, 0, 5, 5] \[Equal] 1, "\", \(nn++\); "\"], \[IndentingNewLine]40, If[subgrpPerm[M, 0, 5, 5] \[Equal] 1, "\", \(nn++\); "\"], \[IndentingNewLine]41, If[subgrpPerm[M, 0, 9, 9] \[Equal] 4, "\", nn += 5; "\"], \[IndentingNewLine]42, If[Length[Position[allFactorGroups[M, 0, 9, 9], "\"]] === 3, mm = 54; nn = ln; "\", mm = 54; nn = ln + 1; "\"], \[IndentingNewLine]43, Switch[trico[M], {3, 1\_27, 3\_8}, D27, _, nn += 13; "\"], \[IndentingNewLine]44, Switch[allFactorGroups[M, 0, 2, 2, {39, 40}], 39, "\"; mm = 64; nn = ln, 40, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]45, If[trico[M] === {9, 3\_9, 9\_2}, "\", nn += 8; "\"], \[IndentingNewLine]46, If[MemberQ[allFactorGroups[M, 0, 2, 2], "\"], "\"; mm = 64; nn = ln + 3, mm = 64; nn = ln; "\"], \[IndentingNewLine]47, If[MemberQ[allFactorGroups[M, 0, 2, 2], "\"], "\"; mm = 64; nn = ln, mm = 64; nn = ln + 1; "\"], \[IndentingNewLine]48, If[subgroups[M, 0, 4, 4] \[Equal] 27, "\", nn += 2; "\"], \[IndentingNewLine]49, Switch[subgroups[M, 0, 8, 8], 7, "\", nn += 24, "\"], \[IndentingNewLine]50, Switch[allFactorGroups[M, 0, 2, 2, {16, 17}], \[IndentingNewLine]17, "\"; mm = 64; nn = ln, 16, "\"; mm = 64; nn = ln + 24], \[IndentingNewLine]51, If[MemberQ[allFactorGroups[M, 0, 2, 2], "\"], "\"; mm = 64; nn = ln + 2, mm = 64; nn = ln; "\"], \[IndentingNewLine]52, Switch[allFactorGroups[M, 0, 2, 2, {2, 9, 14, 40, 42}], \[IndentingNewLine]40, "\"; mm = 64; nn = ln + 85, \[IndentingNewLine]42, "\"; mm = 64; nn = ln + 87, \[IndentingNewLine]_, "\"; mm = 64; nn = ln], \[IndentingNewLine]53, If[subgroups[M, 0, 4, 4] \[Equal] 31, "\", "\"; \(nn++\)], \[IndentingNewLine]54, If[Length[centre[M]] \[Equal] 4, "\"; nn += 43, If[subgroups[M, 0, 4, 4] \[Equal] 35, "\"; nn += 56, "\"]], \[IndentingNewLine]55, Switch[subgroups[M, 0, 32, 0, {16, 17}], 16, nn = ln; "\", 17, nn = ln + 1; "\"], \[IndentingNewLine]56, Switch[subgroups[M, 0, 16, 16], 19, "\"; nn = ln, 11, "\"; nn = ln + 7, 7, "\"; nn = ln + 25], \[IndentingNewLine]57, Switch[allFactorGroups[M, 0, 2, 2, {31, 42}], \[IndentingNewLine]31, "\"; mm = 64; nn = ln, 42, "\"; mm = 64; nn = ln + 9, _, "\"; mm = 64; nn = ln + 6], \[IndentingNewLine]58, Switch[allFactorGroups[M, 0, 2, 2, {22, 43, 28}], 22, "\"; mm = 64; nn = ln, 43, "\"; mm = 64; nn = ln + 9, 28, "\"; mm = 64; nn = ln + 51], \[IndentingNewLine]59, Switch[allFactorGroups[M, 0, 2, 2, {11, 28, 37, 38, 42}], \[IndentingNewLine]37, "\"; mm = 64; nn = ln, \[IndentingNewLine]38, "\"; mm = 64; nn = ln + 1, \[IndentingNewLine]42, "\"; mm = 64; nn = ln + 9, \[IndentingNewLine]11, "\"; mm = 64; nn = ln + 13, \[IndentingNewLine]28, "\"; mm = 64; nn = ln + 64, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 14], \[IndentingNewLine]60, Switch[allFactorGroups[M, 0, 2, 2, {5, 4, 17, 24}], \[IndentingNewLine]5, "\"; mm = 64; nn = ln, \[IndentingNewLine]4, "\"; mm = 64; nn = ln + 67, \[IndentingNewLine]17, "\"; mm = 64; nn = ln + 167, \[IndentingNewLine]24, Switch[allFactorGroups[M, 0, 2, 2, {36, 38}], \[IndentingNewLine]36, "\"; mm = 64; nn = ln + 95, \[IndentingNewLine]38, "\"; mm = 64; nn = ln + 97, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 96], \[IndentingNewLine]_, Switch[allFactorGroups[M, 0, 2, 2, {36, 37, 38}], \[IndentingNewLine]36, "\"; mm = 64; nn = ln + 168, \[IndentingNewLine]37, "\"; mm = 64; nn = ln + 68, \[IndentingNewLine]38, "\"; mm = 64; nn = ln + 69]], \[IndentingNewLine]61, Switch[allFactorGroups[M, 0, 2, 2, {16, 36}], 36, "\"; mm = 64; nn = ln, 16, "\"; mm = 64; nn = ln + 100], \[IndentingNewLine]62, Switch[allFactorGroups[M, 0, 2, 2, {37, 38}], 37, "\"; mm = 64; nn = ln, 38, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]63, Switch[allFactorGroups[M, 0, 2, 2, {31, 34}], 31, "\"; mm = 64; nn = ln, 34, "\"; mm = 64; nn = ln + 6], \[IndentingNewLine]64, Switch[allFactorGroups[M, 0, 2, 2, {31, 32}], 31, "\"; mm = 64; nn = ln + 4, _, "\"; mm = 64; nn = ln], \[IndentingNewLine]65, Switch[allFactorGroups[M, 0, 2, 2, {23, 25, 26, 28}], 26, "\"; mm = 64; nn = ln + 2, _, "\"; mm = 64; nn = ln], \[IndentingNewLine]66, Switch[allFactorGroups[M, 0, 2, 2, {25, 26, 30, 31, 33}], \[IndentingNewLine]31, "\"; mm = 64; nn = ln, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 5], \[IndentingNewLine]67, Switch[allFactorGroups[M, 0, 2, 2, {22, 31}], \[IndentingNewLine]22, "\"; mm = 64; nn = ln, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 7], \[IndentingNewLine]68, Switch[allFactorGroups[M, 0, 2, 2, {28, 33}], \[IndentingNewLine]28, "\"; mm = 64; nn = ln, \[IndentingNewLine]33, "\"; mm = 64; nn = ln + 148, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 149], \[IndentingNewLine]69, Switch[allFactorGroups[M, 0, 2, 2, {2, 24}], \[IndentingNewLine]2, "\"; mm = 64; nn = ln, \[IndentingNewLine]24, "\"; mm = 