Hoop Algebra Supplement.

R.H.Beresford.  21/2/07.

1. Introduction.

This document is supplementary to "HoopAlg&Physics.nb"[in 1], which is a brief account of Hoop Algebras (also available as Hoops&Physics.doc[2]).

Hoops are defined as vector algebras that conserve the symbolic determinant factors of their Cayley multiplication tables as symmetries or "sizes". Symmetry conservation leads, via Noether's theorem, to forces and to particles. Consequently, Hoops subsume and unify all the algebras (including Real, Complex, Quaternion, Octonion, Clifford and Wedge) relevant to physics. They may provide a new paradigm for particle physics, based on vector multiplication by finite Moufang loops as well as coordinate transformations with continuous groups. Using experimental mathematics the author has found many analogies between hoop algebras and particle physics, but (as a 77 yr. old chemical engineer) lacks the skills and time to take the subject much further.
	The Hoops concept has been implemented in MathSource/6198/ [1] as a Mathematica package Hoops.m. This contains over 80 named Hoops (with a few counter-example non-hoop Loops), three main procedures hoopTimes, hoopInverse, hoopPower, and a few subsidiary procedures. Each hoop is defined by a Cayley index-table (a preferred isomorph) and its "shape" (the distinct factors of the inverse symbolic Cayley table). Multiplication, division, and (in many cases) inter-conversion between Cartesian vector {a,b,...} and polar {r,θ...} form are provided.
	The figures in this document are self-explanatory output from a Mathematica session (with results of calculations appearing in bold-face), comprehensible to readers with no knowledge of Mathematica. They use the Hoops.m package. The input data are usually integer vectors, for clarity.  Mathematica users can test them with different data, by installing Hoops.nb and running it (to create the Hoops.m package) in the Addons\ExtraPackages directory, and then using the following instructions:-.

Quit[]

<< "Hoops`"

2. Demonstrations.

Summary.
	Examples demonstrate multiplication and division of vectors in different hoop algebras, the "folding" of loops to signed algebras with conserved sizes and remainders, polar-duals with powers and roots, non-Abelian algebras with rotated roots, and the unification of many algebras by the Hoop concept. Then multi-phase sinusoidal  "Orbits" are introduced, as unital dual sub-algebras with relevance to physics.

2.1 Selecting Hoop Algebras.

When hoops.m has been activated a specific Hoop is selected by Use["H"], where "H" is any Hoop name. This assigns values to the variables {hoopTbl, sh, topol, tovec, mm, nn, gi, gp, plex}. The Cayley table is put into hoopTbl. sh is the shape. topol is the polar form (if known); it defines angles to go with each quadratic size in sh. tovec is the reversion from polar to vector form. This information allows the procedures hoopTimes, hoopInverse, & hoopPower to calculate vector products, inverses and (where appropriate) powers, and roots. (Vector addition is element-by-element for all hoops with the same number of elements.)
	Example 1 shows the index table, the symbolic form, the shape, the polar form and the vector reversion for the C3, C4 & C2 algebras. C2 is unusual, having a “dual” form that does not involve an angle. (Index tables will be shown in later examples if they have not been shown previously and are not too large; later input instructions will be hidden in "closed cells" because they may be incomprehensible to non-Mathematica users.)  Please ignore the commas that separate explanatory messages and the calculated results.

{"Example 1a" // bf ,"C2 Table and properties.\nThe C2 index table is",Use["C2"]; hoopTbl // tf,"\nThe symbolic form is ",s = 
            hoopTbl /. 
              Table[ii -> 
                  alph[[ii]], {ii, mm}]; 
          s // tf,"with determinant",Det[s],s =. ;"\nthis factorises into the shape sh",sh,"\ntopol is",topol,"\ntovec is",tovec,
"\nExample 1b" // bf," C3 Table and properties.\nThe C3 index table is",Use["C3"]; hoopTbl // tf,"\nThe symbolic form is ",s = 
            hoopTbl /. 
              Table[ii -> 
                  alph[[ii]], {ii, mm}]; 
          s // tf,"with determinant",Det[s],s =. ;"\nthis factorises into the shape sh",sh,"\ntopol is",topol,"\ntovec is",tovec,
"\nExample 1c" // bf ," C4 Table and properties.\nThe C4 index table is",Use["C4"]; hoopTbl // tf,"\nThe symbolic form is ",s = 
            hoopTbl /. 
              Table[ii -> 
                  alph[[ii]], {ii, mm}]; 
          s // tf,"with determinant",Det[s],s =. ;"\nthis factorises into the shape sh",sh,"\ntopol is",topol,"\ntovec is",tovec}

