(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 162833, 3369]*) (*NotebookOutlinePosition[ 163881, 3401]*) (* CellTagsIndexPosition[ 163837, 3397]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Hoops Package Creation", "Title", PageWidth->WindowWidth], Cell["R.H.Beresford 21/02/2007.", "Subsubtitle", PageWidth->WindowWidth], Cell["Quit[] (* prevents shadowing *)", "Input"], Cell["\<\ Off[General::spell, General::spell1, Syntax::com] tf = TraditionalForm; \ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[CellGroupData[{ Cell["1. Version, Copyright notice.", "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, -104.188}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[{ "Version 1c. 21/02/07.\n\[Copyright] Copyright 2006,2007", StyleBox[" Roger Herbert Beresford, 30, Brook Rd., Woodhouse Eaves, \ Loughborough, Leics, LE12 8RS, England", FontSlant->"Italic"], ". \nConsiderable effort has gone into checking this work, but some errors \ probably persist. No responsibility will be accepted for any problems arising \ from its use. Permission is granted to distribute this file for any purpose \ except for inclusion in commercial software or program collections. This \ copyright notice must remain intact.\nPlease send any comments to me at ", StyleBox["rhberesford@btinternet.com", FontSlant->"Italic"] }], "Text", PageWidth->WindowWidth, CellMargins->{{Inherited, 0}, {Inherited, Inherited}}, CellSize->{808, Inherited}] }, Open ]], Cell[CellGroupData[{ Cell["2. Revisions.", "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, -104.188}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell["\<\ Version 1 20/04/06. Version 1a. 07/05/06. Minor edits. Introduced BoldFace by \"//bf\". Version 1b. 23/09/06. Minor edits. Exra hoops added, from latest \ GroupLoopHoop.m. Version 1c. 21/02/07. Minor edits. Extended \"as[A_,B_]\"; hoopNames global.\ \ \>", "Text", PageWidth->WindowWidth] }, Open ]], Cell[CellGroupData[{ Cell["3. Keywords.", "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell["\<\ \tAlgebras, Clifford algebras, conservative loops & algebras, \ Frobenius-conservation, generalized signs, Hoops, Hoop shapes & Hoop sizes, \ Moufang Loops, plex-conjugate, partial-division-by-zero, remainders, roots \ of unity, symmetries, zero-vectors.\ \>", "Text", PageWidth->WindowWidth, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, CellSize->{490, Inherited}] }, Closed]], Cell[CellGroupData[{ Cell["4. Warnings.", "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell[TextData[{ "This package is incompatible with GroupLoopHoop.m in ", StyleBox["MathSource", FontSlant->"Italic"], " 4894." }], "Text", PageWidth->WindowWidth] }, Closed]], Cell[CellGroupData[{ Cell["5. Limitations.", "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, -54.375}, {Inherited, Inherited}}, FontFamily->"Times New Roman"], Cell["Many hoops lack polar forms.", "Text", PageWidth->WindowWidth] }, Closed]], Cell[CellGroupData[{ Cell["6. Note on Implementation.", "Subsection", PageWidth->WindowWidth, FontFamily->"Times New Roman"], Cell[TextData[{ "\tData on each hoop is stored as ", StyleBox["Hoop[\"H\"]", FontSlant->"Italic"], "; ", StyleBox["Use[H]", FontSlant->"Italic"], " puts this data into the set of variables used by the procedures ", StyleBox["hoopTimes, hoopInverse, hoopPower", FontSlant->"Italic"], " that implement the Hoop algebra for the \"active\" hoop. ", StyleBox["hoopNames", FontSlant->"Italic"], " lists all the names.", "\n\tFor each hoop ", StyleBox["H", FontSlant->"Italic"], ", the first two elements of ", StyleBox["Hoop[\"H\"] ", FontSlant->"Italic"], "are ", StyleBox["mm & nn,", FontSlant->"Italic"], " i.e. the length of the hoop ", StyleBox["mm", FontSlant->"Italic"], " and its position ", StyleBox["nn", FontSlant->"Italic"], " in the GAP Atlas, as extended in the ", StyleBox["GroupLoopHoop.m", FontFamily->"Times New Roman", FontSlant->"Italic"], " package to include hoops and some other small loops. (This is adapted \ from ([GAP99] The Gap Group. GAP --- Groups, Algorithms, and Programming, \ Version 4.2; Aachen, St Andrews, 1999. http://www-gap. dcs.st-and. \ ac.uk/~gap).\n\tNext is the index table \"protoloop\", i.e. the Cayley table \ isomorph used to define the hoop. This is a list of lists giving the product \ index, ", StyleBox["i.j=k", FontSlant->"Italic"], " in the ", StyleBox["i", FontSlant->"Italic"], "'th row and ", StyleBox["j", FontSlant->"Italic"], "'th column. Tables have {1,2...,mm} in the first row and column, and every \ index occurs (possibly with a sign) once in every row and column. Hoop tables \ have the Moufang division property x(yz) = (xy)z (for all elements x,y,z in \ the table, allowing for signs). By definition, they have the Frobenius \ conservation property Det[A] Det[B] = Det[AB], up to a sign, for the \ multiplication of vectors A & B. (Five tables without the conservation \ property are appended to provide counter-examples. Their names end in ", StyleBox["n", FontSlant->"Italic"], ")\n\tNext comes the factorized determinant for a Cartesian form {a,b,..", StyleBox["m", FontSlant->"Italic"], "}, followed, where known, by the closely related Polar form which includes \ angles that take up the degrees of freedom released by quadratic sizes. (A \ null entry means that the polar form has not been developed. More may be \ available in later versions of ", StyleBox["MathSource", FontSlant->"Italic"], "/4894).\n\tIf it is known, the Polar-Cartesian reversion will come next. \ This gives the Cartesian form for a polar form {a,b,..", StyleBox["m", FontSlant->"Italic"], "}.\n\tTwo lists, ", StyleBox["gi, gp", FontSlant->"Italic"], ", follow. These summarize information about the hoop, needed by the ", StyleBox["hoopInverse, topol", FontSlant->"Italic"], " and ", StyleBox["tovec ", FontSlant->"Italic"], "routines. Finally, a ", StyleBox["Plex", FontSlant->"Italic"], " involution may be supplied. This is the Complex Conjugate for hoops C4c \ (complex), Qr (quaternion) and Oct (octonion) and is an equivalent of the \ Complex Conjugate for some other algebras." }], "Text", PageWidth->WindowWidth] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["7. Set up the Hoops.m package, usage", FontFamily->"Times New Roman"], "." }], "Subsection", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[TextData[{ "7.1 Begin, ", StyleBox["Hoops", FontFamily->"Times New Roman"], " Usage." }], "Subsubsection", PageWidth->WindowWidth, InitializationCell->True], Cell[TextData[{ "BeginPackage[", StyleBox["\"", FontFamily->"Courier New"], StyleBox["Hoops", FontFamily->"Times New Roman"], StyleBox["`\"", FontFamily->"Courier New"], "];" }], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[TextData[{ "Hoops::usage = \"", StyleBox["Hoops", FontFamily->"Times New Roman"], ".m is a package that creates a database of over 80 hoops \ (symmetry-conserving partial-fraction division algebras) or counter-examples \ as index-Tables (Cayley tables with signed indices as elements). It provides \ Hoop procedures for multiplication & division, powers & roots, & some \ polar-dual formulations, of generalized vectors.\";" }], "Input", PageWidth->WindowWidth, CellMargins->{{Inherited, 2}, {Inherited, Inherited}}, InitializationCell->True, FontFamily->"Courier New"] }, Closed]], Cell[CellGroupData[{ Cell["7.2 Alphabetic list of Usage for functions and procedures.", \ "Subsubsection", PageWidth->WindowWidth], Cell[BoxData[ \(a::usage = "\"; b::usage = "\<\>"; c::usage = "\<\>"; d::usage = "\<\>"; e::usage = "\<\>"; f::usage = "\<\>"; g::usage = "\<\>"; h::usage = "\<\>"; i::usage = "\<\>"; j::usage = "\<\>"; k::usage = "\<\>"; l::usage = "\<\>"; m::usage = "\<\>"; o::usage = "\<\>"; p::usage = "\<\>"; q::usage = "\<\>"; r::usage = "\<\>"; s::usage = "\<\>"; t::usage = "\<\>"; u::usage = "\<\>"; v::usage = "\<\>"; w::usage = "\<\>"; x::usage = "\<\>"; y::usage = "\<\>";\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ alph::usage=\"A list of 24 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StyleBox[\"w\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)to convert symbolic functions to \ specific cases.\n Extended so that /.as[v_,w_] replaces elements of v by \ elements of w.\>\""}], ";"}], TraditionalForm]], "Input", InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ \(\(bf::usage = "\";\)\)], "Input",\ PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{\(hoopInverse::usage\), "=", "\"\<\!\(\* StyleBox[\"hoopInverse\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"[\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"A_\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\"]\",\nFontSlant->\"Plain\"]\)\!\(\* StyleBox[\" \",\nFontSlant->\"Plain\"]\)finds the multiplicative left inverse \ Ainv of A. hoopTimes[Ainv,B] divides B by A and creates remainders Rl & Rr to \ conserve any sizes that are lost from A or B\>\""}], ";"}]], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ StyleBox[\(hoopPower::usage = \*"\"\\"";\), ShowStringCharacters->True, NumberMarks->True, FontSlant->"Plain"]], "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ StyleBox[\(hoopTimes::usage\ = \ "\";\), FontSlant->"Plain"]], "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ \(\(gi::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(gp::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(hoopNames::usage = "\";\)\)], \ "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ RowBox[{ RowBox[{\(hoopTbl::usage\), "=", "\"\\"Italic\"]\)e as the indexTable of the active \ hoop.\>\""}], ";"}]], "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ \(\(genCos::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(genSin::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(hmin::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell[BoxData[ \(\(J::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(JJ::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(js::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(mm::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(nn::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(plex::usage = "\";\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(r3::usage = "\";\)\)], "Input", PageWidth->WindowWidth, 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Create Database.", FontFamily->"Times New Roman"]], "Subsection", PageWidth->WindowWidth, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["8", FontFamily->"Times New Roman"], ".1 Begin Private." }], "Subsubsection", PageWidth->WindowWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(Begin["\<`Private`\>"];\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Unprotect[NumberQ, Positive, Abs, Sign, Re, Im];\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["8", FontFamily->"Times New Roman"], ".2 Data File,", StyleBox[" ", FontWeight->"Plain"], StyleBox["alph, hoopNames, Hoop[name].", FontSlant->"Italic"] }], "Subsubsection", PageWidth->WindowWidth, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, InitializationCell->True], Cell[BoxData[ \(\(alph = (*\ Dummy\ variables\ used\ in\ all\ symbolic\ tables\ and\ functions*) \ {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["Data is supplied for the following Hoops:-", "Text", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(hoopNames = {"\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\"};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Many more hoops are listed in [3], but not included here because they do not \ have known polar forms or are too closely related to those that are included \ to be of interest.\ \>", "Text", PageWidth->WindowWidth], Cell[TextData[{ "Nomenclature.\nC", StyleBox["n", FontSlant->"Italic"], " is a cyclic group; C", StyleBox["n", FontSlant->"Italic"], "C", StyleBox["m", FontSlant->"Italic"], " (etc.) are direct compositions; C", StyleBox["n", FontSlant->"Italic"], "iC", StyleBox["m", FontSlant->"Italic"], " are indirect compositions; D", StyleBox["n", FontSlant->"Italic"], " is the dihedral group with 2", StyleBox["n", FontSlant->"Italic"], " elements; Q", StyleBox["n ", FontSlant->"Italic"], "is the generalazed quaternion group with ", StyleBox["n", FontSlant->"Italic"], " elements; Dav, O4 & O8 are tables with 2, 4 & 8 complex planes; P", StyleBox["n", FontSlant->"Italic"], " are tables created from Pauli-\[Sigma] matrices; CL", StyleBox["m", FontSlant->"Italic"], " are Clifford algebras. Names ending in ", StyleBox["r, j, c", FontSlant->"Italic"], " are signed tables obtained from hoops by 2, 3, or 4-fold collapse. D3Mn \ and Q8M2 are nonassociative Moufang loops; the first is not conservative; the \ second is Chein-doubled Q8 and folds to split-octonion algebra Octi." }], "Text", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {2, 1, {{1, 2}, {2, 1}}, {a - b, a + b}, {a - b, a + b}, {\((a + b)\)/2, \((\(-a\) + b)\)/2}, {1, 2, 1, 1}, {1, 1}, {a, \(-b\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {2, 2, {{1, 2}, {2, \(-1\)}}, {\(-a\^2\) - b\^2}, {a\^2 + b\^2, ArcTan[a, b]}, {Sqrt[a] Cos[b], Sqrt[a] Sin[b]}, {1, \(-2\), 1}, {1, 0. }, {a, \(-b\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3\"]={3,1,{{1,2,3},{2,3,1},{3,1,2}},{a+b+c,(1/2)*((a-b)^2+(b-c)^2+(-a+\ c)^2)},{a+b+c,(1/2)*((a-b)^2+(b-c)^2+(-a+c)^2),ArcTan[2*a-b-c,(-Sqrt[3])*(b-c)\ ]},{(1/3)*(a+2*Sqrt[b]*Cos[c]),(1/3)*(a+2*Sqrt[b]*Cos[c+(2*Pi)/3]),(1/3)*(a+2*\ Sqrt[b]*Cos[c-(2*Pi)/3])},{1,3,2,1,1},{1,1,0.