64; nn = ln + 139, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 142], \[IndentingNewLine]70, Switch[allFactorGroups[M, 0, 2, 2, {40, 42}], \[IndentingNewLine]40, "\"; mm = 64; nn = ln, \[IndentingNewLine]42, "\"; mm = 64; nn = ln + 4, \[IndentingNewLine]_, "\"; mm = 64; nn = ln + 32], \[IndentingNewLine]71, Switch[allFactorGroups[M, 0, 2, 2, {40, 42}], \[IndentingNewLine]42, "\"; mm = 64; nn = ln, \[IndentingNewLine]40, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]72, If[conservativeQ[M], "\", nn += 12; "\"], 73, If[Length[centre[M]] \[Equal] 4, "\"; nn += 41, "\"], \[IndentingNewLine]74, Switch[\(collect[allFactorGroups[M, 0, 8, 8, {16, 16}]]\)[\([1, 1]\)], \[IndentingNewLine]11, "\"; mm = 64; nn = ln + 17, \[IndentingNewLine]17, "\"; mm = 64; nn = ln + 15, \[IndentingNewLine]23, "\"; mm = 64; nn = ln + 3, \[IndentingNewLine]_, "\"; mm = 64; nn = ln], \[IndentingNewLine]75, Switch[allFactorGroups[M, 0, 2, 2, {40, 42}], 40, "\"; mm = 64; nn = ln, 42, "\"; mm = 64; nn = ln + 4], \[IndentingNewLine]76, Switch[allFactorGroups[M, 0, 2, 2, {40, 42}], 40, "\"; mm = 64; nn = ln, 42, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]77, Switch[allFactorGroups[M, 0, 2, 2, {40, 42}], 40, "\"; mm = 64; nn = ln, 42, "\"; mm = 64; nn = ln + 2], \[IndentingNewLine]78, Switch[allFactorGroups[M, 0, 2, 2, {31, 42}], 31, "\"; mm = 64; nn = ln, 42, "\"; mm = 64; nn = ln + 12, _, "\"; mm = 64; nn = ln + 11], \[IndentingNewLine]79, Switch[allFactorGroups[M, 0, 2, 2, {18, 34}], 18, "\"; mm = 64; nn = ln + 12, 34, "\"; mm = 64; nn = ln], \[IndentingNewLine]80, Switch[allFactorGroups[M, 0, 2, 2, {34, 39}], 34, "\"; mm = 64; nn = ln, 39, "\"; mm = 64; nn = ln + 6], \[IndentingNewLine]81, Switch[allFactorGroups[M, 0, 2, 2, {19}], 19, "\"; mm = 64; nn = ln, _, "\"; mm = 64; nn = ln + 2], 82, Switch[allFactorGroups[M, 0, 2, 2, {27, 34, 39}], 27, "\"; mm = 64; nn = ln, 34, "\"; mm = 64; nn = ln + 9, 39, "\"; mm = 64; nn = ln + 48], \[IndentingNewLine]83, Switch[\(collect[allFactorGroups[M, 0, 8, 8, {16, 16}]]\)[\([1, 1]\)], \[IndentingNewLine]50, "\"; mm = 64; nn = ln, \[IndentingNewLine]51, "\"; mm = 64; nn = ln, \[IndentingNewLine]59, "\"; mm = 64; nn = ln, \[IndentingNewLine]73, "\"; mm = 64; nn = ln + 1, \[IndentingNewLine]63, "\"; mm = 64; nn = ln + 2, \[IndentingNewLine]49, "\"; mm = 64; nn = ln + 8, \[IndentingNewLine]_, "\"; mm = 64; nn = ln], \[IndentingNewLine]84, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 24, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 6], \[IndentingNewLine]85, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 102, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]86, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 38, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 4], \[IndentingNewLine]87, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 62, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 2], \[IndentingNewLine]88, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 18, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]89, Switch[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)], 58, "\"; mm = 64; nn = ln, 44, "\"; mm = 64; nn = ln + 8, 42, "\"; mm = 64; nn = ln + 2], \[IndentingNewLine]90, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 71, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]91, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 33, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 1], \[IndentingNewLine]92, If[\(collect[allFactorGroups[M, 0, 4, 4]]\)[\([1, 1]\)] \[Equal] 35, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 2], \[IndentingNewLine]93, If[Length[centre[M]] \[Equal] 4, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 3], \[IndentingNewLine]94, If[Length[centre[M]] \[Equal] 4, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 2], \[IndentingNewLine]95, If[Length[centre[M]] \[Equal] 4, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 4], \[IndentingNewLine]96, If[Length[centre[M]] \[Equal] 4, "\"; mm = 64; nn = ln, "\"; mm = 64; nn = ln + 4], \[IndentingNewLine]97, If[Length[fp[M]] \[Equal] 6, nn += 18; "\", "\"], \[IndentingNewLine]98, If[\(! conservativeQ[ M]\), \(nn++\); "\", "\"], \ \[IndentingNewLine]99, If[\(! associativeQ[M]\), nn += 20; "\", "\"]\[IndentingNewLine]]; \ \[IndentingNewLine]If[mm \[NotEqual] Length[loop], glo = M]; gmn = loop[\([mm, nn, 1]\)]]\) subgroups[G_?MatrixQ,o_:0,i1_:2,i2_:49,pp_:associativeQ]:= Module[{d,divs,dl,f=-1,g,i,j,l=Length[G],n=0,p=pp,r,test=0, v=Drop[G[[1]],1]}, If[i1\[NotEqual]2&&GCD[l,i1]\[Equal]1, Print[i1," is not a factor of ",l]; Abort[]]; i=If[NumberQ[i2],Max[i1,i2],i1]; Which[o<0, test={-o};i=i1,p\[Equal]{},p=associativeQ,ListQ[p],test=p]; divs=Select[Drop[Drop[Divisors[l],-1],1]-1,i>#1>=i1-1&];sg={}; globa=G;(* Now find all potential subloops in the range *) d=Flatten[Table[KS[v,divs[[i]]],{i,Length[divs]}],1];dl=Length[d]; If[o>2,Print[{dl," cases"}];]; j=0;While[j++;j-1,f,(* target found? *) o\[Equal]0,n, (* count *) o\[Equal]1,sg,(* subloop list *) True,Print[" Number of subloops = ",n];sg]]; KS[l_List,1,gap_:{},lef_:{}]:=Partition[l,1] KS[l_List,k_Integer?Positive,gap_:{},lef_:{}]:={l}/;(k==Length[l]) KS[l_List,k_Integer?Positive,gap_:{},lef_:{}]:= Module[{f=First[l],g,ngap,nlf}, nlf=Append[lef,f];ngap=Append[gap,f]; g=Length[Intersection[select[nlf],gap]];If[g>0,KS[Drop[l,1],k,ngap,lef], Join[Map[(Prepend[#,f])&,KS[Drop[l,1],k-1,gap,nlf]], KS[Drop[l,1],k,ngap,lef]]]] \!