2.2 Vector multiplication and division.

Generalized vectors (abbreviated to "vecs" - they are not vectors in some algebras) are lists of m elements A={(Subscript[a, 1]),(Subscript[a, 2]),...(Subscript[a, m])}. Their components can be numbers (integer, real, complex, or unsigned), operators, or symbols; the "standard form" uses symbols from a table called alph, i.e. A={a,b,c...}. Vectors can also be thought of as summations of coefficients with associated dimensions, signs or directions, A = (Subscript[a, 1])(Subscript[d, 1]) +(Subscript[a, 2])(Subscript[d, 2]) +...+ (Subscript[a, m])(Subscript[d, m]).
	Generalized Hoop multiplication is effected by hoopTimes[A,B]. The elements of the product AB are the sum of each a[[k]] b[[l]] Sign[hoopTbl[[k,l]]] where Table[[k,l]] contains the (possibly signed) index j. The Moufang property ensures that every vec has a multiplicative inverse Ai=hoopInverse[A]; division of B by A is hoopTimes[Ai,B].
	Hoops are "conservative" - the real factors of the symbolic determinant are "size" functions that are conserved on multiplication, size[AB] = ± size[A]size[B]. (Sizes are actually conjugates, but conjugation has no effect in most cases.) The sizes are listed in sh. The inverse has been found by Cramer's method, so the determinant is a denominator of the inverse; multiple sizes allow inverses to split into partial fractions.
	Example 2 demonstrates multiplication and shape conservation in the C2 and C3 algebras; both have two sizes. Only the output is shown; commas separate a description of the input from the resulting output. Note that the sizes of AB are the product of the sizes of A and B. (Section 2.8 shows that this is untrue for loops in general.)

Use["C2"]; {"Example 2. C2 & C3 multiplication & size \
conservation", "\nThe C2 table is", hoopTbl // tf, "with shape", 
      sh, "\nIf A  = ", A = {3, 4}, "\nand B = ", 
      B = {x, y}, "\nAB = ", 
      AB = hoopTimes[A, B], "\nShape of A  = ", 
      sh /.  as[A], "\nShape of B  = ", 
      sh /.  as[B], "\nShape of AB = ", sh /.  as[AB], 
      Use["C3"]; "\n\nChange to C3.\nIf A  = ", 
      A = {3, 4, 1}, "\nand B = ", B = {5, 1, 2}, "\nAB = ", 
      AB = hoopTimes[A, B], "\nShape of A  = ", 
      sh /.  as[A], "\nShape of B  = ", 
      sh /.  as[B], "\nShape of AB = ", sh /.  as[AB]}

Example 3 calculates the inverses of A & B and shows that their sizes are inverses of the sizes of A & B. Then hoopTimes is used to show that Ainverse.AB recovers B, whilst AB.Binverse recovers A. (hoopInverse calculates left-inverses; as C3 is Abelian the distinction is irrelevant here.)

Use["C3"]; {"Example 3. C3 division", "\nAinverse  = ", 
      Ai = hoopInverse[A], "\nShape of Ainverse  = ", 
      sh /.  as[Ai], "\nBinverse  = ", 
      Bi = hoopInverse[B], "\nShape of Binverse  = ", 
      sh /.  as[Bi], "\nAinverse.AB recovers B", 
      hoopTimes[Ai, AB], "\nAB.Binverse recovers A", 
      hoopTimes[AB, Bi]}

2.3 "Folding" Vectors and Loops.