},{a,c,b}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {3, 2, {{1, 2, 3}, {2, 3, \(-1\)}, {3, \(-1\), \(-2\)}}, {a - b + c, \((1/2)\)*\((\((a + b)\)^2 + \((b + c)\)^2 + \((a - c)\)^2)\)}, {a - b + c, \((1/2)\)*\((\((a + b)\)^2 + \((b + c)\)^2 + \((a - c)\)^2)\), ArcTan[2*a + b - c, \((\(-Sqrt[3]\))\)*\((b + c)\)]}, {\((1/3)\)*\((a + 2*Sqrt[b]*Cos[c])\), \((1/3)\)*\((\(-a\) - 2*Sqrt[b]*Cos[c - 2*Pi/3])\), \((1/3)\)*\((a + 2*Sqrt[b]*Cos[c - 2*Pi/3])\)}, {1, \(-3\), \(-2\), 1, 1}, {1, 1, 0. }, {a, \(-c\), \(-b\)}};\)\)], "Input", PageWidth->WindowWidth], Cell["\<\ Hoop[\"C4C3c\"]={3,4,{{1,2,3},{2,3*I,-1},{3,-1,2*I}},{a+I*b+I*c,((a-I*b)^2-(b-\ c)^2+(a-I*c)^2)/2},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(a+c)^2)/2,ArcTan[2*a+I*b+c,\ Sqrt[3]*((-I)*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/\ 3]),(-a-2*Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,-3,-2,1,1},{1,1,0.},Null};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C3j\"]={3,5,{{1,2,3},{2,3*J^2,J},{3,J,2*^2}},{(a+b J^2+c J^2),((a-b \ J^2)^2+(b J^2-c J^2)^2+(c J^2-a)^2)/2},,,{1,3J,2J,1},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C9j\"]={3,6,{{1,2,3},{2,3*J^2,1},{3,1,2*J}},{a^3-3*a*b*c+c^3*J+b^3*J^2}\ ,,,{1,3,2,1},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3i\"]={3,9,{{1,2,3},{2,3,I},{3,I,2*I}},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(\ a+c)^2)/2},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(a+c)^2)/2,ArcTan[2*a+I*b+c,Sqrt[3]*(\ (-I)*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3]),(-a-2*\ Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,-3*I,-2*I,1,1},{1,1,0.},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C12c\"]={3,8,{{1,2,3},{2,3,-I},{3,-I,-2*I}},{a+I*b-c,((a-I*b)^2+(a+c)^\ 2+(I*b+c)^2)/2},{a+I*b-c,((a-I*b)^2+(a+c)^2+(I*b+c)^2)/2,ArcTan[2*a-I*b+c,\ Sqrt[3]*(I*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(-I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3])\ ,(-a-2*Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,3*I,2*I,1,1},{1,1,0.},};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"K\"]={4,1,{{1,2,3,4},{2,1,4,3},{3,4,1,2},{4,3,2,1}},{a+b-c-d,a-b+c-d,a-\ b-c+d,a+b+c+d},{a+b-c-d,a-b+c-d,a-b-c+d,a+b+c+d},{(a+b+c+d)/4,(a-b-c+d)/4,(-a+\ b-c+d)/4,(-a-b+c+d)/4},{1,2,3,4,1,1,1,1},{1,1,1,1},{a,-b,-c,d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C4\"]={4,2,{{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}},{a+b+c+d,a-b+c-d,(\ a-c)^2+(b-d)^2},{a+b+c+d,a-b+c-d,(a-c)^2+(b-d)^2,ArcTan[a-c,b-d]},{(a+b+2*\ Sqrt[c]*Cos[d])/4,(a-b+2*Sqrt[c]*Sin[d])/4,(a+b-2*Sqrt[c]*Cos[d])/4,(a-b-2*\ Sqrt[c]*Sin[d])/4},{1,4,3,2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Qr\"]={4,3,{{1,2,3,4},{2,-1,4,-3},{3,-4,-1,2},{4,3,-2,-1}},{a^2+b^2+c^\ 2+d^2}, {a^2+b^2+c^2+d^2,ArcTan[a,b],c^2+d^2,ArcTan[c,d]},{Sqrt[a-c]*Cos[b],Sqrt[a-c]*\ Sin[b],Sqrt[c]*Cos[d],Sqrt[c]*Sin[d]},{1,-2,-3,-4,2},{1,0.,1,0.},{a,-b,-c,-d}}\ ;\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Dav\"]={4,6,{{1,2,3,4},{2,-1,4,-3},{3,4,-1,-2},{4,-3,-2,1}},{(b+c)^2+(\ a-d)^2,(b-c)^2+(a+d)^2},{(b+c)^2+(a-d)^2,ArcTan[a-d,b+c],(b-c)^2+(a+d)^2,\ ArcTan[a+d,b-c]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d])/2,(Sqrt[a]*Sin[b]+Sqrt[c]*\ Sin[d])/2,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d])/2,(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d])\ /2},{1,-2,-3,4,1,1},{1,0.,1,0.},{a,-b,-c,d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"P4\"]={4,8,{{1,2,3,4},{2,1,-4*I,3*I},{3,4*I,1,-2*I},{4,-3*I,2*I,1}},{a^\ 2-b^2-c^2-d^2},{Sqrt[(b-c)^2+(c-d)^2+(-b+d)^2]/Sqrt[2],ArcTan[(Sqrt[2]*a)/\ Sqrt[(b-c)^2+(c-d)^2+(-b+d)^2]],(b+c+d)/3,If[c==d,0,ArcTan[2*b-c-d,(-Sqrt[3])*\ (c-d)]]},{a*Tan[b],c+(2*a*Cos[d])/3,c+(2*a*Cos[d+(2*Pi)/3])/3,c+(2*a*Cos[d+(4*\ Pi)/3])/3},{1,2,3,4,2},{1,0.,1,0.},{a,-b,-c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"P16c\"]={4,13,{{1,2,3,4},{2,-1,-4*I,-3*I},{3,4*I,-1,2*I},{4,3*I,-2*I,1}\ },{a^2+b^2+c^2-d^2},{a^2+b^2,If[b!=0,ArcTan[a,b],0],c^2-d^2,If[d!=0,arcTanh[c,\ d],0]},{Sqrt[a]*Cos[b],Sqrt[a]*Sin[b],Sqrt[c]*Cosh[d],Sqrt[c]*Sinh[d]},{1,-2,-\ 3,4,2},{1,0.,1,0.},{a,-b,-c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"O2\"]={4,5,{{1,2,3,4},{2,-1,4,-3},{3,4,1,2},{4,-3,2,-1}},{(a-c)^2+(b-d)\ ^2,(a+c)^2+(b+d)^2},{(a-c)^2+(b-d)^2,ArcTan[a-c,b-d],(a+c)^2+(b+d)^2,ArcTan[a+\ c,b+d]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d])/2,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d])/2,(\ -(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d])/2,(-(Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d])/2},{1,-\ 2,3,-4,1,1},{1,0.,1,0.},{a,-b,-c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL2\"]={4,7,{{1,2,3,4},{2,1,4,3},{3,-4,-1,2},{4,-3,-2,1}},{a^2-b^2+c^2-\ d^2},{a^2-b^2+c^2-d^2,ArcTan[a,c],b^2+d^2,ArcTan[b,d]},{Sqrt[a+c]*Cos[b],Sqrt[\ c]*Cos[d],Sqrt[a+c]*Sin[b],Sqrt[c]*Sin[d]},{1,2,-3,4,2},{1,0.,1,0.},{a,-b,-c,-\ d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C4C2r\"]={4,9,{{1,2,3,4},{2,3,-4,-1},{3,-4,1,-2},{4,-1,-2,3}},{a+b+c-d,\ a-b+c+d,(a-c)^2+(b+d)^2},{a+b+c-d,a-b+c+d,(a-c)^2+(b+d)^2,ArcTan[a-c,b+d]},{(\ a+b+2*Sqrt[c]*Cos[d])/4,(a-b+2*Sqrt[c]*Sin[d])/4,(a+b-2*Sqrt[c]*Cos[d])/4,(-a+\ b+2*Sqrt[c]*Sin[d])/4},{1,-4,3,-2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C4j\"]={4,10, {{1,2,3,4},{2,-3,4,-1},{3,4,1,2},{4,-1,2,-3}},{a+b-c-d,a-b-c+d,(a+c)^2+(b+d)^\ 2},{a+b-c-d,a-b-c+d,(a+c)^2+(b+d)^2,ArcTan[a+c,b+d]},{(a+b+2*Sqrt[c]*Cos[d])/\ 4,(a-b+2*Sqrt[c]*Sin[d])/4,(-a-b+2*Sqrt[c]*Cos[d])/4,(-a+b+2*Sqrt[c]*Sin[d])/\ 4},{1,-4,3,-2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C8c\"]={4,11,{{1,2,3,4},{2,3,4,-1},{3,4,-1,-2},{4,-1,-2,-3}},{a^2+b^2+\ c^2+d^2+Sqrt[2]*(a*b+b*c-a*d+c*d),a^2+b^2+c^2+d^2-Sqrt[2]*(a*b+b*c-a*d+c*d)},,\ ,{1,-4,-3,-2,1,1},{11,1,0.},{a,-b,c,-d}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C5\"]={5,1,{{1,2,3,4,5},{2,3,4,5,1},{3,4,5,1,2},{4,5,1,2,3},{5,1,2,3,4}\ },{a+b+c+d+e,(-(5/4))*(a*b-a*c+b*c-a*d-b*d+c*d+a*e-b*e-c*e+d*e)^2+(1/16)*((a-\ b)^2+(a-c)^2+(b-c)^2+(b-d)^2+(c-d)^2+(-a+d)^2+(c-e)^2+(d-e)^2+(-a+e)^2+(-b+e)^\ 2)^2},,{a+b*Sin[c],a+b*Sin[c+2\[Pi]/5],a+b*Sin[c+4\[Pi]/5],a+b* Sin[c+6\[Pi]/5],a+b*Sin[c+8\[Pi]/5]}/5 ,{1,5,4,3,2,1,1},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"D3\"]={6,1,{{1,2,3,4,5,6},{2,1,6,5,4,3},{3,4,5,6,1,2},{4,3,2,1,6,5},{5,\ 6,1,2,3,4},{6,5,4,3,2,1}},{a+b+c+d+e+f,a-b+c-d+e-f,((a-c)^2-(b-d)^2+(c-e)^2+(-\ a+e)^2-(d-f)^2-(-b+f)^2)/2},{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,ArcTan[2*a-c-e,Sqrt[3]*(-c+e)],((a-c)^2+(c-e)^2+\ (-a+e)^2)/2,ArcTan[2*b-d-f,Sqrt[3]*(-d+f)]},{(a+b+4*Sqrt[e]*Cos[d])/6,(a-b+4*\ Sqrt[-c+e]*Cos[f])/6,(a+b+4*Sqrt[e]*Cos[d+(2*Pi)/3])/6,(a-b+4*Sqrt[-c+e]*Cos[\ f+(2*Pi)/3])/6,(a+b+4*Sqrt[e]*Cos[d-(2*Pi)/3])/6,(a-b+4*Sqrt[-c+e]*Cos[f-(2*\ Pi)/3])/6},{1,2,5,4,3,6,1,1,2},{1,1,1,0.,1,0.},,{(a+c+e)^2-(b+d+f)^2,arcTanh[\ a+c+e,b+d+e],((a-c)^2+(c-e)^2+(-a+e)^2)/2,ArcTan[2*a-c-e,Sqrt[3]*(e-c)],((b-d)\ ^2+(d-f)^2+(-b+f)^2)/2,ArcTan[2*a-c-e,Sqrt[3]*(c-e)]+ArcTan[2*b-d-f,Sqrt[3]*(\ f-d)]},{2*Sqrt[c]*Cos[d]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f]+Sqrt[a]*Sinh[b],2*\ Sqrt[c]*Cos[d+2*Pi/3]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f+2*Pi/3]+Sqrt[a]*Sinh[\ b],2*Sqrt[c]*Cos[d-2*Pi/3]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f-2*Pi/3]+Sqrt[a]*\ Sinh[b]}/3};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C2\"]={6,2,{{1,2,3,4,5,6},{2,3,1,5,6,4},{3,1,2,6,4,5},{4,5,6,1,2,3},{\ 5,6,4,2,3,1},{6,4,5,3,1,2}},{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},{a+b+c+d+e+f,a+b+c-d-\ e-f,((a-b+d-e)^2+(b-c+e-f)^2+(-a+c-d+f)^2)/2,ArcTan[2*a-b-c+2*d-e-f,Sqrt[3]*(-\ b+c-e+f)],((a-b-d+e)^2+(-a+c+d-f)^2+(b-c-e+f)^2)/2,ArcTan[2*a-b-c-2*d+e+f,\ Sqrt[3]*(-b+c+e-f)]},{(a+b+2*Sqrt[c]*Cos[d]+2*Sqrt[e]*Cos[f])/6,(a+b+2*Sqrt[c]\ *Cos[d+(2*Pi)/3]+2*Sqrt[e]*Cos[f+(2*Pi)/3])/6,(a+b+2*Sqrt[c]*Cos[d-(2*Pi)/3]+\ 2*Sqrt[e]*Cos[f-(2*Pi)/3])/6,(a-b+2*Sqrt[c]*Cos[d]-2*Sqrt[e]*Cos[f])/6,(a-b+2*\ Sqrt[c]*Cos[d+(2*Pi)/3]-2*Sqrt[e]*Cos[f+(2*Pi)/3])/6,(a-b+2*Sqrt[c]*Cos[d-(2*\ Pi)/3]-2*Sqrt[e]*Cos[f-(2*Pi)/3])/6},{1,3,2,4,6,5,1,1,1,1},{1,1,1,0.,1,0.},{a,\ c,b,d,f,e}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Q12c\"]={6,3,{{1,2,3,4,5,6},{2,3,1,6,4,5},{3,1,2,5,6,4},{4,5,6,-1,-2,-\ 3}, {5,6,4,-3,-1,-2},{6,4,5,-2,-3,-1}},{(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}, {(a+b+c)^2+(d+e+f)^2,ArcTan[a+b+c,d+e+f], ((a-b)^2+(b-c)^2+(-a+c)^2)/2+((d-e)^2+(e-f)^2+(-d+f)^2)/2, If[c==b,0,ArcTan[2*a-b-c,Sqrt[3]*(c-b)]],((d-e)^2+(e-f)^2+(-d+f)^2)/2, If[e==f,0,ArcTan[2*d-e-f,Sqrt[3]*(f-e)]]}, {(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d])/3,(Sqrt[a]*Cos[b])/3+ (2*Sqrt[c-e]*Cos[d+(2*Pi)/3])/3,(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d-(2*Pi)/\ 3])/3, (2*Sqrt[e]*Cos[f])/3+(Sqrt[a]*Sin[b])/3,(2*Sqrt[e]*Cos[f+(2*Pi)/3])/3+(Sqrt[a]\ *Sin[b])/3, (2*Sqrt[e]*Cos[f-(2*Pi)/3])/3+(Sqrt[a]*Sin[b])/3},{1,3,2,-4,-5,-6,1,2}, {1,2,1,0.,1,0.},{a,c,b,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Q12r\"]={6,4,{{1,2,3,4,5,6},{2,-1,4,-3,6,-5},{3,-6,5,2,-1,4},{4,5,6,-1,\ -2,-3},{5,-4,-1,-6,-3,2},{6,3,-2,5,-4,-1}},{(a-c+e)^2+(b-d+f)^2,((a+c)^2+(b+d)\ ^2+(a-e)^2+(c+e)^2+(b-f)^2+(d+f)^2)/2},{(a-c+e)^2+(b-d+f)^2,If[b+f==d,0,\ ArcTan[a-c+e,b-d+f]],((a+c)^2+(a-e)^2+(c+e)^2)/2+((b+d)^2+(b-f)^2+(d+f)^2)/2,\ If[c==-e,0,ArcTan[2*a+c-e,Sqrt[3]*(c+e)]],((b+d)^2+(b-f)^2+(d+f)^2)/2,If[d==-\ f,0,ArcTan[2*b+d-f,Sqrt[3]*(d+f)]]},{(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d])/\ 3,(2*Sqrt[e]*Cos[f])/3+(Sqrt[a]*Sin[b])/3,-(Sqrt[a]*Cos[b])/3-(2*Sqrt[c-e]*\ Cos[d+(2*Pi)/3])/3,(-2*Sqrt[e]*Cos[f+(2*Pi)/3])/3-(Sqrt[a]*Sin[b])/3,(Sqrt[a]*\ Cos[b])/3+(2*Sqrt[c-e]*Cos[d-(2*Pi)/3])/3,(2*Sqrt[e]*Cos[f-(2*Pi)/3])/3+(Sqrt[\ a]*Sin[b])/3},{1,-2,-5,-4,-3,-6,1,2},{1,2,1,0.,1,0.},};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"D6r\"]={6,5,{{1,2,3,4,5,6},{2,1,4,3,6,5},{3,-6,5,2, \ -1,4},{4,-5,6,1,-2,3},{5,-4,-1,-6,-3,2},{6,-3,-2,-5,-4,1}},{a-b-c+d+e-f,a+b-c-\ d+e+f,(1/2)*((a+c)^2-(b+d)^2+(a-e)^2+(c+e)^2-(b-f)^2-(d+f)^2)},,,{1,2,-5,4,-3,\ 6,1,1,2},{1, 1,1,0.,1,0.},};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True, FontFamily->"Courier New"], Cell["\<\ Hoop[\"KC3c\"]={6,6,{{1,2,3,4,5,6},{2,3,-1,5,6,-4},{3,-1,-2,6,-4,-5},{4,5,6,1,\ 2,3},{5,6,-4,2,3,-1},{6,-4,-5,3,-1,-2}},{a-b+c-d+e-f,a-b+c+d-e+f,((a+b-d-e)^2+\ (b+c-e-f)^2+(a-c-d+f)^2)/2,((a+b+d+e)^2+(a-c+d-f)^2+(b+c+e+f)^2)/2},,,{1,-3,-\ 2,4,-6,-5,1,1,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3Kr\"]={6,7,{{1,2,3,4,5,6},{2,-3,1,5,-6,4},{3,1,-2,6,4,-5},{4,5,6,1,2,\ 3}, {5,-6,4,2,-3,1},{6,4,-5,3,1,-2}},{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+(b-c+e-f)^2+(a+c+d+f)^2)/2},,,{1,3,2,4,6,5,1,1,1,1},{1,1,1,0.,1,\ 0.}, {a,c,b,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C4c\"]:=(*10/1/6*){6,8, {{1,2,3,4,5,6},{2,3,1,5,6,4},{3,1,2,6,4,5},{4,5,6,-1,-2,-3},{5,6,4,-2,-3,-1},{\ 6,4,5,-3,-1,-2}}, {(a+b+c)^2+(d+e+f)^2, \ 3*((a-b+(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]-d+e)^2)/4, \ 3*((a-b-(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]+d-e)^2)/ \ 4(*,((a-b)^2+(b-c)^2+(c-a)^2+(d-e)^2+(e-f)^2+(f-d)^2)^2/ 4-3*(c*(d-e)+a*(e-f)+b*(f-d))^2*)}, {(a+b+c)^2+(d+e+f)^2, If[d+e+f\[Equal]0,If[a+b+c<0,Pi,0],ArcTan[a+b+c,d+e+f]], \ 3*((a-b+(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]-d+e)^2)/4, If[(a+b-2*c)/Sqrt[3]-d+e\[Equal]0,If[a-b+(d+e-2*f)/Sqrt[3]<0,Pi,0], \ ArcTan[a-b+(d+e-2*f)/Sqrt[3],(a+b-2*c)/Sqrt[3]-d+e]]-Pi/6, \ 3*((a-b-(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]+d-e)^2)/4, If[(a+b-2*c)/Sqrt[3]+d-e\[Equal]0, If[a-b-(d+e-2*f)/Sqrt[3]<0,Pi,0], \ ArcTan[a-b-(d+e-2*f)/Sqrt[3],(a+b-2*c)/Sqrt[3]+d-e]]-Pi/6}, \ {Sqrt[a/9]*Cos[b]+Sqrt[c/12]*Cos[d+Pi/6]+Sqrt[c/36]*Sin[d+Pi/6]+Sqrt[e/12]*\ Cos[f+Pi/6]+Sqrt[e/36]*Sin[f+Pi/6], \ Sqrt[a/9]*Cos[b]-Sqrt[c/12]*Cos[d+Pi/6]+Sqrt[c/36]*Sin[d+Pi/6]-Sqrt[e/12]*Cos[\ f+Pi/6]+Sqrt[e/36]*Sin[f+Pi/6], \ Sqrt[a/9]*Cos[b]-Sqrt[c/9]*Sin[d+Pi/6]-Sqrt[e/9]*Sin[f+Pi/6], \ Sqrt[a/9]*Sin[b]+Sqrt[c/36]*Cos[d+Pi/6]-Sqrt[c/12]*Sin[d+Pi/6]-Sqrt[e/36]*Cos[\ f+Pi/6]+Sqrt[e/12]*Sin[f+Pi/6], \ Sqrt[a/9]*Sin[b]+Sqrt[c/36]*Cos[d+Pi/6]+Sqrt[c/12]*Sin[d+Pi/6]-Sqrt[e/36]*Cos[\ f+Pi/6]-Sqrt[e/12]*Sin[f+Pi/6], \ Sqrt[a/9]*Sin[b]-Sqrt[c/9]*Cos[d+Pi/6]+Sqrt[e/9]*Cos[f+Pi/6]}, {1,3,2,-4,-6,-5,1,1,1,0},{1,0.,1,0.,1,0.