\(dico[g_] := Module[{gg = If[Length[g] \[Equal] 1, g\[LeftDoubleBracket]1\[RightDoubleBracket], g, gf], o = 1, n = {}}, gf = Sign[Flatten[gg]]; If[abelianQ[gg], o = 1. ]; If[gg === Abs[gg], Null, o = \(-o\); n = {negco[gg]}]; If[MemberQ[gf, \[ImaginaryI]] || MemberQ[gf, \(-\[ImaginaryI]\)], o = \[ImaginaryI]\ o]; If[MemberQ[gf, \[DoubleStruckCapitalJ]] || MemberQ[gf, \[DoubleStruckCapitalJ]\^2], o = \[DoubleStruckCapitalJ]\ o]; If[Length[gg] \[Equal] 32, n = {subgrpPerm[gg, 0, 4, 4]}]; If[Length[gg] \[Equal] 64, n = {Max[orderCount[gg]], nabco[gg]}]; If[Length[gg] \[Equal] 54 || Length[gg] \[Equal] 72, n = {subgrpPerm[gg, 0, 3, 3]}]; abbr[Distribution[ Table[gg\[LeftDoubleBracket]i, i\[RightDoubleBracket], {i, Length[gg]}]], o, n]]; trico[g_] := abbr[If[Length[g] > 1, Distribution[Table[g[\([i, g[\([i, i]\)]]\)], {i, Length[g]}]], Distribution[ Table[g[\([1, i, g[\([1, i, i]\)]]\)], {i, Length[g[\([1]\)]]}]]]]\n \(nabco[g_] := If[Length[g] \[Equal] 1, Length[g[\([1]\)]]^2 - Length[Position[g[\([1]\)] - Transpose[g[\([1]\)]], 0]], Length[g]^2 - Length[Position[g - Transpose[g], 0]]];\)\[IndentingNewLine] \(negco[g_] := Length[Position[g - Abs[g], a_ /; a \[NotEqual] 0]];\)\) subgrpPerm[g_,a_:0]:=Module[{p=Flatten[{1,Map[Last,Sort[Map[({Random[],#})&,Range[Length[g]-1]]]]+1}]},subgroups[reorder[g,p],a]]; subgrpPerm[g_,a_,b_,c_]:=Module[{p=Flatten[{1,Map[Last,Sort[Map[({Random[],#})&,Range[Length[g]-1]]]]+1}]},subgroups[reorder[g,p],a,b,c]]; subgroupTypes[sg_:sg]:= Module[{c,l=Length[sg],j,g=globa,m=Length[Last[sg]],n,count,types={}}, count=Table[0,{i,m}];Do[count[[Length[sg[[j]] ] ]]++,{j,l}];n=1; Do[c=count[[i]]; If[c>0,If[PrimeQ[i],n+=c; AppendTo[types,{c,StringJoin["C",ToString[i]]}], Do[AppendTo[types,{id[g,sg[[n]]]}];n++,{z,count[[i]]}]]], {i,2,m}]; types] collect[sg_:%]:= Module[{l=Length[sg],c={},s,s1}, Do[If[Length[s=sg[[i]]]>1,AppendTo[c,s],s1=Position[c,s[[1]]]; If[s1\[Equal]{},AppendTo[c,{1,s[[1]]}],s1=s1[[1,1]]; c[[s1,1]]=c[[s1,1]]+1]],{i,l}];c] toPolar[{a__},g_:gmn]:=toPol[{a},g]; toVector[{a__},g_:gmn]:=toVec[{a},g]; tohyPolar[{a__},g_:gmn]:=tohyPol[{a},g]; tohyVector[{a__},g_:gmn]:=tohyVec[{a},g]; hoopTimes[a_,b_,g_,c_]:=(*skeleton*) Module[{m=Length[g],t},t=Table[0,{m}]; Do[Do[ t\[LeftDoubleBracket]g[[l,k]]\[RightDoubleBracket]+= a\[LeftDoubleBracket]l\[RightDoubleBracket] b\[LeftDoubleBracket] k\[RightDoubleBracket] ,{k,m}],{l,m}];t] hoopTimes[a_?VectorQ,b_?VectorQ,g_?MatrixQ]:= hoopTimes[glo=g;mm=Length[g];a,b] hoopTimes[a_?VectorQ,b_?VectorQ,g_]/;Head[g]===String:= hoopTimes[glo=mp[g];gmn=g;mm=Length[g];a,b] hoopTimes[a_?VectorQ,b_?VectorQ]:= Module[{i,j,p,s,ss,st,t,u},u=Table[0,{kk,mm=Length[glo]}];t=u;i=Join[a,t]; j=Join[b,t]; Do[Do[s=glo\[LeftDoubleBracket]ll,kk\[RightDoubleBracket];ss=Sign[s]; st=If[ss===0,s,s/ss]; t\[LeftDoubleBracket]st\[RightDoubleBracket]+= i\[LeftDoubleBracket]ll\[RightDoubleBracket] j\[LeftDoubleBracket] kk\[RightDoubleBracket] ss/.List\[Rule]0,{kk,mm}],{ll,mm}]; Rl=u;Rr=u; If[(p=Position[shape[t],0])\[NotEqual]{}, If[Complement[p,Position[shape[a],0]]\[NotEqual]{},u=hoopInverse[b]; Do[Do[s=glo\[LeftDoubleBracket]ll,kk\[RightDoubleBracket];ss=Sign[s]; st=If[ss===0,s,s/ss]; Rl\[LeftDoubleBracket]st\[RightDoubleBracket]+= u\[LeftDoubleBracket] kk\[RightDoubleBracket] t\[LeftDoubleBracket] ll\[RightDoubleBracket] ss/.List\[Rule]0,{kk,mm}],{ll, mm}];Rl=Chop[a-Rl]]; If[Complement[p,Position[shape[b],0]]\[NotEqual]{},u=hoopInverse[a]; Do[Do[s=glo\[LeftDoubleBracket]ll,kk\[RightDoubleBracket];ss=Sign[s]; st=If[ss===0,s,s/ss]; Rr\[LeftDoubleBracket]st\[RightDoubleBracket]+= t\[LeftDoubleBracket] kk\[RightDoubleBracket]u\[LeftDoubleBracket] ll\[RightDoubleBracket] ss/.List\[Rule]0,{kk,mm}],{ll, mm}];Rr=Chop[b-Rr]]]; t] \!