A key concept in Hoop algebra is the "folding" of a vector via an equivalence relationship "~" such as r[[i]]=a[[i]]-a[[i+m/2]] ~ {a[[i]],a[[i+m/2]]}, which creates a single real number by the "2-folding" of pairs of unsigned numbers. This introduces the sign "-". Similarly,  t[[i]] ~ a[[i]]+J a[[i+m/3]]+JJ a[[i+2m/3]] folds three unsigned numbers to a "terplex" number (with signs J & JJ). One way to fold four unsigned numbers is c[[i]] ~ a[[i]]+i a[[i+m/4]]-a[[i+2m/4]]-i a[[i+3m/4]]. This creates a complex number and the signs i & -i.
	Following Sir William Hamilton, sets m of unsigned numbers are "directors" with "m directions"; they fold to vectors with "m/r dimensions". The multiplication tables also fold; an m×m table with r-fold symmetry can undergo r-folding to an (m/r)×(m/r) table. 
	Example 4 shows that "C4" folds to complex algebra. It demonstrates multiplication, division, raising to a power, and extracting a root. In each case, the result is shown to fold to the corresponding complex number:-

Use["C4"];{"Example 4. C4 folds to Complex algebra.\nThe table ",hoopTbl // tf,"can be written as",{{1, i, -1, -i}, {i, -1, -i, 
                1}, {-1, -i, 1, i}, {-i, 1, i, -1}} // tf,"\nMultiplication using hoopTimes:-\n If A  =",A = {3. , 1. , 1. , 2. },"and B =",B = {4. , 7. , 2. , 3. },"\nmultiplication gives \n AB =",AB = hoopTimes[A, B],
"\nThis 'folds' to the product of 3-1 + (1-2)I\nand 4-2 + (7-3)I to give 31-23 + (32-26)I,\ni.e. (2-I)(2+4I)=" ,(2 - I) (2 + 4  I),"\n\nDivision using hoopInverse:-\nThe inverse of A is",Ainv = hoopInverse[{3, 1, 1, 2}],"\nAinv.A is the unit",hoopTimes[Ainv, A],"\nand Ainv.AB recovers B",Simplify[hoopTimes[Ainv, AB]],"\n\nPowers and Roots:-\nhoopPower[A,3] gives the cube,  \nA3=",A3 = hoopPower[A, 3],"which folds to (2-I)^3=",(2 - I)^3,"\nhoopPower[A3,2/3] gives the 2/3 power of A3",hoopPower[A3, 2/3],"which folds to 3 - 4i and matches the repeated product AA",AA = hoopPower[A, 2],"\nA^3 is the same as the repeated product AAA",AAA = hoopTimes[A, hoopTimes[A, A]],"\nhoopTimes[A,hoopTimes[A,A]] (=(A^3))",hoopTimes[A, hoopTimes[A, A]],hoopTimes[A, hoopTimes[A, A]],
\!\(\*
StyleBox["\"",
FontWeight->"Plain"]\)\nPolar Form",topol /. as[A],"has (r^2)=5, φ=-.4638 OK"}

2.4 Powers & repeated products may differ.

Example 4 checked whether (A^2) = AA and (A^3) = AAA. Powers and repeated products may differ for three reasons, negative sizes (example 5), "angle wrap-round" (example 6), and non-commutativity (section 2.6).
	If a size is negative for A it alternates in sign for repeated products but the power is a continuous function of the exponent, as in Example 5.

Use["C4"];{"Example 5. C4 with negative size., so AA differs from A^2\nIf A  =",A = {1. , 1. , -1. , 0. },
" with shape ",sh /. as[A],"\nhoopPower[A,3] gives the cube,\!\(\*
StyleBox[\(\*
StyleBox[" ",
FontSlant->"Plain"]\\ \)]\)A3= ",A3 = hoopPower[A, 3],"\nwhich folds to 2+11i.\nhoopPower[A3,2/3] gives the 2/3 power of A3",hoopPower[A3, 2/3],"\nwhich folds to 3+4i and matches A^2",A2 = hoopPower[A, 2],
"\nwith shape",sh /. as[A2],"\nwhilst the repeated product AA",AA = hoopTimes[ A, A],
"\nhas the second size positive",sh /. as[AA],"\nA^3 is the same as the repeated product AAA",AAA = hoopTimes[A, hoopTimes[A, A]],"\nThe second size of A^2 is negative, of AA is positive:-\nPolar A   =",topol /.  as[A],"\nPolar A^2 =",topol /.  as[A2],"\nPolar AA  =",topol /.  as[AA],"\nPolar AAA =",topol /.  as[A3],"\n\n  A^1.99",hoopPower[A, 1.99],"\n& A^2.01",hoopPower[A, 2.01],"\n  bracket A^2\nwhilst AA =",AA}

Example 6 shows that if a polar-form angle parameter is large, raising to a power (which multiplies the angle by the exponent) may give an angle greater than 2π. This will become a negative angle and the roots will be rotated. Their sizes will be correct. (Taking roots makes the angle parameter smaller, and so cannot cause wrap-round.)