},{a,b,c,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True, LineSpacing->{1, 0}, FontWeight->"Bold"], Cell[BoxData[ \(\(Hoop["\"] := (*\(10/1\)/6*) \[IndentingNewLine]{6, 8, {{1, 2, 3, 4, 5, 6}, {2, 3, 1, 5, 6, 4}, {3, 1, 2, 6, 4, 5}, {4, 5, 6, \(-1\), \(-2\), \(-3\)}, {5, 6, 4, \(-2\), \(-3\), \(-1\)}, {6, 4, 5, \(-3\), \(-1\), \(-2\)}}, \[IndentingNewLine]{\((a + b + c)\)^2 + \((d + e + f)\)^2, 3*\((\((a - b + \((d + e - 2*f)\)/ Sqrt[3])\)^2 + \((\((a + b - 2*c)\)/Sqrt[3] - d + e)\)^2)\)/4, 3*\((\((a - b - \((d + e - 2*f)\)/ Sqrt[3])\)^2 + \((\((a + b - 2*c)\)/Sqrt[3] + d - e)\)^2)\)/ 4 (*\(,\)\(\((\((a - b)\)^2 + \((b - c)\)^2 + \((c - a)\)^2 + \((d - e)\)^2 + \((e - f)\)^2 + \((f - d)\)^2)\)^2/4 - 3*\((c*\((d - e)\) + a*\((e - f)\) + b*\((f - d)\))\)^2\)*) }, {\((a + b + c)\)^2 + \((d + e + f)\)^2, If[d + e + f \[Equal] 0, If[a + b + c < 0, Pi, 0], ArcTan[a + b + c, d + e + f]], 3*\((\((a - b + \((d + e - 2*f)\)/ Sqrt[3])\)^2 + \((\((a + b - 2*c)\)/Sqrt[3] - d + e)\)^2)\)/4, If[\((a + b - 2*c)\)/Sqrt[3] - d + e \[Equal] 0, If[a - b + \((d + e - 2*f)\)/Sqrt[3] < 0, Pi, 0], ArcTan[a - b + \((d + e - 2*f)\)/Sqrt[3], \((a + b - 2*c)\)/ Sqrt[3] - d + e]] - Pi/6, 3*\((\((a - b - \((d + e - 2*f)\)/ Sqrt[3])\)^2 + \((\((a + b - 2*c)\)/Sqrt[3] + d - e)\)^2)\)/4, If[\((a + b - 2*c)\)/Sqrt[3] + d - e \[Equal] 0, If[a - b - \((d + e - 2*f)\)/Sqrt[3] < 0, Pi, 0], ArcTan[a - b - \((d + e - 2*f)\)/Sqrt[3], \((a + b - 2*c)\)/ Sqrt[3] + d - e]] - Pi/6}, \[IndentingNewLine]{Sqrt[a/9]*Cos[b] + Sqrt[c/12]*Cos[d + Pi/6] + Sqrt[c/36]*Sin[d + Pi/6] + Sqrt[e/12]*Cos[f + Pi/6] + Sqrt[e/36]*Sin[f + Pi/6], \[IndentingNewLine]Sqrt[a/9]*Cos[b] - Sqrt[c/12]*Cos[d + Pi/6] + Sqrt[c/36]*Sin[d + Pi/6] - Sqrt[e/12]*Cos[f + Pi/6] + Sqrt[e/36]*Sin[f + Pi/6], \[IndentingNewLine]Sqrt[a/9]*Cos[b] - Sqrt[c/9]*Sin[d + Pi/6] - Sqrt[e/9]*Sin[f + Pi/6], \[IndentingNewLine]Sqrt[a/9]*Sin[b] + Sqrt[c/36]*Cos[d + Pi/6] - Sqrt[c/12]*Sin[d + Pi/6] - Sqrt[e/36]*Cos[f + Pi/6] + Sqrt[e/12]*Sin[f + Pi/6], \[IndentingNewLine]Sqrt[a/9]*Sin[b] + Sqrt[c/36]*Cos[d + Pi/6] + Sqrt[c/12]*Sin[d + Pi/6] - Sqrt[e/36]*Cos[f + Pi/6] - Sqrt[e/12]*Sin[f + Pi/6], \[IndentingNewLine]Sqrt[a/9]*Sin[b] - Sqrt[c/9]*Cos[d + Pi/6] + Sqrt[e/9]*Cos[f + Pi/6]}, \[IndentingNewLine]{1, 3, 2, \(-4\), \(-6\), \(-5\), 1, 1, 1, 0}, {1, 0. , 1, 0. , 1, 0. }, {a, b, c, \(-d\), \(-e\), \(-f\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True, LineSpacing->{1, 0}, FontWeight->"Bold"], Cell["\<\ Hoop[\"D3C2c\"]={6,9,{{1,2,3,4,5,6},{2,3,1,5,-6,-4},{3,1,2,-6,4,-5},{4,-6,5,1,\ 3,-2},{5,4,-6,2,1,-3},{6,-5,-4,-3,-2,1}},{a+b+c+d+e-f,a+b+c-d-e+f,((a-b)^2+(a-\ c)^2+(b-c)^2-(d-e)^2-(d+f)^2-(e+f)^2)/2},,,{1,3,2,4,5,6,1,1,2},{1,1,1,0.,1,0.}\ ,{a,c,b,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Clm3r\"]={6,10,{{1,2,3,4,5,6},{2,-3,1,5,-6,4},{3,1,-2,6,4,-5},{4,5,6,-\ 1,-2,-3},{5,-6,4,-2,3,-1},{6,4,-5,-3,-1,2}},{(a-b-c)^2+(d-e-f)^2,-3*(-b*d+c*d+\ a*e+c*e-a*f-b*f)^2+((a+b)^2+(b-c)^2+(a+c)^2+(d+e)^2+(e-f)^2+(d+f)^2)^2/4},,,{\ 1,3,2,-4,-6,-5,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Cl3r\"]={6,11,{{1,2,3,4,5,6},{2,3,-1,5,6,-4},{3,-1,-2,6,-4,-5},{4,5,6,-\ 1,-2,-3},{5,6,-4,-2,-3,1},{6,-4,-5,-3,1,2}},{(a-b+c)^2+(d-e+f)^2,-3*(-b*d-c*d+\ a*e-c*e+a*f+b*f)^2+((a+b)^2+(a-c)^2+(b+c)^2+(d+e)^2+(-d+f)^2+(e+f)^2)^2/4},,,{\ 1,-3,-2,-4,6,5,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"g2401c\"]={6,12, {{1,2,3,4,5,6},{2,3,1,6,4,5},{3,1,2,5,6,4},{4,5,6,I,2*I,3*I},{5,6,4,3*I,I,2*I}\ ,{6,4,5,2*I,3*I,I}},{((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},,,{1,3,2,-4*I,-5*I,-6*I,2,1},{1,1,1,0.,1,0.},{a,c,b,-d,-\ e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C8\"]={8,1,{{1,2,3,4,5,6,7,8},{2,3,4,5,6,7,8,1},{3,4,5,6,7,8,1,2}, {4,5,6,7,8,1,2,3},{5,6,7,8,1,2,3,4},{6,7,8,1,2,3,4,5}, {7,8,1,2,3,4,5,6},{8,1,2,3,4,5,6,7}},{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+4*((-b+f)*(c-g)+(a-e)*(d-h))* ((a-e)*(b-f)+(c-g)*(d-h))+((b-f)^2+(d-h)^2)^2},,, {1,8,7,6,5,4,3,2,1,1,1,1},{1,1,1,1,0.,1,0.},{a,-b,c,-d,e,-f,g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C4C2\"]={8,2,{{1,2,3,4,5,6,7,8},{2,3,4,1,6,7,8,5},{3,4,1,2,7,8,5,6}, {4,1,2,3,8,5,6,7},{5,6,7,8,1,2,3,4},{6,7,8,5,2,3,4,1}, {7,8,5,6,3,4,1,2},{8,5,6,7,4,1,2,3}},{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}, {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, ArcTan[a-c-e+g,b-d-f+h],(a-c+e-g)^2+(b-d+f-h)^2, ArcTan[a-c+e-g,b-d+f-h]},{(a+b+c+d+2*Sqrt[e]*Cos[f]+2*Sqrt[g]*Cos[h])/8, (a-b+c-d+2*Sqrt[e]*Sin[f]+2*Sqrt[g]*Sin[h])/8, (a+b+c+d-2*Sqrt[e]*Cos[f]-2*Sqrt[g]*Cos[h])/8, (a-b+c-d-2*Sqrt[e]*Sin[f]-2*Sqrt[g]*Sin[h])/8, (a+b-c-d-2*Sqrt[e]*Cos[f]+2*Sqrt[g]*Cos[h])/8, (a-b-c+d-2*Sqrt[e]*Sin[f]+2*Sqrt[g]*Sin[h])/8, (a+b-c-d+2*Sqrt[e]*Cos[f]-2*Sqrt[g]*Cos[h])/8, (a-b-c+d+2*Sqrt[e]*Sin[f]-2*Sqrt[g]*Sin[h])/8}, {1,4,3,2,5,8,7,6,1,1,1,1,1,1},{1,1,1,1,1,0.,1,0.}, {a,-b,c,-d,e,-f,g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"D4\"]={8,3,{{1,2,3,4,5,6,7,8},{2,1,8,7,6,5,4,3},{3,4,5,6,7,8,1,2}, {4,3,2,1,8,7,6,5},{5,6,7,8,1,2,3,4},{6,5,4,3,2,1,8,7}, {7,8,1,2,3,4,5,6},{8,7,6,5,4,3,2,1}},{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},{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,ArcTan[a-e,c-g],(b-f)^2+(d-h)^2,ArcTan[b-f,d-\ h]},{(a+b+c+d+4*Sqrt[e+g]*Cos[f]), (a+b-c-d+4*Sqrt[g]*Cos[h]),(a-b+c-d+4*Sqrt[e+g]*Sin[f]), (a-b-c+d+4*Sqrt[g]*Sin[h]),(a+b+c+d-4*Sqrt[e+g]*Cos[f]), (a+b-c-d-4*Sqrt[g]*Cos[h]),(a-b+c-d-4*Sqrt[e+g]*Sin[f]), (a-b-c+d-4*Sqrt[g]*Sin[h])}/8,{1,2,7,4,5,6,3,8,1,1,1,1,2},{1,1,1,1,1,0.,1,0.},\ };\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Q8\"]={8,4,{{1,2,3,4,5,6,7,8},{2,5,8,3,6,1,4,7},{3,4,5,6,7,8,1,2}, {4,7,2,5,8,3,6,1},{5,6,7,8,1,2,3,4},{6,1,4,7,2,5,8,3}, {7,8,1,2,3,4,5,6},{8,3,6,1,4,7,2,5}},{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},{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]}, {(a+b+c+d+4*Sqrt[e]*Cos[f]),(a-b-c+d+4*Sqrt[e]*Sin[f]), (a+b-c-d+4*Sqrt[g]*Cos[h]),(a-b+c-d+4*Sqrt[g]*Sin[h]), (a+b+c+d-4*Sqrt[e]*Cos[f]),(a-b-c+d-4*Sqrt[e]*Sin[f]), (a+b-c-d-4*Sqrt[g]*Cos[h]),(a-b+c-d-4*Sqrt[g]*Sin[h])}/8,{1,6,7,8,5,2,3,4,1,1,\ 1,1,2},{1,1,1,1,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"KC2\"]={8,5,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,7,8,5,6}, {4,3,2,1,8,7,6,5},{5,6,7,8,1,2,3,4},{6,5,8,7,2,1,4,3}, {7,8,5,6,3,4,1,2},{8,7,6,5,4,3,2,1}},{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},{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}, {(a+b+c+d+e+f+g+h)/8,(a+b-c-d-e-f+g+h)/8, (a-b+c-d-e+f-g+h)/8,(a-b-c+d+e-f-g+h)/8, (-a+b+c-d+e-f-g+h)/8,(-a+b-c+d-e+f-g+h)/8, (-a-b+c+d-e-f+g+h)/8,(-a-b-c-d+e+f+g+h)/8}, {1,2,3,4,5,6,7,8,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1}, {a,b,c,d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C4C4c\"]={8,8,{{1,2,3,4,5,6,7,8},{2,3,4,1,6,7,8,5},{3,4,1,2,7,8,5,6},{\ 4,1,2,3,8,5,6,7},{5,6,7,8,-1,-2,-3,-4},{6,7,8,5,-2,-3,-4,-1},{7,8,5,6,-3,-4,-\ 1,-2},{8,5,6,7,-4,-1,-2,-3}},{(b-d+e-g)^2+(a-c-f+h)^2,(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,Which[b-d+e-g==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,\ ArcTan[aa$66,bb$66]],(b-d-e+g)^2+(a-c+f-h)^2,Which[b-d-e+g==0,0,aa$66==0,(Pi/\ 2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]],(a-b+c-d)^2+(e-f+g-h)^2,Which[e-f+g-\ h==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]],(a+b+c+d)^2+(e+f+\ g+h)^2,Which[e+f+g+h==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]\ ]}, {(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]),(-(Sqrt[e]*Cos[\ f])+Sqrt[g]*Cos[h]+Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]),(-(Sqrt[a]*Cos[b])-Sqrt[c]*\ Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]),(-(Sqrt[e]*Cos[f])+Sqrt[g]*Cos[h]-Sqrt[\ a]*Sin[b]-Sqrt[c]*Sin[d]),(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[\ g]*Sin[h]),(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]),(-\ (Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]),(Sqrt[a]*Cos[b]\ -Sqrt[c]*Cos[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])}/4,{1,4,3,2,-5,-8,-7,-6,1,1,1,\ 1},{1,0.,1,0.,1,0.,1,0.},{a,-b,c,-d,-e,f,-g,h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"KiC4c\"]={8,9,{{1,2,3,4,5,6,7,8},{2,1,4,3,8,7,6,5},{3,4,1,2,7,8,5,6},{\ 4,3,2,1,6,5,8,7},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-4,-3,-2,-1},{7,8,5,6,-3,-4,-\ 1,-2},{8,7,6,5,-2,-1,-4,-3}},{(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},{(a-c)^2-(b-d)^2+(e-g)^2-(f-h)^2,If[e==g,0,\ ArcTan[a-c,e-g]],(b-d)^2+(f-h)^2,If[f==h,0,ArcTan[b-d,f-h]],(a-b+c-d)^2+(e-f+\ g-h)^2,If[e+g==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==\ g+h,0,ArcTan[a+b+c+d,e+f+g+h]]},{(Sqrt[a+c]*Cos[b])/2+(Sqrt[e]*Cos[f])/4+(\ Sqrt[g]*Cos[h])/4,(Sqrt[c]*Cos[d])/2-(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,-(\ Sqrt[a+c]*Cos[b])/2+(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4, -(Sqrt[c]*Cos[d])/2-(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,(Sqrt[a+c]*Sin[b])/\ 2+(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4,(Sqrt[c]*Sin[d])/2-(Sqrt[e]*Sin[f])/4+\ (Sqrt[g]*Sin[h])/4,-(Sqrt[a+c]*Sin[b])/2+(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/\ 4,-(Sqrt[c]*Sin[d])/2-(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4},{1,2,3,4,-5,-8,-\ 7,-6,1,1,2},{1,0.,1,0.,1,0,1,0.},{a,b,c,d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C4iC4c\"]={8,10,{{1,2,3,4,5,6,7,8},{2,3,4,1,8,5,6,7},{3,4,1,2,7,8,5,6},\ {4,1,2,3,6,7,8,5},{5,6,7,8,-1,-2,-3,-4},{6,7,8,5,-4,-1,-2,-3},{7,8,5,6,-3,-4,-\ 1,-2},{8,5,6,7,-2,-3,-4,-1}},{(a-c)^2+(b-d)^2+(e-g)^2+(f-h)^2,(a-b+c-d)^2+(e-\ f+g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2},,,{1,4,3,2,-5,-6,-7,-8,2,1,1},{1,1,2,1,0.,1,\ 0.},{a,-b,c,-d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"QC2r\"]={8,11,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,-1,-2,-7,-8,5,\ 6},{4,3,-2,-1,-8,-7,6,5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,-5,-\ 6,3,4,-1,-2},{8,7,-6,-5,4,3,-2,-1}},{(a-b)^2+(c-d)^2+(e-f)^2+(g-h)^2,(a+b)^2+(\ c+d)^2+(e+f)^2+(g+h)^2},,,{1,2,-3,-4,-5,-6,-7,-8,2,2},{1,1,1,1,0.,1,0.},{a,b,-\ c,-d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Q8C2r\"]={8,12,{{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}}, {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,(b-c)^2+(a+e)^2+(d+g)^2+(f+h)^2}, {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,(b-c)^2+(a+e)^2,ArcTan[a+e,b-c],(d+g)^2+(f+h)^2,ArcTan[d+g,f+\ h]},{(a+b+c+d+4*Sqrt[e]*Cos[f]), (-a+b+c-d+4*Sqrt[e]*Sin[f]),(-a+b+c-d-4*Sqrt[e]*Sin[f]), (-a-b+c+d+4*Sqrt[g]*Cos[h]),(-a-b-c-d+4*Sqrt[e]*Cos[f]), (-a+b-c+d+4*Sqrt[g]*Sin[h]),(a+b-c-d+4*Sqrt[g]*Cos[h]), (a-b+c-d+4*Sqrt[g]*Sin[h])}/8,{1,3,2,-7,5,-8,-4,-6,1,1,1,1,2},{1,1,1,1,1,0.,1,\ 0.},{a,-b,-c,-d,e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"O4\"]={8,13,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,7,8,5,6},{4,\ 3,2,1,8,7,6,5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,5,6,-3,-4,-1,-\ 2},{8,7,6,5,-4,-3,-2,-1}},{(a-b-c+d)^2+(e-f-g+h)^2,(a-b+c-d)^2+(e-f+g-h)^2,(a+\ b-c-d)^2+(e+f-g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2}, {(a-b-c+d)^2+(e-f-g+h)^2,If[e+h==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==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==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==0,0,\ ArcTan[a+b+c+d,e+f+g+h]]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[\ g]*Cos[h])/4,(-(Sqrt[a]*Cos[b])-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/\ 4,(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(Sqrt[a]*\ Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(Sqrt[a]*Sin[b]+Sqrt[c]\ *Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4,(-(Sqrt[a]*Sin[b])-Sqrt[c]*Sin[d]+\ Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4,(-(Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d]-Sqrt[e]*\ Sin[f]+Sqrt[g]*Sin[h])/4,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]\ *Sin[h])/4},{1,2,3,4,-5,-6,-7,-8,1,1,1,1},{1,0.,1,0.,1,0.,1,0.},{a,b,c,d,-e,-\ f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"P8\"]={8,16,{{1,2,3,4,5,6,7,8},{2,1,-4*I,3*I,6,5,-8*I,7*I},{3,4*I,1,-2*\ I,7,8*I,5,-6*I},{4,-3*I,2*I,1,8,-7*I,6*I,5},{5,6,7,8,1,2,3,4},{6,5,-8*I,7*I,2,\ 1,-4*I,3*I},{7,8*I,5,-6*I,3,4*I,1,-2*I},{8,-7*I,6*I,5,4,-3*I,2*I,1}},{(a+e)^2-\ (b+f)^2-(c+g)^2-(d+h)^2,(a-e)^2-(b-f)^2-(c-g)^2-(d-h)^2},{(a+e)^2-(b+f)^2,\ arcTanh[a+e,b+f],(c+g)^2+(d+h)^2,ArcTan[c+g,d+h],(a-e)^2-(b-f)^2,arcTanh[a-e,\ b-f],(c-g)^2+(d-h)^2,ArcTan[c-g,d-h]},{(Sqrt[a]*Cosh[b]+Sqrt[e]*Cosh[f])/2,(\ Sqrt[a]*Sinh[b]+Sqrt[e]*Sinh[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*\ Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[a]*Cosh[b]-Sqrt[e]*Cosh[f])/2,(Sqrt[a]*Sinh[b]-\ Sqrt[e]*Sinh[f])/2,(Sqrt[c]*Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]-Sqrt[g]*\ Sin[h])/2},{1,2,3,4,5,6,7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}\ };\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C2D4r\"]={8,18,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,-7,-8,-5,-\ 6},{4,3,2,1,-8,-7,-6,-5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,5,6,\ 3,4,1,2},{8,7,6,5,4,3,2,1}},{(a+b)^2-(c+d)^2+(e+f)^2-(g+h)^2,(a-b)^2-(c-d)^2+(\ e-f)^2-(g-h)^2},{(a+b)^2+(e+f)^2,If[e==-f,0,ArcTan[a+b,e+f]],(c+d)^2+(g+h)^2,\ If[g==-h,0,ArcTan[c+d,g+h]],(a-b)^2+(e-f)^2,If[e==f,0,ArcTan[a-b,e-f]],(c-d)^\ 2+(g-h)^2,If[g==h,0,ArcTan[c-d,g-h]]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(\ Sqrt[a]*Cos[b]-Sqrt[e]*Cos[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*\ Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(Sqrt[a]*Sin[b]-\ Sqrt[e]*Sin[f])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[c]*Sin[d]-Sqrt[g]*\ Sin[h])/2},{1,2,3,4,-5,-6,7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,b,-c,-d,-e,-f,-g,-\ h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL3\"]={8,19,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,-4,1,-2,7,-8,5,-6}\ ,{4,-3,2,-1,8,-7,6,-5},{5,-6,-7,8,1,-2,-3,4},{6,-5,-8,7,2,-1,-4,3},{7,8,-5,-6,\ 3,4,-1,-2},{8,7,-6,-5,4,3,-2,-1}},{4*(-(d*e)+c*f-b*g+a*h)^2+(a^2-b^2-c^2+d^2-\ e^2+f^2+g^2-h^2)^2},,,{1,2,3,-4,5,-6,-7,-8,2},{2,1,0.,1,0.