\(\(hmin = 4 $MachineEpsilon;\)\n \(hoopInverse[v_?VectorQ] := Module[{arrsz, abcsz, ls, inv, var, vec, s, temp}, inv = Table[0, {ii, mm}]; vec = Take[Flatten[Append[v, inv]], mm]; arrsz = shape[vec]; ls = Length[arrsz]; abcsz = hoopShape[]; Do[s = Sign[gi\[LeftDoubleBracket]ii\[RightDoubleBracket]]; var = alph\[LeftDoubleBracket]gi[\([ii]\)]/s\[RightDoubleBracket]; temp = 0; Do[Which[ Abs[Chop[arrsz\[LeftDoubleBracket]jj\[RightDoubleBracket], hmin]] === 0, Null, True, temp += s\ gi\[LeftDoubleBracket] mm + jj\[RightDoubleBracket] \[PartialD]\_var abcsz\[LeftDoubleBracket]jj\[RightDoubleBracket]/ abcsz\[LeftDoubleBracket]jj\[RightDoubleBracket]], {jj, ls}]; inv\[LeftDoubleBracket]ii\[RightDoubleBracket] += temp, {ii, mm}]; Chop[Simplify[inv/mm /. fromAlph[vec]]]];\)\) \!\(\(hoopPower[A_, r_: 1/2] := Module[{a = {}, lg, l, s = shape[A]}, If[r < 0, hoopPower[hoopInverse[A], \(-r\)], l = toPolar[A]; lg = Length[gp]; Do[AppendTo[a, If[IntegerQ[gp[\([ii]\)]], If[Head[l[\([ii]\)]] === Complex, l[\([ii]\)]\^r, \(Abs[l[\([ii]\)]]\^r\) Sign[l[\([ii]\)]], l[\([ii]\)]\^r], r l[\([ii]\)]]], {ii, lg}]; Chop[toVector[a]]]];\)\) hoopTest[x_:glo,t_:Real,r_:{-1.,1.},o_:0]:= Module[{A,Am,Ap,B,BA,Bi,C,ms,no=" No other",pa,pb,pab,q=0,res,result,sx, sz=2^-16,xq,X}, X=Which[Head[x]===TraditionalForm,x[[1]],Head[x]===String,mp[x], IntegerQ[x]&&IntegerQ[T],t=Real;mp[x,T],True,x]; If[MatrixQ[X]===False,X=makeProtoloop[x]];If[X===glo,,glo=caylindex[X]]; gmn=id[glo];res=gmn; hoopFactor[];ms=minsign[glo]; If[Head[sh["Unrecognized"]]===List,sh["Unrecognized"]=.]; If[Head[sh[gmn]]===sh,sh[gmn]=hoopShape[]]; If[o>0,Print[{"Hoop =",res,X,mm}]];While[A=Table[Random[t,r],{i,mm}]//N; Position[0,pa=Chop[Expand[shape[A]],sz]/.ms]\[NotEqual]{},]; While[B=Table[Random[t,r],{i,mm}]//N; Position[0,Chop[pb=Expand[shape[B]],sz]/.ms]\[NotEqual]{},]; AB=hoopTimes[A,B];pba=Chop[Expand[shape[AB]]/.ms,sz]; Am=Chop[(Expand[pa*pb]-pba)/.ms,sz];Ap=Chop[(Expand[pa*pb]+pba)/.ms,sz]; If[o>2,Print[{"A=",A,"\nB=",B,"\nAB=",AB,"pa,pb,pa pb,pba,pa pb-pba",pa, pb,pa pb,pba,Am,Union[Am],Ap,Union[Ap]}]]; If[(Flatten[Union[Am]]==={0})||(Flatten[Union[Ap]]==={0}), res="Conservative "<>res, If[gmn=="Unrecognized",Print[gmn<>" Factors NOT conserved"], Print[gmn<>" Isomorph. Protoloop factors NOT conserved"]]; Goto[exit]]; Ai=Chop[Together[Expand[hoopInverse[A]]]/.ms,sz]; pba=Chop[Together[Expand[(AiAB=hoopTimes[Ai,AB])-B]]/.ms,sz]; If[o>0,Print[{"A Inverse=",Ai,"AiAB=",AiAB,"AiAB-B=",pba}]]; If[Union[pba]==={0},no="";res=res<>", division"]; If[Head[toPol[A,gmn]]===List, If[Union[Chop[toVector[toPolar[A]]-A,sz]]==={0},no=""; res=res<>", polar-vector"]; If[Union[Chop[hoopPower[hoopPower[A],2]-A,sz]]==={0},no=""; res=res<>", +ve Powers"]; If[abelianQ[glo], While[q<10,q++;A=Table[Random[t,r],{i,mm}]; B=Table[Random[t,r],{i,mm}];xq=Table[Random[t,r],{i,mm}]//N; sx=shape[xq];C=hoopTimes[A,hoopTimes[xq,xq]]+hoopTimes[B,xq]; result=Chop[genQuadratic[A,B,C]]; If[Union[Chop[sx-shape[result[[1]]]]]==={0}|| Union[Chop[sx-shape[result[[2]]]]]==={0}, res=res<>", quadratic eqn.";q=10]]]; If[Union[Chop[hoopPower[hoopPower[A,-1/3],-3]-A]]==={0},no=""; res=res<>", -ve Powers"]]; If[Head[tohyPol[A,gmn]]===List, If[Union[Chop[tohyVector[tohyPolar[A]]-A,sz]]==={0},no=""; res=res<>", hyPolar"]];Label[exit];res<>no<>" test(s) passed"] \!\(\* RowBox[{\(genQuadratic[A_?VectorQ, B_?VectorQ, C_?VectorQ] := Module[{a, b}, If[Length[gp] \[NotEqual] Length[A], Print["\"]; Abort[], \[IndentingNewLine]a = hoopInverse[A]; b = hoopRoot[ hoopTimes[B, B] + 4\ hoopTimes[A, C]]; {hoopTimes[ a, \((\(-B\) + b)\)], hoopTimes[a, \((\(-B\) - b)\)]}/2]]\), ";", RowBox[{ StyleBox[\(hoopRoot[A_?VectorQ]\), ShowStringCharacters->True, NumberMarks->True], StyleBox[" ", ShowStringCharacters->True, NumberMarks->True], StyleBox[":=", ShowStringCharacters->True, NumberMarks->True], RowBox[{ StyleBox["Module", ShowStringCharacters->True, NumberMarks->True], StyleBox["[", ShowStringCharacters->True, NumberMarks->True], RowBox[{ StyleBox[\({a = {}, l = toPolar[A]}\), ShowStringCharacters->True, NumberMarks->True], StyleBox[",", ShowStringCharacters->True, NumberMarks->True], StyleBox[\( (*\ gp\ shows\ whether\ radius\ or\ angle\ *) \), ShowStringCharacters->True, NumberMarks->True], "\[IndentingNewLine]", RowBox[{ RowBox[{ StyleBox["Do", ShowStringCharacters->True, NumberMarks->True], StyleBox["[", ShowStringCharacters->True, NumberMarks->True], RowBox[{ StyleBox[ RowBox[{"AppendTo", "[", RowBox[{"a", ",", RowBox[{"If", "[", RowBox[{\(IntegerQ[gp[\([i]\)]]\), ",", RowBox[{ SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{ StyleBox["l", ShowStringCharacters->True, NumberMarks->True], "[", \([i]\), "]"}], "]"}], \(1/2\)], \(Sign[l[\([i]\)]]\)}], ",", \(l[\([i]\)]/2\)}], "]"}]}], "]"}], ShowStringCharacters->True, NumberMarks->True], ",", "\[IndentingNewLine]", \({i, Length[gp]}\)}], "]"}], ";", RowBox[{ StyleBox["Chop", ShowStringCharacters->True, NumberMarks->True], StyleBox["[", ShowStringCharacters->True, NumberMarks->True], RowBox[{ StyleBox["toVector", ShowStringCharacters->True, NumberMarks->True], StyleBox["[", ShowStringCharacters->True, NumberMarks->True], "a", StyleBox["]", ShowStringCharacters->True, NumberMarks->True]}], "]"}]}]}], "]"}]}]}]\) \!