},{a,-b,-c,-d,-e,-f,\ -g,h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Octr\"]=(*Octonion Algebra*) {8,21,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,-8,7},{3,-4,-1,2,7,8,-5,-6},{4,3,-2,-\ 1,8,-7,6,-5},{5,-6,-7,-8,-1,2,3,4},{6,5,-8,7,-2,-1,-4,3},{7,8,5,-6,-3,4,-1,-2}\ ,{8,-7,6,5,-4,-3,2,-1}},{a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2},{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]},{Sqrt[a-c-e-g]*Cos[b],Sqrt[a-c-e-g]*Sin[b],Sqrt[c]*Cos[d],Sqrt[c]\ *Sin[d],Sqrt[e]*Cos[f],Sqrt[e]*Sin[f],Sqrt[g]*Cos[h],Sqrt[g]*Sin[h]},{1,-2,-3,\ -4,-5,-6,-7,-8,4},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"KiC4r\"]={8,22,{{1,2,3,4,5,6,7,8},{2,1,-8,-7,6,5,-4,-3},{3,-4,-1,2,7,-\ 8,-5,6},{4,-3,6,-5,8,-7,2,-1},{5,6,7,8,1,2,3,4},{6,5,-4,-3,2,1,-8,-7},{7,-8,-\ 5,6,3,-4,-1,2},{8,-7,2,-1,4,-3,6,-5}},{(a-e)^2-(b-f)^2+(c-g)^2-(d-h)^2,(a+b+e+\ f)^2+(c-d+g-h)^2,(a-b+e-f)^2+(c+d+g+h)^2},,,{1,2,-3,-8,5,6,-7,-4,2,1,1},{1,1,\ 1,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL21\"]={8,27,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,8,-7},{3,-4,1,-2,7,-8,\ 5,-6},{4,3,2,1,8,7,6,5},{5,-6,-7,8,1,-2,-3,4},{6,5,-8,-7,2,1,-4,-3},{7,8,-5,-\ 6,3,4,-1,-2},{8,-7,-6,5,4,-3,-2,1}},{-(d-e)^2-(c+f)^2+(b+g)^2+(a-h)^2,-(d+e)^\ 2-(c-f)^2+(b-g)^2+(a+h)^2},{(b+g)^2+(a-h)^2,ArcTan[a-h,b+g],(d-e)^2+(c+f)^2,\ ArcTan[c+f,d-e],(b-g)^2+(a+h)^2,ArcTan[a+h,b-g],(d+e)^2+(c-f)^2,ArcTan[c-f,d+\ e]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(\ Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(-(Sqrt[c]*\ Sin[d])+Sqrt[g]*Sin[h])/2,(Sqrt[c]*Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]-\ Sqrt[e]*Sin[f])/2,(-(Sqrt[a]*Cos[b])+Sqrt[e]*Cos[f])/2},{1,-2,3,4,5,6,-7,8,2,\ 2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL03\"]={8,28,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,8,-7},{3,-4,-1,2,7,-8,\ -5,6},{4,3,-2,-1,8,7,-6,-5},{5,-6,-7,8,-1,2,3,-4},{6,5,-8,-7,-2,-1,4,3},{7,8,\ 5,6,-3,-4,-1,-2},{8,-7,6,-5,-4,3,-2,1}},{(d+e)^2+(c-f)^2+(b+g)^2+(a-h)^2,(d-e)\ ^2+(c+f)^2+(b-g)^2+(a+h)^2},{(b+g)^2+(a-h)^2,ArcTan[a-h,b+g],(d+e)^2+(c-f)^2,\ ArcTan[c-f,d+e],(b-g)^2+(a+h)^2,ArcTan[a+h,b-g],(d-e)^2+(c+f)^2,ArcTan[c+f,d-\ e]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(\ Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[c]*\ Sin[d]-Sqrt[g]*Sin[h])/2,(-(Sqrt[c]*Cos[d])+Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]-\ Sqrt[e]*Sin[f])/2,(-(Sqrt[a]*Cos[b])+Sqrt[e]*Cos[f])/2},{1,-2,-3,-4,-5,-6,-7,\ 8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Octi\"]=(*Split-Octonion*){8,30,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,-8,-7},\ {3,-4,-1,2,7,8,-5,-6},{4,-3,-2,1,8,7,6,5},{5,-6,-7,-8,-1,2,3,4},{6,-5,-8,-7,-\ 2,1,-4,-3},{7,8,5,-6,-3,4,-1,-2},{8,7,6,-5,-4,3,2,1}},{a^2-b^2+c^2-d^2+e^2-f^\ 2+g^2-h^2},{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]},{Sqrt[a]*Cos[b],Sqrt[c]*Cos[d],Sqrt[a]*Sin[b]\ ,Sqrt[c]*Sin[d],Sqrt[e]*Cos[f],Sqrt[g]*Cos[h],Sqrt[e]*Sin[f],Sqrt[g]*Sin[h]},{\ 1,2,-3,4,-5,6,-7,8,4},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,-h}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C3\"]={9,2,{{1,2,3,4,5,6,7,8,9},{2,3,1,5,6,4,8,9,7},{3,1,2,6,4,5,9,7,\ 8},{4,5,6,7,8,9,1,2,3},{5,6,4,8,9,7,2,3,1},{6,4,5,9,7,8,3,1,2},{7,8,9,1,2,3,4,\ 5,6},{8,9,7,2,3,1,5,6,4},{9,7,8,3,1,2,6,4,5}},{a+b+c+d+e+f+g+h+i,((-a+c+e-f+g-\ h)^2+(a-b-d+f+h-i)^2+(b-c+d-e-g+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-b+d-e+g-h)^2+(b-c+e-f+h-i)^2+(-a+c-d+f-g+i)^2)/2,((a+b+c-d-\ e-f)^2+(a+b+c-g-h-i)^2+(d+e+f-g-h-i)^2)/2},{a+b+c+d+e+f+g+h+i,((-a+c+e-f+g-h)^\ 2+(a-b-d+f+h-i)^2+(b-c+d-e-g+i)^2)/2,ArcTan[2*a-b-c-d-e+2*f-g+2*h-i,-(Sqrt[3]*\ (b-c+d-e-g+i))],((b-c-d+f+g-h)^2+(a-b+e-f-g+i)^2+(a-c-d+e-h+i)^2)/2,ArcTan[2*\ a-b-c-d+2*e-f-g-h+2*i,-(Sqrt[3]*(b-c-d+f+g-h))],((a-b+d-e+g-h)^2+(b-c+e-f+h-i)\ ^2+(-a+c-d+f-g+i)^2)/2,ArcTan[2*a-b-c+2*d-e-f+2*g-h-i,-(Sqrt[3]*(b-c+e-f+h-i))\ ],((a+b+c-d-e-f)^2+(a+b+c-g-h-i)^2+(d+e+f-g-h-i)^2)/2,ArcTan[2*a+2*b+2*c-d-e-\ f-g-h-i,-(Sqrt[3]*(d+e+f-g-h-i))]},{(a+2*Sqrt[b]*Cos[c]+2*Sqrt[d]*Cos[e]+2*\ Sqrt[f]*Cos[g]+2*Sqrt[h]*Cos[i])/9,(a+2*Sqrt[h]*Cos[i]+2*Sqrt[b]*Cos[c+(2*Pi)/\ 3]+2*Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(2*Pi)/3])/9,(a+2*Sqrt[h]*Cos[i]+\ 2*Sqrt[b]*Cos[c+(4*Pi)/3]+2*Sqrt[d]*Cos[e+(4*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3])\ /9,(a+2*Sqrt[f]*Cos[g]+2*Sqrt[b]*Cos[c+(2*Pi)/3]+2*Sqrt[h]*Cos[i+(2*Pi)/3]+2*\ Sqrt[d]*Cos[e+(4*Pi)/3])/9,(a+2*Sqrt[d]*Cos[e]+2*Sqrt[f]*Cos[g+(2*Pi)/3]+2*\ Sqrt[h]*Cos[i+(2*Pi)/3]+2*Sqrt[b]*Cos[c+(4*Pi)/3])/9,(a+2*Sqrt[b]*Cos[c]+2*\ Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[h]*Cos[i+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3])/\ 9,(a+2*Sqrt[f]*Cos[g]+2*Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[b]*Cos[c+(4*Pi)/3]+2*\ Sqrt[h]*Cos[i+(4*Pi)/3])/9,(a+2*Sqrt[b]*Cos[c]+2*Sqrt[f]*Cos[g+(2*Pi)/3]+2*\ Sqrt[d]*Cos[e+(4*Pi)/3]+2*Sqrt[h]*Cos[i+(4*Pi)/3])/9,(a+2*Sqrt[d]*Cos[e]+2*\ Sqrt[b]*Cos[c+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3]+2*Sqrt[h]*Cos[i+(4*Pi)/3])/\ 9},{1,3,2,7,9,8,4,6,5,1,1,1,1,1},{1,1,0.,1,0.,1,0.,1,0.},{a,c,b,g,i,h,d,f,e}};\ \ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 1, {{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, 6, 5, 7, 9, 8, 10, 12, 11, 1, 3, 2}, {5, 4, 6, 8, 7, 9, 11, 10, 12, 2, 1, 3}, {6, 5, 4, 9, 8, 7, 12, 11, 10, 3, 2, 1}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 12, 11, 1, 3, 2, 4, 6, 5, 7, 9, 8}, {11, 10, 12, 2, 1, 3, 5, 4, 6, 8, 7, 9}, {12, 11, 10, 3, 2, 1, 6, 5, 4, 9, 8, 7}}, {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, \ \((\((a - b + g - h)\)\^2 + \((a - c + g - i)\)\^2 + \((\(-b\) + c - h \ + i)\)\^2 - \((d - e + j - k)\)\^2 - \((d - f + j - l)\)\^2 - \((\(-e\) + f - \ k + l)\)\^2)\)/ 2, \ \((\((a - b - g + h)\)\^2 + \((\(-b\) + c + h - i)\)\^2 + \ \((a - c - g + i)\)\^2 + \((d - e - j + k)\)\^2 + \((\(-e\) + f + k - l)\)\^2 \ + \((d - f - j + l)\)\^2)\)/2}, {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, ArcTan[a + b + c - g - h - i, d + e + f - j - k - l], \ \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \ \((\(-a\) + c - g + i)\)\^2)\)/2, ArcTan[2\ a - b - c + 2\ g - h - i, \(-\@3\)\ \((b - c + h - i)\)], \ \((\((d - e + j - k)\)\^2 + \((e - f + k - \ l)\)\^2 + \((\(-d\) + f - j + l)\)\^2)\)/2, ArcTan[2\ d - e - f + 2\ j - k - l, \(-\@3\)\ \((e - f + k - l)\)], \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\ \)\^2 + \((b - c - h + i)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ g + h + i, \(-\@3\)\ \((b - c - h + i)\)], \ \((\((d - e - j + k)\)\^2 + \((\(-d\) + f + j - \ l)\)\^2 + \((e - f - k + l)\)\^2)\)/2, ArcTan[2\ d - e - f - 2\ j + k + l, \(-\@3\)\ \((e - f - k + l)\)]}, {\ \((a + b + 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f] + 4\ \@i\ Cos[j])\)/ 2, \ \((a + b + 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 4\ \@i\ Cos[j + \(2\ \[Pi]\)\/3])\), \ \((a + b + 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 4\ \@i\ Cos[j - \(2\ \[Pi]\)\/3])\), \ \((a - b + 4\ \@g\ Cos[h] + 4\ \@k\ Cos[l] + 2\ \@c\ Sin[d])\), \ \((a - b + 4\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 4\ \@k\ Cos[l + \(2\ \[Pi]\)\/3] + 2\ \@c\ Sin[d])\), \ \((a - b + 4\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 4\ \@k\ Cos[l - \(2\ \[Pi]\)\/3] + 2\ \@c\ Sin[d])\), \ \((a + b - 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f] - 4\ \@i\ Cos[j])\), \ \((a + b - 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 4\ \@i\ Cos[j + \(2\ \[Pi]\)\/3])\), \ \((a + b - 2\ \@c\ Cos[d] + 4\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 4\ \@i\ Cos[j - \(2\ \[Pi]\)\/3])\), \ \((a - b + 4\ \@g\ Cos[h] - 4\ \@k\ Cos[l] - 2\ \@c\ Sin[d])\), \ \((a - b + 4\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 4\ \@k\ Cos[l + \(2\ \[Pi]\)\/3] - 2\ \@c\ Sin[d])\), \ \((a - b + 4\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 4\ \@k\ Cos[l - \(2\ \[Pi]\)\/3] - 2\ \@c\ Sin[d])\)}/ 12, {1, 3, 2, 10, 11, 12, 7, 9, 8, 4, 5, 6, 1, 1, 1, 2, 2}, {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 2, {{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, 7, 8, 9, 10, 11, 12, 1, 2, 3}, {5, 6, 4, 8, 9, 7, 11, 12, 10, 2, 3, 1}, {6, 4, 5, 9, 7, 8, 12, 10, 11, 3, 1, 2}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9}, {11, 12, 10, 2, 3, 1, 5, 6, 4, 8, 9, 7}, {12, 10, 11, 3, 1, 2, 6, 4, 5, 9, 7, 8}}, {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, \((\((a - b + d - e + g - h + j - k)\)\^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 + \((\(-a\) + c + \ d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/ 2, \((a - b - g + h)\)\^2 + \((a - b - g + h)\)\ \((b - c - h + i)\) + \((b - c - h + i)\)\^2 + \((d - e - j + k)\)\^2 + \ \((d - e - j + k)\)\ \((e - f - k + l)\) + \((e - f - k + l)\)\^2 + \@3\ \((\((b - c - h + i)\)\ \((d - e - j + k)\) - \((a - b - g + h)\)\ \((e - f - k + l)\))\), \((a - b - g + h)\)\^2 + \((a - b - g + h)\)\ \((b - c - h + i)\) + \((b - c - h + i)\)\^2 + \((d - e - j + k)\)\^2 + \ \((d - e - j + k)\)\ \((e - f - k + l)\) + \((e - f - k + l)\)\^2 - \@3\ \((\((b - c - h + i)\)\ \((d - e - j + k)\) - \((a - b - g + h)\)\ \((e - f - k + l)\))\)}, {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, ArcTan[a + b + c - g - h - i, d + e + f - j - k - l], \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e \ - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/2, ArcTan[ 2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - l, \(-\@3\)\ \((b - c + e - f + h - i + k - l)\)], \((\((a - b - d + e + g - h - j + k)\)\^2 + \ \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + l, \(-\@3\)\ \((b - c - e + f + h - i - k + l)\)], \((\((a - b + d - e - g + h - j + k)\)\^2 + \ \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + l, \(-\@3\)\ \((b - c + e - f - h + i - k + l)\)], \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - \ c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}, {\ \((a + b + 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] + 2\ \@k\ Cos[l])\), \ \((a + b + 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a + b + 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \ \((a - b + 2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] - 2\ \@k\ Cos[l] + 2\ \@c\ Sin[d])\), \ \((a - b + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3] + 2\ \@c\ Sin[d])\), \ \((a - b + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3] + 2\ \@c\ Sin[d])\), \ \((a + b - 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] - 2\ \@k\ Cos[l])\), \ \((a + b - 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a + b - 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \((a - b + 2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] + 2\ \@k\ Cos[l] - 2\ \@c\ Sin[d])\), \ \((a - b + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3] - 2\ \@c\ Sin[d])\), \((a - b + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3] - 2\ \@c\ Sin[d])\)}/ 12, {1, 3, 2, 10, 12, 11, 7, 9, 8, 4, 6, 5, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }, {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 3, {{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, 11, 9, 1, 8, 12, 10, 5, 3, 7, 2, 6}, {5, 12, 7, 2, 9, 10, 11, 6, 1, 8, 3, 4}, {6, 10, 8, 3, 7, 11, 12, 4, 2, 9, 1, 5}, {7, 5, 12, 10, 2, 9, 1, 11, 6, 4, 8, 3}, {8, 6, 10, 11, 3, 7, 2, 12, 4, 5, 9, 1}, {9, 4, 11, 12, 1, 8, 3, 10, 5, 6, 7, 2}, {10, 8, 6, 7, 11, 3, 4, 2, 12, 1, 5, 9}, {11, 9, 4, 8, 12, 1, 5, 3, 10, 2, 6, 7}, {12, 7, 5, 9, 10, 2, 6, 1, 11, 3, 4, 8}}, {a + b + c + d + e + f + g + h + i + j + k + l, \ \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - \ f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/ 2, \((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - e - h + k)\) - \((a - d + g - j)\)\ \((b - e - h + k)\)\ \((c + f - i - l)\) - \((a - d - g + j)\)\ \((b + e - h - k)\)\ \((c - f + i - l)\) - \((a + d - g - j)\)\ \((b - e + h - k)\)\ \((c - f - i + l)\) + \((c + f - i - l)\)\ \((c - f + i - l)\)\ \((c - f - i + l)\)}, , , {1, 3, 2, 4, 9, 11, 7, 12, 5, 10, 6, 8, 1, 1, 3}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 4, \[IndentingNewLine]{{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, 6, 5, 1, 3, 2, 10, 12, 11, 7, 9, 8}, {5, 4, 6, 2, 1, 3, 11, 10, 12, 8, 7, 9}, {6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 12, 11, 7, 9, 8, 4, 6, 5, 1, 3, 2}, {11, 10, 12, 8, 7, 9, 5, 4, 6, 2, 1, 3}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}}, {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 + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \((\(-a\ \) + c - g + i)\)\^2 - \((\(-d\) + e - j + k)\)\^2 - \((d - f + j - l)\)\^2 - \ \((\(-e\) + f - k + l)\)\^2)\)/ 2, \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\)\^2 + \ \((b - c - h + i)\)\^2 - \((\(-d\) + e + j - k)\)\^2 - \((\(-e\) + f + k - l)\ \)\^2 - \((d - f - j + l)\)\^2)\)/2}, {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 + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \((\(-a\ \) + c - g + i)\)\^2)\)/2, ArcTan[2\ a - b - c + 2\ g - h - i, \(-\@3\)\ \((b - c + h - i)\)], \((\((\(-d\) + e - j + k)\)\^2 + \((d - f + j - l)\ \)\^2 + \((\(-e\) + f - k + l)\)\^2)\)/2, ArcTan[\(-2\)\ d + e + f - 2\ j + k + l, \(-\@3\)\ \((\(-e\) + f - k + l)\)], \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\ \)\^2 + \((b - c - h + i)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ g + h + i, \(-\@3\)\ \((b - c - h + i)\)], \((\((\(-d\) + e + j - k)\)\^2 + \((\(-e\) + f + k \ - l)\)\^2 + \((d - f - j + l)\)\^2)\)/2, ArcTan[\(-2\)\ d + e + f + 2\ j - k - l, \(-\@3\)\ \((\(-e\) + f + k - l)\)]}, {\ \((a + b + c + d + 4\ \@e\ Cos[f] + 4\ \@i\ Cos[j])\), \ \((a + b + c + d + 4\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 4\ \@i\ Cos[j + \(2\ \[Pi]\)\/3])\), \ \((a + b + c + d + 4\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 4\ \@i\ Cos[j - \(2\ \[Pi]\)\/3])\), \ \((a - b + c - d - 4\ \@g\ Cos[h] - 4\ \@k\ Cos[l])\), \ \((a - b + c - d - 4\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 4\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a - b + c - d - 4\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 4\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \ \((a + b - c - d + 4\ \@e\ Cos[f] - 4\ \@i\ Cos[j])\), \ \((a + b - c - d + 4\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 4\ \@i\ Cos[j + \(2\ \[Pi]\)\/3])\), \ \((a + b - c - d + 4\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 4\ \@i\ Cos[j - \(2\ \[Pi]\)\/3])\), \ \((a - b - c + d - 4\ \@g\ Cos[h] + 4\ \@k\ Cos[l])\), \ \((a - b - c + d - 4\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 4\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a - b - c + d - 4\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 4\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\)}/12, {1, 3, 2, 4, 5, 6, 7, 9, 8, 10, 11, 12, 1, 1, 1, 1, 2, 2}, {1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 5, {{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, 10, 11, 12, 7, 8, 9}, {5, 6, 4, 2, 3, 1, 11, 12, 10, 8, 9, 7}, {6, 4, 5, 3, 1, 2, 12, 10, 11, 9, 7, 8}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 11, 12, 7, 8, 9, 4, 5, 6, 1, 2, 3}, {11, 12, 10, 8, 9, 7, 5, 6, 4, 2, 3, 1}, {12, 10, 11, 9, 7, 8, 6, 4, 5, 3, 1, 2}}, {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 + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \((\((a - b + d - e + g - h + j - k)\)\^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 + \((\(-a\) + c \ + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/ 2, \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - \ d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^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 + 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 + d + e + f - g - h - i - j - k - l, a + b + c - d - e - f - g - h - i + j + k + l, \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f \ + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/2, ArcTan[2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - l, \(-\@3\)\ \((b - c + e - f + h - i + k - l)\)], \((\((a - b - d + e + g - h - j + k)\)\^2 + \ \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + l, \(-\@3\)\ \((b - c - e + f + h - i - k + l)\)], \((\((a - b + d - e - g + h - j + k)\)\^2 + \ \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + l, \(-\@3\)\ \((b - c + e - f - h + i - k + l)\)], \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - \ c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + \ l)\)\^2)\)/2, ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}, {\((a + b + c + d + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] + 2\ \@k\ Cos[l])\), \ \((a + b + c + d + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a + b + c + d + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \ \((a - b + c - d + 2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] - 2\ \@k\ Cos[l])\), \ \((a - b + c - d + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a - b + c - d + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] + 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \ \((a + b - c - d + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] - 2\ \@k\ Cos[l])\), \ \((a + b - c - d + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a + b - c - d + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] + 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] - 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\), \ \((a - b - c + d + 2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] + 2\ \@k\ Cos[l])\), \ \((a - b - c + d + 2\ \@e\ Cos[f + \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h + \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j + \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l + \(2\ \[Pi]\)\/3])\), \ \((a - b - c + d + 2\ \@e\ Cos[f - \(2\ \[Pi]\)\/3] - 2\ \@g\ Cos[h - \(2\ \[Pi]\)\/3] - 2\ \@i\ Cos[j - \(2\ \[Pi]\)\/3] + 2\ \@k\ Cos[l - \(2\ \[Pi]\)\/3])\)}/12, {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 0.`, 1, 0.`, 1, 0.`, 1, 0.`}, {a, b, c, \(-d\), \(-e\), \(-f\), g, h, i, \(-j\), \(-k\), \(-l\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 7, {{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, \(-1\), \(-2\), \(-3\), \(-10\), \(-11\), \(-12\), 7, 8, 9}, {5, 6, 4, \(-2\), \(-3\), \(-1\), \(-11\), \(-12\), \(-10\), 8, 9, 7}, {6, 4, 5, \(-3\), \(-1\), \(-2\), \(-12\), \(-10\), \(-11\), 9, 7, 8}, {7, 8, 9, 10, 11, 12, \(-1\), \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, 7, 11, 12, 10, \(-2\), \(-3\), \(-1\), \(-5\), \(-6\), \(-4\)}, \ {9, 7, 8, 12, 10, 11, \(-3\), \(-1\), \(-2\), \(-6\), \(-4\), \(-5\)}, {10, 11, 12, \(-7\), \(-8\), \(-9\), 4, 5, 6, \(-1\), \(-2\), \(-3\)}, {11, 12, 10, \(-8\), \(-9\), \(-7\), 5, 6, 4, \(-2\), \(-3\), \(-1\)}, {12, 10, 11, \(-9\), \(-7\), \(-8\), 6, 4, 5, \(-3\), \(-1\), \(-2\)}}, {\((a + b + c)\)\^2 + \((d + e + \ f)\)\^2 + \((g + h + i)\)\^2 + \((j + k + l)\)\^2, \((\((a - b)\)\^2 + \((b - \ c)\)\^2 + \((\(-a\) + c)\)\^2 + \((d - e)\)\^2 + \((e - f)\)\^2 + \((\(-d\) + \ f)\)\^2 + \((g - h)\)\^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 + \((b\ \ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^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)\)}, , , {1, 3, 2, \(-4\), \(-6\), \(-5\), \(-7\), \(-9\), \(-8\), \(-10\), \(-12\ \), \(-11\), 2, 2}, , {a, c, b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \ \(-l\), \(-k\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 8, {{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, 10\ \[ImaginaryI], 11\ \[ImaginaryI], 12\ \[ImaginaryI], \(-7\)\ \[ImaginaryI], \(-8\)\ \ \[ImaginaryI], \(-9\)\ \[ImaginaryI]}, {5, 6, 4, 2, 3, 1, 11\ \[ImaginaryI], 12\ \[ImaginaryI], 10\ \[ImaginaryI], \(-8\)\ \[ImaginaryI], \(-9\)\ \ \[ImaginaryI], \(-7\)\ \[ImaginaryI]}, {6, 4, 5, 3, 1, 2, 12\ \[ImaginaryI], 10\ \[ImaginaryI], 11\ \[ImaginaryI], \(-9\)\ \[ImaginaryI], \(-7\)\ \ \[ImaginaryI], \(-8\)\ \[ImaginaryI]}, {7, 8, 9, \(-10\)\ \[ImaginaryI], \(-11\)\ \[ImaginaryI], \(-12\)\ \ \[ImaginaryI], 1, 2, 3, 4\ \[ImaginaryI], 5\ \[ImaginaryI], 6\ \[ImaginaryI]}, {8, 9, 7, \(-11\)\ \[ImaginaryI], \(-12\)\ \[ImaginaryI], \(-10\)\ \ \[ImaginaryI], 2, 3, 1, 5\ \[ImaginaryI], 6\ \[ImaginaryI], 4\ \[ImaginaryI]}, {9, 7, 8, \(-12\)\ \[ImaginaryI], \(-10\)\ \[ImaginaryI], \(-11\)\ \ \[ImaginaryI], 3, 1, 2, 6\ \[ImaginaryI], 4\ \[ImaginaryI], 5\ \[ImaginaryI]}, {10, 11, 12, 7\ \[ImaginaryI], 8\ \[ImaginaryI], 9\ \[ImaginaryI], \(-4\)\ \[ImaginaryI], \(-5\)\ \[ImaginaryI], \ \(-6\)\ \[ImaginaryI], 1, 2, 3}, {11, 12, 10, 8\ \[ImaginaryI], 9\ \[ImaginaryI], 7\ \[ImaginaryI], \(-5\)\ \[ImaginaryI], \(-6\)\ \[ImaginaryI], \ \(-4\)\ \[ImaginaryI], 2, 3, 1}, {12, 10, 11, 9\ \[ImaginaryI], 7\ \[ImaginaryI], 8\ \[ImaginaryI], \(-6\)\ \[ImaginaryI], \(-4\)\ \[ImaginaryI], \ \(-5\)\ \[ImaginaryI], 3, 1, 2}}, {\((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + h + i)\)\ \^2 - \((j + k + l)\)\^2, \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\ \)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((\(-d\) + f)\)\^2 - \((g - \ h)\)\^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 + \((b\ \ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^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)\)}, , , {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 2, 2}, {a, c, b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \ \(-l\), \(-k\)}, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 14, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 4, 5, 6, 1, 12, 7, 8, 9, 10, 11}, {3, 4, 5, 6, 1, 2, 11, 12, 7, 8, 9, 10}, {4, 5, 6, 1, 2, 3, 10, 11, 12, 7, 8, 9}, {5, 6, 1, 2, 3, 4, 9, 10, 11, 12, 7, 8}, {6, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 7}, {7, 8, 9, 10, 11, 12, \(-1\), \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, 10, 11, 12, 7, \(-6\), \(-1\), \(-2\), \(-3\), \(-4\), \(-5\)}, {9, 10, 11, 12, 7, 8, \(-5\), \(-6\), \(-1\), \(-2\), \(-3\), \(-4\)}, {10, 11, 12, 7, 8, 9, \(-4\), \(-5\), \(-6\), \(-1\), \(-2\), \(-3\)}, {11, 12, 7, 8, 9, 10, \(-3\), \(-4\), \(-5\), \(-6\), \(-1\), \(-2\)}, {12, 7, 8, 9, 10, 11, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-1\)}}, {\((a - b \ + c - d + e - f)\)\^2 + \((g - h + i - j + k - l)\)\^2, 1\/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)\), 1\/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)\), \((a + b + c + d + e + f)\)\^2 + \((g + h + i + j + k + \ l)\)\^2}, , , {1, 6, 5, 4, 3, 2, \(-7\), \(-8\), \(-9\), \(-10\), \(-11\), \(-12\), 1, 2, 2, 1}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 19, {{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, \(-10\), 11, 12, \(-7\), 8, 9}, {5, 6, 4, 2, 3, 1, \(-11\), 12, 10, 8, 9, \(-7\)}, {6, 4, 5, 3, 1, 2, \(-12\), 10, 11, 9, \(-7\), 8}, {7, \(-8\), \(-9\), \(-10\), \(-11\), \(-12\), 1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, \(-7\), 11, 12, 10, \(-2\), 3, 1, 5, 6, 4}, {9, \(-7\), 8, 12, 10, 11, \(-3\), 1, 2, 6, 4, 5}, {10, 11, 12, \(-7\), 8, 9, \(-4\), 5, 6, 1, 2, 3}, {11, 12, 10, 8, 9, \(-7\), \(-5\), 6, 4, 2, 3, 1}, {12, 10, 11, 9, \(-7\), 8, \(-6\), 4, 5, 3, 1, 2}}, {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 - d - e - f + g - h - i + j + k + l, a + b + c + d + e + f - g + h + i + j + k + l, 1\/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)\), 1\/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)\), 1\/2\ \((\((a - b - d + e - g - h - j + k)\)\^2 + \((\(-a\) + c + \ d - f + g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), 1\/2\ \((\((a - b + d - e + g + h - j + k)\)\^2 + \((\(-a\) + c - \ d + f - g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)}, , , \ {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}, , {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {12, 20, {{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, 10, 11, 12, 7, 8, 9}, {5, \(-6\), 4, 2, \(-3\), 1, 11, \(-12\), 10, 8, \(-9\), 7}, {6, 4, \(-5\), 3, 1, \(-2\), 12, 10, \(-11\), 9, 7, \(-8\)}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, \(-9\), 7, 11, \(-12\), 10, 2, \(-3\), 1, 5, \(-6\), 4}, {9, 7, \(-8\), 12, 10, \(-11\), 3, 1, \(-2\), 6, 4, \(-5\)}, {10, 11, 12, 7, 8, 9, 4, 5, 6, 1, 2, 3}, {11, \(-12\), 10, 8, \(-9\), 7, 5, \(-6\), 4, 2, \(-3\), 1}, {12, 10, \(-11\), 9, 7, \(-8\), 6, 4, \(-5\), 3, 1, \(-2\)}}, {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 - d + e + f + g - h - i - j + k + l, a - b - c + d - e - f - g + h + i - j + k + l, 1\/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)\), 1\/2\ \((\((a + b - d - e + g + h - j - k)\)\^2 + \((a + c - d - \ f + g + i - j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), 1\/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)\), 1\/2\ \((\((a + b + d + e - g - h - j - k)\)\^2 + \((a + c + d + \ f - g - i - j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)}, , , {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}, , {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 2, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15}, {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1}, {7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2}, {8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9}, {15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10}, {16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11}}, {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, \((b - d - e + g - j + l + m - \ o)\)\^2 + \((a - c + f - h - i + k - n + p)\)\^2, \((b - d + e - g - j + l - \ m + o)\)\^2 + \((a - c - f + h - i + k + n - p)\)\^2}, , , {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}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 3, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13}, {5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 4, 3, 2}, {6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1}, {7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 3, 2, 1, 4}, {8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6}, {12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5}, {13, 16, 15, 14, 1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 10}, {14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9}, {15, 14, 13, 16, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12}, {16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11}}, {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}, , , {1, 2, 3, 4, 13, 16, 15, 14, 9, 10, 11, 12, 5, 8, 7, 6, 1, 1, 1, 1, 1, 1, 2, 2}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 4, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15}, {5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 4, 3, 2}, {6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3}, {7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 3, 2, 1, 4}, {8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7}, {13, 16, 15, 14, 1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 10}, {14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11}, {15, 14, 13, 16, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12}, {16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9}}, {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}, , , {1, 4, 3, 2, 13, 14, 15, 16, 9, 12, 11, 10, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 2, 2}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 7, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3}, {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2}, {4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5}, {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7}, {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6}, {8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11}, {11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15}, {15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 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, \((a - e + i - m)\)\^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}, , , {1, 2, 15, 4, 13, 6, 11, 8, 9, 10, 7, 12, 5, 14, 3, 16, 1, 1, 1, 1, 2, 2}, gp, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 9, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 9, 16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11}, {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2}, {4, 11, 2, 9, 16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13}, {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 13, 4, 11, 2, 9, 16, 7, 14, 5, 12, 3, 10, 1, 8, 15}, {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6}, {8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 5, 12, 3, 10, 1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 5, 12, 3}, {11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 5}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7}, {15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, {16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9}}, {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, \((a - i)\)\^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}, , , {1, 10, 15, 12, 13, 14, 11, 16, 9, 2, 7, 4, 5, 6, 3, 8, 1, 1, 1, 1, 2, 2}, gp, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 10, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 12}, {6, 7, 8, 5, 2, 3, 4, 1, 14, 15, 16, 13, 10, 11, 12, 9}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 11, 12, 9, 10}, {8, 5, 6, 7, 4, 1, 2, 3, 16, 13, 14, 15, 12, 9, 10, 11}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, 15, 16, 13, 10, 11, 12, 9, 6, 7, 8, 5, 2, 3, 4, 1}, {15, 16, 13, 14, 11, 12, 9, 10, 7, 8, 5, 6, 3, 4, 1, 2}, {16, 13, 14, 15, 12, 9, 10, 11, 8, 5, 6, 7, 4, 1, 2, 3}}, {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}, {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, \[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]]}, { (*a*) a + b + c + d + e + f + g + h + 2 Sqrt[i]\ Cos[j] + 2 Sqrt[k]\ Cos[l] + 2 Sqrt[m]\ Cos[n] + 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*b*) a + b - c - d - e - f + g + h + 2 Sqrt[i]\ Sin[j] + 2 Sqrt[k]\ Sin[l] + 2 Sqrt[m]\ Sin[n] + 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*c*) a + b + c + d + e + f + g + h - 2 Sqrt[i]\ Cos[j] - 2 Sqrt[k]\ Cos[l] - 2 Sqrt[m]\ Cos[n] - 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*d*) a + b - c - d - e - f + g + h - 2 Sqrt[i]\ Sin[j] - 2 Sqrt[k]\ Sin[l] - 2 Sqrt[m]\ Sin[n] - 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*e*) a - b + c - d - e + f - g + h - 2 Sqrt[i]\ Cos[j] + 2 Sqrt[k]\ Cos[l] - 2 Sqrt[m]\ Cos[n] + 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*f*) a - b - c + d + e - f - g + h - 2 Sqrt[i]\ Sin[j] + 2 Sqrt[k]\ Sin[l] - 2 Sqrt[m]\ Sin[n] + 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*g*) a - b + c - d - e + f - g + h + 2 Sqrt[i]\ Cos[j] - 2 Sqrt[k]\ Cos[l] + 2 Sqrt[m]\ Cos[n] - 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*h*) a - b - c + d + e - f - g + h + 2 Sqrt[i]\ Sin[j] - 2 Sqrt[k]\ Sin[l] + 2 Sqrt[m]\ Sin[n] - 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*i*) a - b - c + d - e + f + g - h - 2 Sqrt[i]\ Cos[j] - 2 Sqrt[k]\ Cos[l] + 2 Sqrt[m]\ Cos[n] + 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*j*) a - b + c - d + e - f + g - h - 2 Sqrt[i]\ Sin[j] - 2 Sqrt[k]\ Sin[l] + 2 Sqrt[m]\ Sin[n] + 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*k*) a - b - c + d - e + f + g - h + 2 Sqrt[i]\ Cos[j] + 2 Sqrt[k]\ Cos[l] - 2 Sqrt[m]\ Cos[n] - 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*l*) a - b + c - d + e - f + g - h + 2 Sqrt[i]\ Sin[j] + 2 Sqrt[k]\ Sin[l] - 2 Sqrt[m]\ Sin[n] - 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*m*) a + b - c - d + e + f - g - h + 2 Sqrt[i]\ Cos[j] - 2 Sqrt[k]\ Cos[l] - 2 Sqrt[m]\ Cos[n] + 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*n*) a + b + c + d - e - f - g - h + 2 Sqrt[i]\ Sin[j] - 2 Sqrt[k]\ Sin[l] - 2 Sqrt[m]\ Sin[n] + 2 Sqrt[o]\ Sin[p], \[IndentingNewLine] (*o*) a + b - c - d + e + f - g - h - 2 Sqrt[i]\ Cos[j] + 2 Sqrt[k]\ Cos[l] + 2 Sqrt[m]\ Cos[n] - 2 Sqrt[o]\ Cos[p], \[IndentingNewLine] (*p*) a + b + c + d - e - f - g - h - 2 Sqrt[i]\ Sin[j] + 2 Sqrt[k]\ Sin[l] + 2 Sqrt[m]\ Sin[n] - 2 Sqrt[o]\ Sin[p]}/16, {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}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. }, };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 11, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15}, {3, 8, 5, 2, 7, 4, 1, 6, 11, 16, 13, 10, 15, 12, 9, 14}, {4, 7, 6, 1, 8, 3, 2, 5, 12, 15, 14, 9, 16, 11, 10, 13}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 12}, {6, 5, 8, 7, 2, 1, 4, 3, 14, 13, 16, 15, 10, 9, 12, 11}, {7, 4, 1, 6, 3, 8, 5, 2, 15, 12, 9, 14, 11, 16, 13, 10}, {8, 3, 2, 5, 4, 7, 6, 1, 16, 11, 10, 13, 12, 15, 14, 9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7}, {11, 16, 13, 10, 15, 12, 9, 14, 3, 8, 5, 2, 7, 4, 1, 6}, {12, 15, 14, 9, 16, 11, 10, 13, 4, 7, 6, 1, 8, 3, 2, 5}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, 13, 16, 15, 10, 9, 12, 11, 6, 5, 8, 7, 2, 1, 4, 3}, {15, 12, 9, 14, 11, 16, 13, 10, 7, 4, 1, 6, 3, 8, 5, 2}, {16, 11, 10, 13, 12, 15, 14, 9, 8, 3, 2, 5, 4, 7, 6, 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, \((a - e + i - m)\)\^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}, , , {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}, , {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 12, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 5, 4, 7, 6, 1, 8, 3, 10, 13, 12, 15, 14, 9, 16, 11}, {3, 8, 5, 2, 7, 4, 1, 6, 11, 16, 13, 10, 15, 12, 9, 14}, {4, 3, 6, 5, 8, 7, 2, 1, 12, 11, 14, 13, 16, 15, 10, 9}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 12}, {6, 1, 8, 3, 2, 5, 4, 7, 14, 9, 16, 11, 10, 13, 12, 15}, {7, 4, 1, 6, 3, 8, 5, 2, 15, 12, 9, 14, 11, 16, 13, 10}, {8, 7, 2, 1, 4, 3, 6, 5, 16, 15, 10, 9, 12, 11, 14, 13}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 13, 12, 15, 14, 9, 16, 11, 2, 5, 4, 7, 6, 1, 8, 3}, {11, 16, 13, 10, 15, 12, 9, 14, 3, 8, 5, 2, 7, 4, 1, 6}, {12, 11, 14, 13, 16, 15, 10, 9, 4, 3, 6, 5, 8, 7, 2, 1}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, 9, 16, 11, 10, 13, 12, 15, 6, 1, 8, 3, 2, 5, 4, 7}, {15, 12, 9, 14, 11, 16, 13, 10, 7, 4, 1, 6, 3, 8, 5, 2}, {16, 15, 10, 9, 12, 11, 14, 13, 8, 7, 2, 1, 4, 3, 6, 5}}, {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}, , , {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}, gp, {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"P16\"]={16,13, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,8,15,6,5,12,3,10,9,16,7,14,13,\ 4,11},{3,16,1,6,7,4,5,10,11,8,9,14,15,12,13,2},{4,7,14,1,8,11,2,5,12,15,6,9,\ 16,3,10,13},{5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4},{6,5,12,3,10,9,16,7,14,\ 13,4,11,2,1,8,15},{7,4,5,10,11,8,9,14,15,12,13,2,3,16,1,6},{8,11,2,5,12,15,6,\ 9,16,3,10,13,4,7,14,1},{9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8},{10,9,16,7,14,\ 13,4,11,2,1,8,15,6,5,12,3},{11,8,9,14,15,12,13,2,3,16,1,6,7,4,5,10},{12,15,6,\ 9,16,3,10,13,4,7,14,1,8,11,2,5},{13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12},{14,\ 13,4,11,2,1,8,15,6,5,12,3,10,9,16,7},{15,12,13,2,3,16,1,6,7,4,5,10,11,8,9,14},\ {16,3,10,13,4,7,14,1,8,11,2,5,12,15,6,9}},{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},,,{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},gp,{a,b,c,-d,e,-f,-g,-h,i,-j,-k,-l,-m,-n,-o,p}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 14, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 12}, {6, 5, 8, 7, 2, 1, 4, 3, 14, 13, 16, 15, 10, 9, 12, 11}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 11, 12, 9, 10}, {8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6}, {12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, 13, 16, 15, 10, 9, 12, 11, 6, 5, 8, 7, 2, 1, 4, 3}, {15, 16, 13, 14, 11, 12, 9, 10, 7, 8, 5, 6, 3, 4, 1, 2}, {16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 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}, {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}, {\ \((a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p)\), \((a - b - c + d - e + f + g - h + i + j + k - l + m - n - o - p)\), \ \((a - b + c - d + e - f + g - h + i + j - k + l - m - n + o - p)\), \ \((a + b - c - d - e - f + g + h + i + j - k - l - m + n - o + p)\), \ \((a + b + c + d - e - f - g - h + i - j - k - l + m + n + o - p)\), \ \((a - b - c + d + e - f - g + h + i - j - k + l + m - n - o + p)\), \((a - b + c - d - e + f - g + h + i - j + k - l - m - n + o + p)\), \ \((a + b - c - d + e + f - g - h + i - j + k + l - m + n - o - p)\), \((a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - p)\), \((a - b - c + d - e + f + g - h - i - j - k + l - m + n + o + p)\), \((a - b + c - d + e - f + g - h - i - j + k - l + m + n - o + p)\), \((a + b - c - d - e - f + g + h - i - j + k + l + m - n + o - p)\), \ \((a + b + c + d - e - f - g - h - i + j + k + l - m - n - o + p)\), \ \((a - b - c + d + e - f - g + h - i + j + k - l - m + n + o - p)\), \ \((a - b + c - d - e + f - g + h - i + j - k + l + m + n - o - p)\), \ \((a + b - c - d + e + f - g - h - i + j - k - l + m - n + o + p)\)}/ 16, {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}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 15, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 9, 12, 3, 14, 5, 8, 15, 10, 1, 4, 11, 6, 13, 16, 7}, {3, 4, 9, 10, 15, 16, 5, 6, 11, 12, 1, 2, 7, 8, 13, 14}, {4, 11, 2, 9, 16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13}, {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 13, 8, 15, 2, 9, 4, 11, 14, 5, 16, 7, 10, 1, 12, 3}, {7, 16, 13, 6, 3, 12, 9, 2, 15, 8, 5, 14, 11, 4, 1, 10}, {8, 7, 14, 13, 4, 3, 10, 9, 16, 15, 6, 5, 12, 11, 2, 1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8}, {10, 1, 4, 11, 6, 13, 16, 7, 2, 9, 12, 3, 14, 5, 8, 15}, {11, 12, 1, 2, 7, 8, 13, 14, 3, 4, 9, 10, 15, 16, 5, 6}, {12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 5}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {14, 5, 16, 7, 10, 1, 12, 3, 6, 13, 8, 15, 2, 9, 4, 11}, {15, 8, 5, 14, 11, 4, 1, 10, 7, 16, 13, 6, 3, 12, 9, 2}, {16, 15, 6, 5, 12, 11, 2, 1, 8, 7, 14, 13, 4, 3, 10, 9}}, {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}, , , {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}, , {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"r31r\"]={16,17, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,\ 16,15},{3,-4,1,-2,7,-8,5,-6,11,-12,9,-10,15,-16,13,-14},{4,-3,2,-1,8,-7,6,-5,\ 