\(\(c2simeq[a_, b_, c_] := c2simeq[a\[LeftDoubleBracket]1\[RightDoubleBracket], a\[LeftDoubleBracket]2\[RightDoubleBracket], b\[LeftDoubleBracket]1\[RightDoubleBracket], b\[LeftDoubleBracket]2\[RightDoubleBracket], c\[LeftDoubleBracket]1\[RightDoubleBracket], c\[LeftDoubleBracket]2\[RightDoubleBracket], a\[LeftDoubleBracket]3\[RightDoubleBracket], a\[LeftDoubleBracket]4\[RightDoubleBracket], b\[LeftDoubleBracket]3\[RightDoubleBracket], b\[LeftDoubleBracket]4\[RightDoubleBracket], c\[LeftDoubleBracket]3\[RightDoubleBracket], c\[LeftDoubleBracket]4\[RightDoubleBracket]];\)\n c2simeq[a1_, a2_, b1_, b2_, c1_, c2_, a3_, a4_, b3_, b4_, c3_, c4_] := Module[{a1m2b = \((a1 - a2)\)\ \((b3 - b4)\) - \((a3 - a4)\)\ \((b1 - b2)\), a1p2b = \((a1 + a2)\)\ \((b3 + b4)\) - \((a3 + a4)\)\ \((b1 + b2)\), a1m2c = \((a1 - a2)\)\ \((c3 - c4)\) - \((a3 - a4)\)\ \((c1 - c2)\), a1p2c = \((a1 + a2)\)\ \((c3 + c4)\) - \((a3 + a4)\)\ \((c1 + c2)\), b1m2 = \((b1 - b2)\)\ \((c3 - c4)\) - \((b3 - b4)\)\ \((c1 - c2)\), b1p2 = \((b1 + b2)\)\ \((c3 + c4)\) - \((b3 + b4)\)\ \((c1 + c2)\), x4}, Which[a1p2b == 0 && a1p2c == 0 || a1m2b == 0, Flatten[{Simplify[{c1\ \((a4\ b2 - a1\ b3)\) + c2\ \((a2\ b3 - a4\ b1)\) + c3\ \((a1\ b1 - a2\ b2)\) + x4\ \((a4\ \((b1\^2 - b2\^2)\) + \((a1\ b2 - a2\ b1)\)\ b3 + \((a2\ b2 - a1\ b1)\)\ b4)\), c1\ \((a2\ b3 - a3\ b2)\) + c2\ \((a3\ b1 - a1\ b3)\) + c3\ \((a1\ b2 - a2\ b1)\) + x4\ \((a3\ \((b2\^2 - b1\^2)\) + \((a1\ b1 - a2\ b2)\)\ b3 + \((a2\ b1 - a1\ b2)\)\ b4)\), \(1\/a2\) \((\((\(-a1\)\ a3 \ + a2\ a4)\)\ \((a1\ \((c2 - b1\ x4)\) - a2\ \((c1 - b2\ x4)\))\) + \((a1\^2 - a2\^2)\)\ \((a3\ c2 - a2\ c3 - a3\ b1\ x4 + a2\ b4\ x4)\))\)}/\((\((a1\ a3 - a2\ a4)\)\ b1 + \((a1\ a4 - a2\ a3)\)\ b2 + \((a2\^2 - a1\^2)\)\ b3)\)], {x4}}], \[IndentingNewLine]a1p2b == 0, {{0, 0}, {0, 0}}, \[IndentingNewLine]True, {\(-b1m2\), b1m2, a1m2c, \(-a1m2c\)}/ a1m2b\ 2 + {\(-b1p2\), \(-b1p2\), a1p2c, a1p2c}/a1p2b\ 2]]\) simc2[a_,b_,o_:0]:= Module[{l=Length[a],ai,aijp,aijn,dp,dn,p={},n={},at=Transpose[a],sp={}, sn={},i=1,bp={},bip,bn={},bin,X,x},x=Unique[];x=Table[x,{i,l}]; Do[ai=at[[i]];bip=b[[i,1]];bin=b[[i,2]]; AppendTo[bp,bip+bin];AppendTo[bn,bip-bin]; Do[aijp=ai[[j,1]];aijn=ai[[j,2]]; p={p,aijp+aijn};n={n,aijp-aijn},{j,l}],{i,l}]; p=Partition[Flatten[p],l];dp=Chop[2 Det[p]]; n=Partition[Flatten[n],l];dn=Chop[2 Det[n]]; If[o===2,Print[{"n",n,dn,"\np",p,dp}]]; While[i\[LessEqual]l,bip=Det[Insert[Drop[p,{i}],bp,i]]; bin=Det[Insert[Drop[n,{i}],bn,i]]; sp={sp,bip,bip};sn={sn,bin,-bin};i+=1]; X=If[dp===0,x,Partition[Flatten[sp],2]/dp]+ If[dn===0,x,Partition[Flatten[sn],2]/dn]; If[o===1,X,X=Transpose[X];X[[1]]-X[[2]]]] QPower[{a_,b_,c_,d_},r_:.5]:= Module[{s,t=ArcTan[a,Sqrt[b^2+c^2+d^2]],u=ArcTan[b,Sqrt[c^2+d^2]], v=ArcTan[c,d]},s=Sin[r t]; (a^2+b^2+c^2+d^2)^(r/2){Cos[r t],s Cos[u],s Sin[u]Cos[v], s Sin[u]Sin[v]}]; fp[g_,p_:power2]/;Head[g]===TraditionalForm:=fp[g[[1]],p]; fp[g_?indexTableQ,p_:pow2]:= Module[{pp=p,lg=Length[g],lp=Length[p]}, pp=If[lp1&&fp[[1]]\[Equal]-1,-pa,pa,pa]]; fa[g_]/;Head[g]===TraditionalForm:=fa[g[[1]]]; fa[g_?indexTableQ]:= If[Length[g]<13, Flatten[If[(pa=Factor[Det[gmap[g,alph]]])[[1]]\[Equal]-1,-pa,pa,pa]], Print["Too Long for fa"];Abort[]]; alph={a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y, z,\[Alpha],\[Beta],\[Gamma],\[Delta],\[CurlyEpsilon],\[Zeta],\[Eta],\ \[Theta],\[CapitalTheta],\[Iota],\[Kappa],\[Lambda],\[Mu],\[Nu],\[Xi],\ \[CapitalXi],\[CurlyPi],\[CapitalPi],\[Rho],\[CurlyRho],\[Sigma],\ \[CapitalSigma],\[Tau],\[Upsilon],\[CapitalUpsilon],\[Phi],\[CurlyPhi],\ \[CapitalPhi],\[Chi],\[Psi],\[CapitalPsi],\[CapitalOmega],\[CapitalKoppa],\ \[Sampi],\[CapitalSampi],\[EmptySet],\[Mho],a,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p, q,r,s,t,u,v,w,x,y, z,\[Alpha],\[Beta],\[Gamma],\[Delta],\[CurlyEpsilon],\[Zeta],\[Eta],\ \[Theta],\[CapitalTheta],\[Iota],\[Kappa],\[Lambda],\[Mu],\[Nu],\[Xi],\ \[CapitalXi],\[CurlyPi],\[CapitalPi],\[Rho],\[CurlyRho],\[Sigma],\ \[CapitalSigma],\[Tau],\[Upsilon],\[CapitalUpsilon],\[Phi],\[CurlyPhi],\ \[CapitalPhi],\[Chi],\[Psi],\[CapitalPsi],\[Omega],\[CapitalOmega],\ \[CapitalKoppa],\[Sampi],\[CapitalSampi],\[EmptySet],\[Mho],\[WeierstrassP],a, b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y, z,\[Alpha],\[Beta],\[Gamma],\[Delta],\[CurlyEpsilon],\[Zeta],\[Eta],\ \[Theta],\[CapitalTheta],\[Iota],\[Kappa],\[Lambda],\[Mu],\[Nu],\[Xi],\ \[CapitalXi],\[CurlyPi],\[CapitalPi],\[Rho],\[CurlyRho],\[Sigma],\ \[CapitalSigma],\[Tau],\[Upsilon],\[CapitalUpsilon],\[Phi],\[CurlyPhi],\ \[CapitalPhi],\[Chi],\[Psi],\[CapitalPsi],\[Omega],\[CapitalOmega],\ \[CapitalKoppa],\[Sampi],\[CapitalSampi],\[EmptySet],\[Mho],a,a,b,c,d,e,f,g,h, i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y, z,\[Alpha],\[Beta],\[Gamma],\[Delta],\[CurlyEpsilon],\[Zeta],\[Eta],\ \[Theta],\[CapitalTheta],\[Iota],\[Kappa],\[Lambda],\[Mu],\[Nu],\[Xi],\ \[CapitalXi],\[CurlyPi],\[CapitalPi],\[Rho],\[CurlyRho],\[Sigma],\ \[CapitalSigma],\[Tau],\[Upsilon],\[CapitalUpsilon],\[Phi],\[CurlyPhi],\ \[CapitalPhi],\[Chi],\[Psi],\[CapitalPsi],\[Omega],\[CapitalOmega],\ \[CapitalKoppa],\[Sampi],\[CapitalSampi],\[EmptySet],\[Mho],a}; (*unique Length 64, useable lengths 260*) \!