12,-11,10,-9,16,-15,14,-13},{5,-6,-7,8,-1,2,3,-4,13,-14,-15,16,-9,10,11,-12},{\ 6,-5,-8,7,-2,1,4,-3,14,-13,-16,15,-10,9,12,-11},{7,8,-5,-6,-3,-4,1,2,15,16,-\ 13,-14,-11,-12,9,10},{8,7,-6,-5,-4,-3,2,1,16,15,-14,-13,-12,-11,10,9},{9,-10,-\ 11,12,-13,14,15,-16,1,-2,-3,4,-5,6,7,-8},{10,-9,-12,11,-14,13,16,-15,2,-1,-4,\ 3,-6,5,8,-7},{11,12,-9,-10,-15,-16,13,14,3,4,-1,-2,-7,-8,5,6},{12,11,-10,-9,-\ 16,-15,14,13,4,3,-2,-1,-8,-7,6,5},{13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4},{\ 14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3},{15,-16,13,-14,11,-12,9,-10,7,-8,5,-6,\ 3,-4,1,-2},{16,-15,14,-13,12,-11,10,-9,8,-7,6,-5,4,-3,2,-1}},{(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*(-(h*i-g*j+f*k-e*l-d*m+\ c*n-b*o+a*p)^2+(a*i-b*j-c*k+d*l+e*m-f*n-g*o+h*p)^2-(a*e-b*f-c*g+d*h+i*m-j*n-k*\ o+l*p)^2+(a*c-b*d-e*g+f*h+i*k-j*l-m*o+n*p)^2+(a*b+c*d-e*f-g*h+i*j+k*l-m*n-o*p)\ ^2)},,,{1,2,3,-4,-5,6,7,8,9,-10,-11,-12,13,14,15,-16,4},gp,{a,b,c,-d,e,-f,-g,-\ h,i,-j,-k,-l,-m,-n,-o,p}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 21, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15}, {3, \(-4\), 1, \(-2\), 7, \(-8\), 5, \(-6\), 11, \(-12\), 9, \(-10\), 15, \(-16\), 13, \(-14\)}, {4, \(-3\), 2, \(-1\), 8, \(-7\), 6, \(-5\), 12, \(-11\), 10, \(-9\), 16, \(-15\), 14, \(-13\)}, {5, \(-6\), \(-7\), 8, 1, \(-2\), \(-3\), 4, 13, \(-14\), \(-15\), 16, 9, \(-10\), \(-11\), 12}, {6, \(-5\), \(-8\), 7, 2, \(-1\), \(-4\), 3, 14, \(-13\), \(-16\), 15, 10, \(-9\), \(-12\), 11}, {7, 8, \(-5\), \(-6\), 3, 4, \(-1\), \(-2\), 15, 16, \(-13\), \(-14\), 11, 12, \(-9\), \(-10\)}, {8, 7, \(-6\), \(-5\), 4, 3, \(-2\), \(-1\), 16, 15, \(-14\), \(-13\), 12, 11, \(-10\), \(-9\)}, {9, \(-10\), \(-11\), 12, \(-13\), 14, 15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 7, \(-8\)}, {10, \(-9\), \(-12\), 11, \(-14\), 13, 16, \(-15\), 2, \(-1\), \(-4\), 3, \(-6\), 5, 8, \(-7\)}, {11, 12, \(-9\), \(-10\), \(-15\), \(-16\), 13, 14, 3, 4, \(-1\), \(-2\), \(-7\), \(-8\), 5, 6}, {12, 11, \(-10\), \(-9\), \(-16\), \(-15\), 14, 13, 4, 3, \(-2\), \(-1\), \(-8\), \(-7\), 6, 5}, {13, 14, 15, 16, \(-9\), \(-10\), \(-11\), \(-12\), 5, 6, 7, 8, \(-1\), \(-2\), \(-3\), \(-4\)}, {14, 13, 16, 15, \(-10\), \(-9\), \(-12\), \(-11\), 6, 5, 8, 7, \(-2\), \(-1\), \(-4\), \(-3\)}, {15, \(-16\), 13, \(-14\), \(-11\), 12, \(-9\), 10, 7, \(-8\), 5, \(-6\), \(-3\), 4, \(-1\), 2}, {16, \(-15\), 14, \(-13\), \(-12\), 11, \(-10\), 9, 8, \(-7\), 6, \(-5\), \(-4\), 3, \(-2\), 1}}, {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 \ + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\^2 - 4\ \((\((h\ i - g\ j + f\ k - e\ l - d\ m + c\ n - b\ o + a\ p)\ \)\^2 + \((a\ i - b\ j - c\ k + d\ l - e\ m + f\ n + g\ o - h\ p)\)\^2 + \((a\ \ e - b\ f - c\ g + d\ h + i\ m - j\ n - k\ o + l\ p)\)\^2 + \((a\ c - b\ d + \ e\ g - f\ h + i\ k - j\ l + m\ o - n\ p)\)\^2 + \((a\ b + c\ d + e\ f + g\ h \ + i\ j + k\ l + m\ n + o\ p)\)\^2)\)}, , , {1, 2, 3, \(-4\), 5, \(-6\), \(-7\), \(-8\), 9, \(-10\), \(-11\), \(-12\), \(-13\), \(-14\), \(-15\), 16, 4}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL31\"]={16,22, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,-1,4,-3,6,-5,8,-7,10,-9,12,-11,\ 14,-13,16,-15},{3,-4,1,-2,7,-8,5,-6,11,-12,9,-10,15,-16,13,-14},{4,3,2,1,8,7,\ 6,5,12,11,10,9,16,15,14,13},{5,-6,-7,8,1,-2,-3,4,13,-14,-15,16,9,-10,-11,12},{\ 6,5,-8,-7,2,1,-4,-3,14,13,-16,-15,10,9,-12,-11},{7,8,-5,-6,3,4,-1,-2,15,16,-\ 13,-14,11,12,-9,-10},{8,-7,-6,5,4,-3,-2,1,16,-15,-14,13,12,-11,-10,9},{9,-10,-\ 11,12,-13,14,15,-16,1,-2,-3,4,-5,6,7,-8},{10,9,-12,-11,-14,-13,16,15,2,1,-4,-\ 3,-6,-5,8,7},{11,12,-9,-10,-15,-16,13,14,3,4,-1,-2,-7, -8,5,6},{12,-11,-10,9,-16,15,14,-13,4,-3,-2,1,-8,7,6,-5},{13,14,15,16,-9,-10,-\ 11,-12,5,6,7,8,-1,-2,-3,-4},{14,-13,16,-15,-10,9,-12,11,6,-5,8,-7,-2,1,-4,3},{\ 15,-16,13,-14,-11,12,-9,10,7,-8,5,-6,-3,4,-1,2},{16,15,14,13,-12,-11,-10,-9,8,\ 7,6,5,-4,-3,-2,-1}},{(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*(-(-(h*i)-g*j+f*k+e*l-d*m-c*n+b*o+a*p)^2-(-(g*i)+h*j+e*k-f*l-c*m+\ d*n+a*o-b*p)^2+(-(f*i)+e*j+h*k-g*l-b*m+a*n+d*o-c*p)^2+(-(d*i)+c*j-b*k+a*l-h*m+\ g*n-f*o+e*p)^2+(-(d*e)+c*f-b*g+a*h+l*m-k*n+j*o-i*p)^2)},,,{1,-2,3,4,5,6,-7,8,\ 9,10,-11,12,-13,14,-15,-16,4},gp,{a,c,b,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 23, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, \(-1\), 4, \(-3\), 6, \(-5\), 8, \(-7\), 10, \(-9\), 12, \(-11\), 14, \(-13\), 16, \(-15\)}, {3, \(-4\), \(-1\), 2, 7, \(-8\), \(-5\), 6, 11, \(-12\), \(-9\), 10, 15, \(-16\), \(-13\), 14}, {4, 3, \(-2\), \(-1\), 8, 7, \(-6\), \(-5\), 12, 11, \(-10\), \(-9\), 16, 15, \(-14\), \(-13\)}, {5, \(-6\), \(-7\), 8, 1, \(-2\), \(-3\), 4, 13, \(-14\), \(-15\), 16, 9, \(-10\), \(-11\), 12}, {6, 5, \(-8\), \(-7\), 2, 1, \(-4\), \(-3\), 14, 13, \(-16\), \(-15\), 10, 9, \(-12\), \(-11\)}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 11, 12, 9, 10}, {8, \(-7\), 6, \(-5\), 4, \(-3\), 2, \(-1\), 16, \(-15\), 14, \(-13\), 12, \(-11\), 10, \(-9\)}, {9, \(-10\), \(-11\), 12, \(-13\), 14, 15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 7, \(-8\)}, {10, 9, \(-12\), \(-11\), \(-14\), \(-13\), 16, 15, 2, 1, \(-4\), \(-3\), \(-6\), \(-5\), 8, 7}, {11, 12, 9, 10, \(-15\), \(-16\), \(-13\), \(-14\), 3, 4, 1, 2, \(-7\), \(-8\), \(-5\), \(-6\)}, {12, \(-11\), 10, \(-9\), \(-16\), 15, \(-14\), 13, 4, \(-3\), 2, \(-1\), \(-8\), 7, \(-6\), 5}, {13, 14, 15, 16, \(-9\), \(-10\), \(-11\), \(-12\), 5, 6, 7, 8, \(-1\), \(-2\), \(-3\), \(-4\)}, {14, \(-13\), 16, \(-15\), \(-10\), 9, \(-12\), 11, 6, \(-5\), 8, \(-7\), \(-2\), 1, \(-4\), 3}, {15, \(-16\), \(-13\), 14, \(-11\), 12, 9, \(-10\), 7, \(-8\), \(-5\), 6, \(-3\), 4, 1, \(-2\)}, {16, 15, \(-14\), \(-13\), \(-12\), \(-11\), 10, 9, 8, 7, \(-6\), \(-5\), \(-4\), \(-3\), 2, 1}}, {\((a\^2 + b\^2 + c\^2 + d\^2 - e\^2 - f\^2 - g\^2 - h\^2 \ - i\^2 - j\^2 - k\^2 - l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\^2 - 4\ \((\((\(-h\)\ i - g\ j + f\ k + e\ l - d\ m - c\ n + b\ o + \ a\ p)\)\^2 + \((\(-g\)\ i + h\ j + e\ k - f\ l - c\ m + d\ n + a\ o - b\ p)\)\ \^2 + \((\(-f\)\ i + e\ j - h\ k + g\ l - b\ m + a\ n - d\ o + c\ p)\)\^2 - \ \((\(-d\)\ i + c\ j - b\ k + a\ l - h\ m + g\ n - f\ o + e\ p)\)\^2 - \((\(-d\ \)\ e + c\ f - b\ g + a\ h + l\ m - k\ n + j\ o - i\ p)\)\^2)\)}, , , {1, \ \(-2\), \(-3\), \(-4\), 5, 6, 7, \(-8\), 9, 10, 11, \(-12\), \(-13\), 14, 15, 16, 4}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {16, 24, \[IndentingNewLine]{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, \(-1\), 4, \(-3\), 6, \(-5\), 8, \(-7\), 10, \(-9\), 12, \(-11\), 14, \(-13\), 16, \(-15\)}, {3, \(-4\), \(-1\), 2, 7, \(-8\), \(-5\), 6, 11, \(-12\), \(-9\), 10, 15, \(-16\), \(-13\), 14}, {4, 3, \(-2\), \(-1\), 8, 7, \(-6\), \(-5\), 12, 11, \(-10\), \(-9\), 16, 15, \(-14\), \(-13\)}, {5, \(-6\), \(-7\), 8, \(-1\), 2, 3, \(-4\), 13, \(-14\), \(-15\), 16, \(-9\), 10, 11, \(-12\)}, {6, 5, \(-8\), \(-7\), \(-2\), \(-1\), 4, 3, 14, 13, \(-16\), \(-15\), \(-10\), \(-9\), 12, 11}, {7, 8, 5, 6, \(-3\), \(-4\), \(-1\), \(-2\), 15, 16, 13, 14, \(-11\), \(-12\), \(-9\), \(-10\)}, {8, \(-7\), 6, \(-5\), \(-4\), 3, \(-2\), 1, 16, \(-15\), 14, \(-13\), \(-12\), 11, \(-10\), 9}, {9, \(-10\), \(-11\), 12, \(-13\), 14, 15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 7, \(-8\)}, {10, 9, \(-12\), \(-11\), \(-14\), \(-13\), 16, 15, 2, 1, \(-4\), \(-3\), \(-6\), \(-5\), 8, 7}, {11, 12, 9, 10, \(-15\), \(-16\), \(-13\), \(-14\), 3, 4, 1, 2, \(-7\), \(-8\), \(-5\), \(-6\)}, {12, \(-11\), 10, \(-9\), \(-16\), 15, \(-14\), 13, 4, \(-3\), 2, \(-1\), \(-8\), 7, \(-6\), 5}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, \(-13\), 16, \(-15\), 10, \(-9\), 12, \(-11\), 6, \(-5\), 8, \(-7\), 2, \(-1\), 4, \(-3\)}, {15, \(-16\), \(-13\), 14, 11, \(-12\), \(-9\), 10, 7, \(-8\), \(-5\), 6, 3, \(-4\), \(-1\), 2}, {16, 15, \(-14\), \(-13\), 12, 11, \(-10\), \(-9\), 8, 7, \(-6\), \(-5\), 4, 3, \(-2\), \(-1\)}}, {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + \ f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\ \^2 - 4\ \((\((e\ i + f\ j + g\ k + h\ l + a\ m + b\ n + c\ o + d\ p)\)\^2 + \ \((c\ i + d\ j + a\ k + b\ l - g\ m - h\ n - e\ o - f\ p)\)\^2 + \((b\ i + a\ \ j - d\ k - c\ l - f\ m - e\ n + h\ o + g\ p)\)\^2 + \((a\ i - b\ j - c\ k + d\ \ l - e\ m + f\ n + g\ o - h\ p)\)\^2 + \((\(-d\)\ e + c\ f - b\ g + a\ h - l\ \ m + k\ n - j\ o + i\ p)\)\^2)\)}, , , {1, \(-2\), \(-3\), \(-4\), \(-5\), \ \(-6\), \(-7\), 8, 9, 10, 11, \(-12\), 13, \(-14\), \(-15\), \(-16\), 4}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"CL04\"]={16,25, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,-1,4,-3,6,-5,8,-7,10,-9,12,-11,\ 14,-13,16,-15},{3,-4,-1,2,7,-8,-5,6,11,-12,-9,10,15,-16,-13,14},{4,3,-2,-1,8,\ 7,-6,-5,12,11,-10,-9,16,15,-14,-13},{5,-6,-7,8,-1,2,3,-4,13,-14,-15,16,-9,10,\ 11,-12},{6,5,-8,-7,-2,-1,4,3,14,13,-16,-15,-10,-9,12,11},{7,8,5,6,-3,-4,-1,-2,\ 15,16,13,14,-11,-12,-9,-10},{8,-7,6,-5,-4,3,-2,1,16,-15,14,-13,-12,11,-10,9},{\ 9,-10,-11,12,-13,14,15,-16,-1,2,3,-4,5,-6,-7,8},{10,9,-12,-11,-14,-13,16,15,-\ 2,-1,4,3,6,5,-8,-7},{11,12,9,10,-15,-16,-13,-14,-3,-4,-1,-2,7,8,5,6},{12,-11,\ 10,-9,-16,15,-14,13,-4,3,-2,1,8,-7,6,-5},{13,14,15,16,9,10,11,12,-5,-6,-7,-8,-\ 1,-2,-3,-4},{14,-13,16,-15,10,-9,12,-11,-6,5,-8,7,-2,1,-4,3},{15,-16,-13,14,\ 11,-12,-9,10,-7,8,5,-6,-3,4,1,-2},{16,15,-14,-13,12,11,-10,-9,-8,-7,6,5,-4,-3,\ 2,1}}, {(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*((-(h*\ i)-g*j+f*k+e*l-d*m-c*n+b*o+a*p)^2+(-(g*i)+h*j+e*k-f*l-c*m+d*n+a*o-b*p)^2+(-(f*\ i)+e*j-h*k+g*l-b*m+a*n-d*o+c*p)^2+(-(d*i)+c*j-b*k+a*l+h*m-g*n+f*o-e*p)^2+(-(d*\ e)+c*f-b*g+a*h-l*m+k*n-j*o+i*p)^2)},,,{1,-2,-3,-4,-5,-6,-7,8,-9,-10,-11,12,-\ 13,14,15,16,4},gp,{a,c,b,-d,-e,-f}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"O8\"]={16,27, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,\ 16,15},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{4,3,2,1,8,7,6,5,12,11,10,9,\ 16,15,14,13},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{6,5,8,7,2,1,4,3,14,13,\ 16,15,10,9,12,11},{7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10},{8,7,6,5,4,3,2,1,\ 16,15,14,13,12,11,10,9},{9,10,11,12,13,14,15,16,-1,-2,-3,-4,-5,-6,-7,-8},{10,\ 9,12,11,14,13,16,15,-2,-1,-4,-3,-6,-5,-8,-7},{11,12,9,10,15,16,13,14,-3,-4,-1,\ -2,-7,-8,-5,-6},{12,11,10,9,16,15,14,13,-4,-3,-2,-1,-8,-7,-6,-5},{13,14,15,16,\ 9,10,11,12,-5,-6,-7,-8,-1,-2,-3,-4},{14,13,16,15,10,9,12,11,-6,-5,-8,-7,-2,-1,\ -4,-3},{15,16,13,14,11,12,9,10,-7,-8,-5,-6,-3,-4,-1,-2},{16,15,14,13,12,11,10,\ 9,-8,-7,-6,-5,-4,-3,-2,-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},{(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],(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],(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],(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],(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],(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],(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],(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]},{(Sqrt[a]*Cos[b]+\ Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]+\ Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]\ +Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/\ 8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]-\ Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]\ -Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]-\ Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]\ -Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[\ b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]+\ Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]\ -Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/\ 8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]-\ Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]\ +Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]+\ Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]\ +Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[\ b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]+\ Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]\ -Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/\ 8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]-\ Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]\ -Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]-\ Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]\ -Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[\ b]+Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]+\ Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8},{1,2,3,4,5,6,7,8,-9,-10,-11,-12,-13,-14,-15,\ -16,1,1,1,1,1,1,1,1},{1,0.,1,0.,1,0.,1,0.,1,0.,1,0.,1,0.,1,0.},{a,b,c,-d,e,-f,\ -g,-h,i,-j,-k,-l,-m,-n,-o,p}};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"Q8M2\"]={16,34, {{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}, {2,5,8,3,6,1,4,7,10,13,12,15,14,9,16,11}, {3,4,5,6,7,8,1,2,11,16,13,10,15,12,9,14}, {4,7,2,5,8,3,6,1,12,11,14,13,16,15,10,9}, {5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12}, {6,1,4,7,2,5,8,3,14,9,16,11,10,13,12,15}, {7,8,1,2,3,4,5,6,15,12,9,14,11,16,13,10}, {8,3,6,1,4,7,2,5,16,15,10,9,12,11,14,13}, {9,14,15,16,13,10,11,12,1,6,7,8,5,2,3,4}, {10,9,12,15,14,13,16,11,2,1,8,3,6,5,4,7}, {11,16,9,10,15,12,13,14,3,4,1,6,7,8,5,2}, {12,11,14,9,16,15,10,13,4,7,2,1,8,3,6,5}, {13,10,11,12,9,14,15,16,5,2,3,4,1,6,7,8}, {14,13,16,11,10,9,12,15,6,5,4,7,2,1,8,3}, {15,12,13,14,11,16,9,10,7,8,5,2,3,4,1,6}, {16,15,10,13,12,11,14,9,8,3,6,5,4,7,2,1}}, {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-e)^2+(b-f)^2+(c-g)^2+(d-h)^2-(j-n)^2 -(k-o)^2-(l-p)^2-(i-m)^2},,, {1,6,7,8,5,2,3,4,9,10,11,12,13,14,15,16,1,1,1,1,1,1,1,1,4},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ Hoop[\"C3C4C2\"] = {24,9,{ {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}, {2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16,20,21,19,23,24,22}, {3,1,2,6,4,5,9,7,8,12,10,11,15,13,14,18,16,17,21,19,20,24,22,23}, {4,5,6,7,8,9,10,11,12,1,2,3,16,17,18,19,20,21,22,23,24,13,14,15}, {5,6,4,8,9,7,11,12,10,2,3,1,17,18,16,20,21,19,23,24,22,14,15,13}, {6,4,5,9,7,8,12,10,11,3,1,2,18,16,17,21,19,20,24,22,23,15,13,14}, {7,8,9,10,11,12,1,2,3,4,5,6,19,20,21,22,23,24,13,14,15,16,17,18}, {8,9,7,11,12,10,2,3,1,5,6,4,20,21,19,23,24,22,14,15,13,17,18,16}, {9,7,8,12,10,11,3,1,2,6,4,5,21,19,20,24,22,23,15,13,14,18,16,17}, {10,11,12,1,2,3,4,5,6,7,8,9,22,23,24,13,14,15,16,17,18,19,20,21}, {11,12,10,2,3,1,5,6,4,8,9,7,23,24,22,14,15,13,17,18,16,20,21,19}, {12,10,11,3,1,2,6,4,5,9,7,8,24,22,23,15,13,14,18,16,17,21,19,20}, {13,14,15,16,17,18,19,20,21,22,23,24,1,2,3,4,5,6,7,8,9,10,11,12}, {14,15,13,17,18,16,20,21,19,23,24,22,2,3,1,5,6,4,8,9,7,11,12,10}, {15,13,14,18,16,17,21,19,20,24,22,23,3,1,2,6,4,5,9,7,8,12,10,11}, {16,17,18,19,20,21,22,23,24,13,14,15,4,5,6,7,8,9,10,11,12,1,2,3}, {17,18,16,20,21,19,23,24,22,14,15,13,5,6,4,8,9,7,11,12,10,2,3,1}, {18,16,17,21,19,20,24,22,23,15,13,14,6,4,5,9,7,8,12,10,11,3,1,2}, {19,20,21,22,23,24,13,14,15,16,17,18,7,8,9,10,11,12,1,2,3,4,5,6}, {20,21,19,23,24,22,14,15,13,17,18,16,8,9,7,11,12,10,2,3,1,5,6,4}, {21,19,20,24,22,23,15,13,14,18,16,17,9,7,8,12,10,11,3,1,2,6,4,5}, {22,23,24,13,14,15,16,17,18,19,20,21,10,11,12,1,2,3,4,5,6,7,8,9}, {23,24,22,14,15,13,17,18,16,20,21,19,11,12,10,2,3,1,5,6,4,8,9,7}, {24,22,23,15,13,14,18,16,17,21,19,20,12,10,11,3,1,2,6,4,5,9,7,8}}, {a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p+q+r+s+t+u+v+w+x, a+b+c-d-e-f+g+h+i-j-k-l+m+n+o-p-q-r+s+t+u-v-w-x, a+b+c+d+e+f+g+h+i+j+k+l-m-n-o-p-q-r-s-t-u-v-w-x, a+b+c-d-e-f+g+h+i-j-k-l-m-n-o+p+q+r-s-t-u+v+w+x, (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, (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, ((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, ((a-b+d-e+g-h+j-k-m+n-p+q-s+t-v+w)^2+(-a+c-d+f-g+i-j+l+m-o+p-r+s-u+v-x)^2+ \ (b-c+e-f+h-i+k-l-n+o-q+r-t+u-w+x)^2)/2, ((a-b-d+e+g-h-j+k+m-n-p+q+s-t-v+w)^2+(-a+c+d-f-g+i+j-l-m+o+p-r-s+u+v-x)^2+ (b-c-e+f+h-i-k+l+n-o-q+r+t-u-w+x)^2)/2, ((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, (1/4)*((a-b-d+e-g+h+j-k+m-n-p+q-s+t+v-w)^4+ (a-b+d-e-g+h-j+k+m-n+p-q-s+t-v+w)^4+(-a+c-d+f+g-i+j-l-m+o-p+r+s-u+v-x)^4+ (b-c-e+f-h+i+k-l+n-o-q+r-t+u+w-x)^4+(-a+c+d-f+g-i-j+l-m+o+p-r+s-u-v+x)^4+ (b-c+e-f-h+i-k+l+n-o+q-r-t+u-w+x)^4)- ((a-b-g+h+m-n-s+t)^2+(-a+c+g-i-m+o+s-u)^2+(b-c-h+i+n-o-t+u)^2)* ((d-e-j+k+p-q-v+w)^2+(-d+f+j-l-p+r+v-x)^2+(e-f-k+l+q-r-w+x)^2), (1/4)*((a-b+d-e-g+h-j+k-m+n-p+q+s-t+v-w)^4+ (a-b-d+e-g+h+j-k-m+n+p-q+s-t-v+w)^4+(-a+c+d-f+g-i-j+l+m-o-p+r-s+u+v-x)^4+ (b-c+e-f-h+i-k+l-n+o-q+r+t-u+w-x)^4+(-a+c-d+f+g-i+j-l+m-o+p-r-s+u-v+x)^4+ (b-c-e+f-h+i+k-l-n+o+q-r+t-u-w+x)^4)- ((a-b-g+h-m+n+s-t)^2+(b-c-h+i-n+o+t-u)^2+(-a+c+g-i+m-o-s+u)^2)* ((d-e-j+k-p+q+v-w)^2+(e-f-k+l-q+r+w-x)^2+(-d+f+j-l+p-r-v+x)^2)},,, \ {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},,};\ \>", "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ \tFive non-Moufang tables are appended. They do not have the Frobenius \ property. They are not Hoops, as indicated by the final \"n\" in their names.\ \ \>", "Text", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] := {mm = 4, nn = 14, hoopTbl = {{1, 2, 3, 4}, {2, 1, 4, \(-3\)}, {3, \(-4\), 1, 2}, {4, 3, \(-2\), 1}}, sh = {a\^2 - b\^2 - c\^2 - d\^2, a\^2 + b\^2 + c\^2 + d\^2}, , , gi = {1, 2, 3, 4, 1, 1}, , plex = {a, \(-b\), \(-c\), \(-d\)}};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] := {mm = 6, nn = 13, hoopTbl = {{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}}, sh = {a + b + c - d - e - f, a + b + c + d + e + f, 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 \ - \((d - e)\)\^2 - \((e - f)\)\^2 - \((\(-d\) + f)\)\^2)\), 1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 \ + \((d - e)\)\^2 + \((e - f)\)\^2 + \((\(-d\) + f)\)\^2)\)}, , , gi = {1, 3, 2, 5, 4, 6, 1, 1, 1, 1}, , a + b + c - d - e - f};\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] := {mm = 8, nn = 36, hoopTbl = {{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}}, sh = {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, \((a - e)\)\^2 + \((b - f)\)\^2 + \((c - g)\)\^2 + \((d - h)\)\^2}, \ , , gi = {1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ The Alt8n table passes the alternativeQ and moufangQ test (i.e. up to a sign) \ but is not conservative.\ \>", "Text", PageWidth->WindowWidth, InitializationCell->True], Cell[BoxData[ \(\(Hoop["\"] = {mm = 8, nn = 33, hoopTbl = {{1, 2, 3, 4, 5, 6, 7, 8}, {2, \(-1\), 4, \(-3\), 6, \(-5\), 8, \(-7\)}, {3, 4, 1, 2, 7, 8, 5, 6}, {4, \(-3\), 2, \(-1\), 8, \(-7\), 6, \(-5\)}, {5, \(-6\), \(-7\), \(-8\), \(-1\), 2, 3, 4}, {6, 5, \(-8\), 7, \(-2\), \(-1\), 4, \(-3\)}, {7, \(-8\), \(-5\), \(-6\), \(-3\), 4, 1, 2}, {8, 7, \(-6\), 5, \(-4\), \(-3\), 2, \(-1\)}}, , , , gi = {1, \(-2\), 3, \(-4\), \(-5\), \(-6\), 7, \(-8\), 1}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell["\<\ The D3Mn loop is the smallest noncommutative Moufang loop, created by Chein \ doubling of the D3 group. It is not conservative.\ \>", "Text", PageWidth->WindowWidth], Cell[CellGroupData[{ Cell[BoxData[ \(id["\"]\)], "Input", PageWidth->WindowWidth], Cell[BoxData[ \("D3Mn"\)], "Output", PageWidth->WindowWidth] }, Open ]], Cell[BoxData[ \(gd[]\)], "Input", PageWidth->WindowWidth], Cell[BoxData[ \(glo // tf\)], "Input", PageWidth->WindowWidth], Cell[BoxData[ \(\(Hoop["\"] = {mm = 12, nn = 28, hoopTbl = {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 1, 6, 5, 4, 3, 8, 7, 10, 9, 12, 11}, {3, 4, 5, 6, 1, 2, 9, 12, 11, 8, 7, 10}, {4, 3, 2, 1, 6, 5, 10, 11, 12, 7, 8, 9}, {5, 6, 1, 2, 3, 4, 11, 10, 7, 12, 9, 8}, {6, 5, 4, 3, 2, 1, 12, 9, 8, 11, 10, 7}, {7, 8, 11, 10, 9, 12, 1, 2, 5, 4, 3, 6}, {8, 7, 10, 11, 12, 9, 2, 1, 6, 3, 4, 5}, {9, 10, 7, 12, 11, 8, 3, 6, 1, 2, 5, 4}, {10, 9, 12, 7, 8, 11, 4, 5, 2, 1, 6, 3}, {11, 12, 9, 8, 7, 10, 5, 4, 3, 6, 1, 2}, {12, 11, 8, 9, 10, 7, 6, 3, 4, 5, 2, 1}}, , , , gi = {1, 5, 4, 3, 6.7, 8, 9, 10, 11, 12}, , };\)\)], "Input", PageWidth->WindowWidth, InitializationCell->True] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["9.", FontFamily->"Times New Roman"], " ", StyleBox["Exported functions.", FontFamily->"Times New Roman"] }], "Subsection", PageWidth->WindowWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True], Cell[CellGroupData[{ Cell[TextData[{ "9.1. Main Procedures; ", StyleBox["as, hoopTimes, hoopInverse, hoopPower, Use", FontSlant->"Italic"], "." }], "Subsubsection", PageWidth->WindowWidth, InitializationCell->True], Cell[TextData[{ "\tSymbolic functions such as ", StyleBox["topol. tovec", FontSlant->"Italic"], " are given new values by e.g. ", StyleBox["topol/.as[{3.,1.,4.}].", FontSlant->"Italic"] }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \(as[A_] := (*Relace\ alph\ dummy\ variables\ by\ the\ coefficients\ in\ \ A\ *) \[IndentingNewLine]Table[ alph[\([ii]\)] \[Rule] A[\([ii]\)], {ii, Min[mm, Length[A]]}]; as[A_, B_] := (*Relace\ B\ variables\ by\ the\ coefficients\ in\ A\ \ *) \[IndentingNewLine]Table[ B[\([ii]\)] \[Rule] A[\([ii]\)], {ii, Min[mm, Length[A], Length[B]]}];\)], "Input", InitializationCell->True, FormatType->TraditionalForm], Cell[TextData[{ "\t", StyleBox["hoopTimes", FontSlant->"Italic"], StyleBox[" multiplies two vectors and (if ", FontSlant->"Plain"], StyleBox["c", FontSlant->"Italic"], StyleBox["==0) sets the remainders ", FontSlant->"Plain"], StyleBox["Rl & Rr", FontSlant->"Italic"], ". ", StyleBox["Lists of zeroes", FontSlant->"Plain"], " ", StyleBox["t, Rl, Rr", FontSlant->"Italic"], StyleBox[" are initialised. A & B are padded with zeroes and truncated to \ the correct length.\n\tThe product ", FontSlant->"Plain"], StyleBox["t", FontSlant->"Italic"], StyleBox[" is built up by adding each (signed) product to the appropriate \ element. If disparate zero-sizes occur, some sizes will be lost from the \ product AB and AB/A will not recover B; left and right remainders are created \ so that sizes are conserved and ", FontSlant->"Plain"], "B=AB/A+Rr", StyleBox[" & ", FontSlant->"Plain"], "A=AB/B+Rl", StyleBox[".", FontSlant->"Plain"] }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \(hoopTimes[AA_, BB_, c_: 0] := (*Multiply\ two\ vectors, \ calculating\ remainders\ if\ c \[Equal] 0*) \[IndentingNewLine]Module[{t = Table[0, {mm}], R, A, B}, Rl = t; Rr = t; R = t; A = Take[Flatten[Append[AA, t]], mm]; B = Take[Flatten[Append[BB, t]], mm]; \[IndentingNewLine]Do[ Do[t\[LeftDoubleBracket]Abs[ hoopTbl[\([ll, kk]\)]]\[RightDoubleBracket] += A\[LeftDoubleBracket] ll\[RightDoubleBracket]\ B\[LeftDoubleBracket] kk\[RightDoubleBracket] Sign[hoopTbl[\([ll, kk]\)]]\ , {kk, mm}], {ll, mm}]; (*\(Calculate\ remainders\ Rl\ &\)\ Rr\ if\ any\ sizes\ \ are\ lost*) \[IndentingNewLine]If[ c \[Equal] 0, \[IndentingNewLine]R = Chop[Simplify[B - hoopTimes[hoopInverse[A], t, 1]], hmin]; Rl = Chop[Simplify[A - hoopTimes[t, hoopInverse[B], 1]], hmin], ]; Rr = R; t]\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[TextData[{ "\tDivision is effected by the same procedure, ", StyleBox["hoopTimes", FontSlant->"Italic"], ", as multiplication, using the multiplicative inverse of the divisor. The \ inverse always exists because all known hoops have the Moufang property; it \ is calculated by ", StyleBox["hoopInverse", FontSlant->"Italic"], " so that Ainv.A={1,0,..}. \n\t", StyleBox["inv", FontSlant->"Italic"], " is set up as a table of zeroes, and then filled with symbolic \ partial-fractions ", StyleBox["D[sh[[jj]],var]/sh[[jj]]", FontSlant->"Italic"], " with sizes (defined in ", StyleBox["sh", FontSlant->"Italic"], ") as denominators and their derivations as numerators. ", StyleBox["gi", FontSlant->"Italic"], " contains the signed list of locations of 1's in the Cayley table; this \ identifies the differentiation variable ", StyleBox["var", FontSlant->"Italic"], ". In the ", StyleBox["jj", FontSlant->"Italic"], " loop, a substitution rule ", StyleBox["as", FontSlant->"Italic"], " attempts to calculate the numeric value of each shape for A. Any that \ differ from zero by less than ", StyleBox["hmin ", FontSlant->"Italic"], "(arbitrarily set at 4 times the \"Machine Epsilon\") are ignored - the \ calculations continue in a constrained sub-algebra by omitting the \"infinite\ \" partial fractions. ", StyleBox["gi", FontSlant->"Italic"], " also contains ", StyleBox["gi[[mm+jj]],", FontSlant->"Italic"], " the multiplicity of each size; this is a multiplier for the signed \ partial fraction. Finally ", StyleBox["inv", FontSlant->"Italic"], " is divided by ", StyleBox["mm", FontSlant->"Italic"], ", and the symbols are replaced by the elements of A. In effect, Cramer's \ method has been used, so each term of the result has the determinant as a \ divisor. If the determinant factorizes, each term is expressed as a sum of \ partial fraction with factors (sizes) as denominators. " }], "Text", PageWidth->WindowWidth], Cell[BoxData[ \(hoopInverse[ A_] := (*Calculate\ inverse\ of\ \(\(A\)\(.\)\)\ \ *) \[IndentingNewLine]Module[{inv = Table[0, {ii, mm}], ls = Length[sh], var, t}, \[IndentingNewLine]Do[ (*\ ii\ elements\ *) ss = Sign[gi[\([ii]\)]]; t = 0; var = alph[\([gi[\([ii]\)]/ss]\)]; (*Diff' n\ variable*) \[IndentingNewLine]Do[ (*jj\ shapes*) If[ Chop[Simplify[sh[\([jj]\)] /. \ as[A]], hmin] === 0, Null (*Omits\ any\ near - zero\ cases*) , t += ss*gi[\([mm + jj]\)]*\((D[sh[\([jj]\)], var]/ sh[\([jj]\)])\)], {jj, ls}]; \[IndentingNewLine]inv[\([ii]\)] += t, {ii, mm}]; Simplify[inv/mm /. \ as[A]]]\)], "Input", PageWidth->WindowWidth, InitializationCell->True], Cell[TextData[{ "\t", StyleBox["hoopPower", FontSlant->"Italic"], " ", StyleBox["calculates ", FontSlant->"Plain"], Cell[BoxData[ \(TraditionalForm\`A\^p\)]], StyleBox[", giving ", FontSlant->"Plain"], Cell[BoxData[ \(TraditionalForm\`\@A\)]], StyleBox[" if ", FontSlant->"Plain"], StyleBox["p", FontSlant->"Italic"], StyleBox[" is not specified", FontSlant->"Plain"], ".", StyleBox[" The polar-vector interconversions ", FontSlant->"Plain"], StyleBox["topol, tovec", FontSlant->"Italic"], StyleBox[" are needed for the algebra (not all are available, many may not \ exist). 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