\(\* RowBox[{\(alpha = {"\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\<\[Alpha]\>", "\<\[Beta]\>", \ "\<\[Gamma]\>", "\<\[Delta]\>", "\<\[CurlyEpsilon]\>", "\<\[Zeta]\>", "\<\ \[Eta]\>", "\<\[Theta]\>", "\<\[CapitalTheta]\>", "\<\[Iota]\>", \ "\<\[Kappa]\>", "\<\[Lambda]\>", "\<\[Mu]\>", "\<\[Nu]\>", "\<\[Xi]\>", "\<\ \[CapitalXi]\>", "\<\[CurlyPi]\>", "\<\[CapitalPi]\>", "\<\[Rho]\>", "\<\ \[CurlyRho]\>", "\<\[Sigma]\>", "\<\[CapitalSigma]\>", "\<\[Tau]\>", "\<\ \[Upsilon]\>", "\<\[CapitalUpsilon]\>", "\<\[Phi]\>", "\<\[CurlyPhi]\>", "\<\ \[CapitalPhi]\>", "\<\[Chi]\>", "\<\[Psi]\>", "\<\[CapitalPsi]\>", \ "\<\[Omega]\>", "\<\[CapitalOmega]\>", "\<\[CapitalKoppa]\>", "\<\[Sampi]\>", \ "\<\[CapitalSampi]\>", "\<\[EmptySet]\>", "\<\[Mho]\>", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", "\", \ "\", "\<\[Alpha]\>", "\<\[Beta]\>", "\<\[Gamma]\>", "\<\[Delta]\>", "\<\ \[CurlyEpsilon]\>"}; (*unique\ Length\ 64, \ useable\ lengths\ 96*) \[IndentingNewLine]pow2 = Flatten[Prepend[Table[2\^i, {i, 0, 50}], {a}]];\), "\n", \(power2 = Flatten[Prepend[Table[2\^i, {i, 0, 50}], {"\", 3, 7}]]; power3 = Flatten[Prepend[Table[3\^i, {i, 0, 32}], {"\", 2, 4}]];\), "\[IndentingNewLine]", \(cliff = {e, a, b, ab, c, ac, bc, abc, d, ad, bd, abd, cd, acd, bcd, abcd, f, af, bf, abf, cf, acf, bcf, abcf, df, adf, bdf, abdf, cdf, acdf, bcdf, abcdf}; (*\ scalar, \ univector, \ bivector, \ trivector, \ etc . \ Clifford\ algebra\ elements\ *) \n fromAlph[Null] := {};\), "\[IndentingNewLine]", \(fromAlph[vec_?VectorQ, a_: alph] := Table[a[\([ii]\)] \[Rule] vec[\([ii]\)], {ii, Length[vec]}]; fromZtoCliff[a_, B_: cliff, Z_: Range[32]] := a /. Flatten[ Table[{Z[\([ii]\)] \[Rule] B[\([ii]\)], \(-Z[\([ii]\)]\) \[Rule] \(-B[\([ii]\)]\)}, {ii, Length[B]}]]; linearFactor[f_, l_: mm] := Module[{res = Take[alph, l], po = fromAlph[Take[pow2, l]]}, Do[If[Plus @@ \((res /. po)\) - f >= 2 pow2[\([ii]\)], res[\([ii]\)] *= \(-1\)], {ii, l, 0, \(-1\)}]; Plus @@ res]\), "\[IndentingNewLine]", FormBox[\(SetAttributes[linearFactor, Listable]\), "TraditionalForm"]}]\) (* Permutation Lists *) s3={{2,1,3},{3,1,2}};s4={{2,1,3,4},{1,2,4,3},{1,4,2,3},{3,4,1,2},{2,3,4,1}}; s5={{2,1,3,4,5},{1,2,4,5,3},{2,3,4,1,5},{1,4,5,2,3},{3,4,5,1,2},{2,3,4,5, 1},{3,4,1,2,5},{1,2,4,3,5}}; (* vecs pr1, pr2 that usually give good factorizations *) pr1={1,2,3,5,7,11,13,17,19,23,29,37,41,43,47,61,67,53,73,79,83,59,31,103,107, 109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199, 211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311, 313,317,331}; pr2={1,3,7,5,7,11,13,17,19,23,31,29,61,37,43,41,53,47,61,59,73,67,79,71,83,89, 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181, 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281, 283,293}; \!\(\(u2 = {{{\(-1\), 0}, {0, \(-1\)}}, {{1, 0}, {0, \(-1\)}}, {{0, \(-\[ImaginaryI]\)}, {\[ImaginaryI], 0}}, {{0, 1}, {1, 0}}, {{0, \[DoubleStruckCapitalJ]}, {\[DoubleStruckCapitalJ]\^2, 0}}, {{\[DoubleStruckK], 0}, {0, \[DoubleStruckK]}}, {{0, \[DoubleStruckN]\^4}, {\ \[DoubleStruckN]\^3, 0}}, {{\[DoubleStruckM], 0}, {0, \[DoubleStruckM]}}, {{0, \[DoubleStruckCapitalI]}, {\ \[DoubleStruckCapitalI]\^3, 0}}, {{\[DoubleStruckCapitalJ], 0}, {0, \[DoubleStruckCapitalJ]\^2}}, {{\[DoubleStruckCapitalY], 0}, {0, \[DoubleStruckCapitalY]\^2}}, {{\[ImaginaryI], 0}, {0, \[ImaginaryI]}}, {{0, \[ImaginaryI]}, {\[ImaginaryI], 0}}, {{0, \(-1\)}, {1, 0}}, {{0, \[DoubleStruckCapitalJ]\^2}, \ {\(-\[DoubleStruckCapitalJ]\), 0}}, {{\[DoubleStruckCapitalI], 0}, {0, \[DoubleStruckCapitalI]\^3}}, {{0, \(-\[DoubleStruckO]\^3\ \)}, {\[DoubleStruckO]\^5, 0}}, {{\[DoubleStruckP], 0}, {0, \[DoubleStruckP]\^4}}, {{\[DoubleStruckH], 0}, {0, \[DoubleStruckH]\^5}}, {{\[DoubleStruckCapitalJ]\^2, 0}, {0, \(-\[DoubleStruckCapitalJ]\)}}, {{\[DoubleStruckG], 0}, {0, \[DoubleStruckG]\^6}}, {{\[DoubleStruckO], 0}, {0, \[DoubleStruckO]\^7}}, {{\[DoubleStruckN], 0}, {0, \[DoubleStruckN]\^8}}, {{0, 1}, {\(-1\), 0}}, {{0, \[DoubleStruckCapitalJ]}, {\(-\[DoubleStruckCapitalJ]^2\ \), 0}}, {{0, \[DoubleStruckK]}, {\(-\[DoubleStruckK]\), 0}}, {{0, \[DoubleStruckL]}, {\(-\[DoubleStruckL]\), 0}}, {{0, \[DoubleStruckM]}, {\(-\[DoubleStruckM]\), 0}}, \ {{0, 1}, {\[ImaginaryI], 0}}, {{1, 0}, {0, \[DoubleStruckS]}}, {{0, 1}, {\[DoubleStruckS], 0}}, {{0, 1}, {\[DoubleStruckCapitalJ], 0}}, {{\[DoubleStruckCapitalY], 0}, {0, 1}}, {{\[ImaginaryI], 0}, {0, \(-\[ImaginaryI]\)}}, {{0, 1}, {\[DoubleStruckCapitalI], 0}}};\)\) \!\(\(u3 = {{{\(-1\), 0, 0}, {0, \(-1\), 0}, {0, 0, \(-1\)}}, {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}, {{1, 0, 0}, {0, \[DoubleStruckK], 0}, {0, 0, \[DoubleStruckK]}}, {{\(-1\), 0, 0}, {0, \(-\[DoubleStruckK]\), 0}, {0, 0, \(-\[DoubleStruckK]\)}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 1}}, {{0, \(-1\), 0}, {\(-1\), 0, 0}, {0, 0, \(-1\)}}, {{0, \[DoubleStruckCapitalJ], 0}, {\[DoubleStruckCapitalJ]\^2, 0, 0}, {0, 0, 1}}, {{1, 0, 0}, {0, \[DoubleStruckCapitalJ], 0}, {0, 0, \[DoubleStruckCapitalJ]\^2}}, {{0, 0, \[DoubleStruckCapitalJ]}, {\[DoubleStruckCapitalJ], 0, 0}, {0, \[DoubleStruckCapitalJ], 0}}, {{0, 0, 1}, {1, 0, 0}, {0, 1, 0}}, {{0, 0, \[ImaginaryI]}, {\[ImaginaryI], 0, 0}, {0, \(-1\), 0}}, {{\[DoubleStruckCapitalJ], 0, 0}, {0, \[DoubleStruckCapitalJ], 0}, {0, 0, \[DoubleStruckCapitalJ]}}, {{0, 0, 1}, {0, 1, 0}, {\(-1\), 0, 0}}, {{0, 0, \[ImaginaryI]}, {0, 1, 0}, {\[ImaginaryI], 0, 0}}, {{1, 0, 0}, {0, \[ImaginaryI], 0}, {0, 0, \(-\[ImaginaryI]\)}}, {{1, 0, 0}, {0, \[DoubleStruckP]\^2, 0}, {0, 0, \[DoubleStruckP]\^3}}, {{\(-\[DoubleStruckCapitalJ]\), 0, 0}, {0, \(-\[DoubleStruckCapitalJ]\), 0}, {0, 0, \(-\[DoubleStruckCapitalJ]\)}}, {{\(-1\), 0, 0}, {0, \(-\[DoubleStruckCapitalJ]\), 0}, {0, 0, \(-\[DoubleStruckCapitalJ]\^2\)}}, \ {{\[DoubleStruckCapitalJ]\^2, 0, 0}, {0, 0, \[DoubleStruckCapitalJ]\^2}, {0, \[DoubleStruckCapitalJ]\^2, 0}}, {{0, 0, \(-\[ImaginaryI]\)}, {\(-\[ImaginaryI]\), 0, 0}, {0, 1, 0}}, {{0, 1, 0}, {0, 0, 1}, {\(-1\), 0, 0}}, {{\[DoubleStruckCapitalJ], 0, 0}, {0, \(-\[DoubleStruckCapitalJ]\), 0}, {0, 0, \(-\[DoubleStruckCapitalJ]\)}}, {{0, 0, \(-\[DoubleStruckCapitalJ]\)}, {\(-\[DoubleStruckCapitalJ]\), 0, 0}, {0, \(-\[DoubleStruckCapitalJ]\), 0}}, {{1, 0, 0}, {0, \[DoubleStruckG]\^2, 0}, {0, 0, \[DoubleStruckG]\^5}}, {{0, 1, 0}, {\[ImaginaryI], 0, 0}, {0, 0, \[ImaginaryI]}}, {{\[ImaginaryI], 0, 0}, {0, 0, 1}, {0, \[ImaginaryI], 0}}, {{\[DoubleStruckO], 0, 0}, {0, 1, 0}, {0, 0, \[DoubleStruckO]\^7}}, {{1, 0, 0}, {0, \[DoubleStruckN]\^4, 0}, {0, 0, \[DoubleStruckN]\^5}}, {{\(-1\), 0, 0}, {0, \(-\[DoubleStruckP]\^2\), 0}, {0, 0, \(-\[DoubleStruckP]\^3\)}}, {{\(-1\), 0, 0}, {0, \(-\[DoubleStruckG]\^2\), 0}, {0, 0, \(-\[DoubleStruckG]\^5\)}}, {{\(-1\), 0, 0}, {0, \(-\[DoubleStruckN]\^4\), 0}, {0, 0, \(-\[DoubleStruckN]\^5\)}}, {{1, 0, 0}, {0, \[DoubleStruckCapitalY], 0}, {0, 0, \[DoubleStruckCapitalY]\^2}}, {{\[DoubleStruckN], 0, 0}, {0, \[DoubleStruckN], 0}, {0, 0, \[DoubleStruckN]\^7}}, {{0, \(-1\), 0}, {1, 0, 0}, {0, 0, 1}}, {{0, \(-1\), 0}, {0, 0, \(-1\)}, {1, 0, 0}}, {{1, 0, 0}, {0, 0, \(-1\)}, {0, 1, 0}}, {{0, \[DoubleStruckP], 0}, {\[DoubleStruckP]\^3, 0, 0}, {0, 0, \[DoubleStruckP]\^2}}, {{\(-1\), 0, 0}, {0, 1, 0}, {0, 0, \(-1\)}}, {{0, 0, \(-1\)}, {0, 1, 0}, {1, 0, 0}}, {{1, 0, 0}, {0, \(-1\), 0}, {0, 0, 1}}, {{0, 0, 1}, {0, \(-1\), 0}, {\(-1\), 0, 0}}, {{0, 0, \(-1\)}, {0, \(-1\), 0}, {1, 0, 0}}, {{\[ImaginaryI], 0, 0}, {0, \[ImaginaryI], 0}, {0, 0, \[ImaginaryI]}}, \(-{{\[ImaginaryI], 0, 0}, {0, \[ImaginaryI], 0}, {0, 0, \[ImaginaryI]}}\), {{0, 1, 0}, {0, 0, \[ImaginaryI]}, {1, 0, 0}}, {{0, \[ImaginaryI], 0}, {0, 0, \[ImaginaryI]}, {\[ImaginaryI], 0, 0}}, {{0, \[DoubleStruckS], 0}, {0, 0, 1}, {1, 0, 0}}};\)\) \!\(\(u4 = {{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, \(-1\), 0}, {0, 0, 0, \(-1\)}}, {{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, \(-1\), 0}, {0, 0, 0, \(-1\)}}, {{1, 0, 0, 0}, {0, 0, \(-1\), 0}, {0, \(-1\), 0, 0}, {0, 0, 0, 1}}, {{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}}, {{\(-1\), 0, 0, 0}, {0, 1, 0, 0}, {0, 0, \(-1\), 0}, {0, 0, 0, 1}}, {{1, 0, 0, 0}, {0, \(-1\), 0, 0}, {0, 0, \(-1\), 0}, {0, 0, 0, 1}}, {{0, 0, 1, 0}, {0, 0, 0, \(-1\)}, {1, 0, 0, 0}, {0, \(-1\), 0, 0}}, {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, \[DoubleStruckCapitalJ], 0}, {0, 0, 0, \[DoubleStruckCapitalJ]\^2}}, {{0, 0, \(-1\), 0}, {1, 0, 0, 0}, {0, \(-1\), 0, 0}, {0, 0, 0, 1}}, {{1, 0, 0, 0}, {0, 0, 0, \[DoubleStruckCapitalJ]}, {0, \[DoubleStruckCapitalJ], 0, 0}, {0, 0, \[DoubleStruckCapitalJ], 0}}, {{1, 0, 0, 0}, {0, \[DoubleStruckCapitalJ], 0, 0}, {0, 0, \[DoubleStruckCapitalJ], 0}, {0, 0, 0, \[DoubleStruckCapitalJ]}}, {{1, 0, 0, 0}, {0, \(-1\), 0, 0}, {0, 0, \[ImaginaryI], 0}, {0, 0, 0, \[ImaginaryI]}}, {{0, 0, \(-1\), 0}, {0, 0, 0, \(-1\)}, {0, 1, 0, 0}, {1, 0, 0, 0}}, {{0, 0, 0, \(-1\)}, {0, 0, 1, 0}, {0, \(-1\), 0, 0}, {1, 0, 0, 0}}, {{0, 0, \(-1\), 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, \(-1\), 0, 0}}, {{0, \(-1\), 0, 0}, {1, 0, 0, 0}, {0, 0, 0, \(-1\)}, {0, 0, 1, 0}}, {{1, 0, 0, 0}, {0, \[DoubleStruckP], 0, 0}, {0, 0, \[DoubleStruckP]\^2, 0}, {0, 0, 0, \[DoubleStruckP]\^2}}, {{0, 1, 0, 0}, {0, 0, \(-1\), 0}, {\(-1\), 0, 0, 0}, {0, 0, 0, \(-1\)}}, {{1, 0, 0, 0}, {0, \[DoubleStruckH], 0, 0}, {0, 0, \[DoubleStruckH]\^2, 0}, {0, 0, 0, \[DoubleStruckH]\^3}}, {{\[DoubleStruckG], 0, 0, 0}, {0, \[DoubleStruckG], 0, 0}, {0, 0, \[DoubleStruckG]\^2, 0}, {0, 0, 0, \[DoubleStruckG]\^3}}, {{1, 0, 0, 0}, {0, 0, \(-\[ImaginaryI]\), 0}, {0, \(-1\), 0, 0}, {0, 0, 0, \[ImaginaryI]}}, {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, \[DoubleStruckO]\^3, 0}, {0, 0, 0, \[DoubleStruckO]\^5}}, {{1, 0, 0, 0}, {0, \[DoubleStruckN], 0, 0}, {0, 0, \[DoubleStruckN]\^4, 0}, {0, 0, 0, \[DoubleStruckN]\^4}}, {{1, 0, 0, 0}, {0, 0, \(-\[DoubleStruckP]\^2\), 0}, {0, \(-\[DoubleStruckP]\), 0, 0}, {0, 0, 0, \[DoubleStruckP]\^2}}, {{\(-1\), 0, 0, 0}, {0, \(-\[DoubleStruckP]\), 0, 0}, {0, 0, \(-\[DoubleStruckP]\^2\), 0}, {0, 0, 0, \(-\[DoubleStruckP]\^2\)}}, {{0, 0, \(-\[ImaginaryI]\), 0}, {1, 0, 0, 0}, {0, \(-1\), 0, 0}, {0, 0, 0, \[ImaginaryI]}}, {{0, 0, \[DoubleStruckCapitalJ], 0}, {0, 0, 0, \[DoubleStruckCapitalJ]\^2}, {\(-1\), 0, 0, 0}, {0, \(-1\), 0, 0}}, {{0, 0, \(-\[DoubleStruckP]\^2\), 0}, {1, 0, 0, 0}, {0, \(-\[DoubleStruckP]\), 0, 0}, {0, 0, 0, \[DoubleStruckP]\^2}}, {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, \(-1\), 0, 0}, {\(-1\), 0, 0, 0}}, {{0, 0, 0, \(-\[ImaginaryI]\)}, {0, 0, \[ImaginaryI], 0}, {0, \[ImaginaryI], 0, 0}, {\(-\[ImaginaryI]\), 0, 0, 0}}, {{0, 0, 1, 0}, {0, 0, 0, \(-1\)}, {\(-1\), 0, 0, 0}, {0, 1, 0, 0}}, {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}, {{0, \[DoubleStruckCapitalJ], 0, 0}, {\[DoubleStruckCapitalJ], 0, 0, 0}, {0, 0, 0, 1}, {0, 0, \[DoubleStruckCapitalJ], 0}}, {{0, 0, \[DoubleStruckCapitalJ], 0}, {0, 0, 0, \[DoubleStruckCapitalJ]\^2}, {\[DoubleStruckCapitalJ]\^2, 0, 0, 0}, {0, \[DoubleStruckCapitalJ], 0, 0}}, {{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, \(-1\)}, {0, 0, \(-1\), 0}}, {{0, 1, 0, 0}, {0, 0, 0, \(-1\)}, {\(-1\), 0, 0, 0}, {0, 0, 1, 0}}, \[IndentingNewLine]{{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {\(-1\), 0, 0, 0}}, {{0, \(-\[ImaginaryI]\), 0, 0}, {\[ImaginaryI], 0, 0, 0}, {0, 0, 0, \(-\[ImaginaryI]\)}, {0, 0, \[ImaginaryI], 0}}, {{0, 0, \(-\[ImaginaryI]\), 0}, {0, 0, 0, \(-\[ImaginaryI]\)}, {\[ImaginaryI], 0, 0, 0}, {0, \[ImaginaryI], 0, 0}}, {{0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}}, {{0, 0, \(-1\), 0}, {0, 0, 0, \(-1\)}, {1, 0, 0, 0}, {0, 1, 0, 0}}, {{0, 0, 0, 1}, {0, 0, \(-1\), 0}, {0, \(-1\), 0, 0}, {1, 0, 0, 0}}, {{0, 1, 0, 0}, {\(-1\), 0, 0, 0}, {0, 0, 0, 1}, {0, 0, \(-1\), 0}}, {{0, 0, 0, \(-1\)}, {0, 0, \(-1\), 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}, {{0, \[DoubleStruckS], 0, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}, {0, 0, 1, 0}}};\)\) \!\(genCos[x_, m_: 4, j_: 0] := Which[m == 0, j \((\[ExponentialE]\^\(\(\ \)\(x\)\) + \[ExponentialE]\^\(-x\))\)/2, m == 2, Re[\[ExponentialE]\^x + \(\((\(-1\))\)\^j\) \[ExponentialE]\^\(\ \(\ \)\(-x\)\)]/2, \[IndentingNewLine]True, Re[\(\((\(-1\))\)\^\(2 \((m - j)\)/ m\)\) \[ExponentialE]\^\(\[ImaginaryI]\ \ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]]\[IndentingNewLine] genSin[x_, m_: 4, j_: 0] := Which[m == 0, \((\[ExponentialE]\^\(\(\ \)\(x\)\) - \[ExponentialE]\^\(-x\))\)/ 2, m == 2, Re[\(\((\(-1\))\)\^j\) \[ExponentialE]\^\(\(\ \)\(x\)\) - \ \[ExponentialE]\^\(-x\)]/2, \[IndentingNewLine]True, Im[\(\((\(-1\))\)\^\(2 \((m - j)\)/ m\)\) \[ExponentialE]\^\(\[ImaginaryI]\ \ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]]\) End[]; Protect[Evaluate[$Context <> "*"]]; Unprotect[ a1,a2,a2,a3,a4,ab,aa,abc,abef,agek,ai,ap,ar,as,azx,ayz,A,Ai,AiBA, AiAB,Am,Ap, b1,b2,b3,b4,bb,bc,B,AB,BA,c1,c2,c3,c4,cr,d1,da,dab,db,de, dozaltest,e1,e2,ef,f2,fa,ff,fi,fj,fn,fp,G,g1,g2,gf,gg,glo,globa,genel,gi, gmn,gp,h1,hoopList,hh,i1,i2,ii,im,in,jj,kk,l22a,l22b,l3,l4,l4p,l4q,l4r, l4s,la,ll,lm,lp,lr,ls,lt,m1,mi,matel,mm,nn,nb,no,np,nw,nx,ok,o1,o2,o3,o4, oo,p1,p2,p22,p23,p26,pa,pb,pab,papb,pba,pi,po,permel,plex,pp,ps,q22,q23, q24,q26,qq,res,result,\[DoubleStruckR],R1,R2,Rl,Rr,r22,r23,r26,rhs,rr,sab, sh,sg,s1,s2,s5,s22,s23,s24,s26,si,sij,sm,ss,strep,temp,test,tes,tf,ts,uu, var,vec,vf,vv,x2,x4,xy,y2,z,L3,L1a,L1b,L1c,L1d,L22,L23a,L23b,L23c,L23d, L26a,L26b,L26c,L26d,L4,L24Q,L24\[Sigma],L4Q,L4\[Sigma],L49,L49a,L49b,L49c, L4a3,L4b3,L4c3,L4d3,L4a6,L4b6,L4c6, L4d6(*,\[Alpha],\[Beta],\[Gamma],\[Delta],\[Epsilon],\[Zeta],\[Eta],\ \[Theta],\[CurlyTheta],\[Iota],\[Kappa],\[CurlyKappa],\[Lambda],\[Mu],\[Nu],\ \[Xi],\[Rho],\[Sigma],\[Tau],\[Phi],\[CurlyPhi],\[Chi],\[Psi],\[Omega]*)]; EndPackage[];