Use["C4"];{"Example 6. C4 with large angle. (A^3)^(1/3) differs from A.\nIf A  =",A = {2. , 2. , 1. , 0. },"which folds to 1+2i\nhoopPower[A,3] gives the cube,  A3= ",A3 = hoopPower[A, 3],"\nwhich folds to -11-2i\nhoopPower[A3,1/3] gives rA3, the 1/3 power of A3",rA3 = hoopPower[A3, 1/3],"\nwhich folds to 1.232-1.866i (which cubes correctly)\nrA3 differs from A, but has the same sizes","\nThe angle of A^3 exceeds 2Pi and so is negated:-\nPolar A   =",topol /.  as[A],"\nPolar AAA =",topol /.  as[A3],"\nPolar rA3 =",topol /.  as[rA3]}

2.5 Signed Tables, Generalized Signs.

Many Hoops are "signed tables" with signed products in the body of the multiplication table, but the elements are unsigned (unlike the second table in Example 4, where two elements are "labeled" -1 & -i). As these products are not members of the defining set, such tables are not loops. Signed tables are equivalence relations, folded from unsigned Moufang loops with r-fold symmetry. The commonest cases involve negation, where r = 2 and the signed table is obtained from the top left quarter of the Moufang loop. Complex algebra is the "C4c" hoop, folded from C4. Compare the following table with the 2×2 quarter-tables in Example 4. Index 3 becomes -1, corresponding to index 2 becoming i with i.i =-1. Example 7 is equivalent to the complex multiplication (3+1 i)(x+y i) = (3x-y)+i(x+3y). (C4c is demonstrated in more detail later, in Figure S1.)

Use["C4c"]; {"Example 7. C4c multiplication & division is Complex \
Algebra", "\nThe table is", hoopTbl // tf, "\nA  = ", 
      A = {3, 1}, "\nB  = ", B = {x, y}, "\nAB = ", 
      AB = hoopTimes[A, B], "\nAinv = ", 
      Ainv = hoopInverse[A], "\nAinv.AB = B ", 
      Simplify[hoopTimes[hoopInverse[A], 
          AB]], "\nComplex AB (3+\[ImaginaryI])(x+\[ImaginaryI]y) = (3x-y) \
+ \[ImaginaryI](x+3y)"}

The Davenport Algebra is another signed table. It is folded from the following C4C2 isomorph, with 5→-1, 6→-2, 7→-3, and gives two copies of the complex field, with only the origin in common. It has application in binocular vision.

\(Use["Dav"]; {"Example 8.  Dav division", "
The table is ", 
      hoopTbl // tf, "
A=  ", A = {3, 4, 6, 1}, "
B=  ", 
      B = {w, x, y, z}, "
AB =", 
      AB = hoopTimes[A, B], "
Ainv = ", 
hoopInverse[
        A], "
Ainv.AB = B ", 

      Simplify[hoopTimes[hoopInverse[A], AB]]}\)

Some hoops have complex products, with i occurring in the body of the multiplication table. The g2401c algebra has 6 elements; it is "4-folded" from the group g2401, i.e. the first group with 24 elements in the GAP Group Atlas (not shown):-

\(Use["g2401c"]; {"Example 9. g2401c division
", "The table is ", hoopTbl // tf, "
A= ", A = {3, 4, 6, 1, 2, 7}, "
B= ", 
      B = {u, v, w, x, y, z}, "
AB = ", 
      AB = hoopTimes[A, B], "
Ainv = ", 
hoopInverse[
        A], "
Ainv.AB = B ", 

      Simplify[hoopTimes[hoopInverse[A], AB]]}\)

A few hoops have J & (J^2) (primitive cube roots of +1) as signs. "C9J" is 3-folded from the following C9 isomorph by 9→(J^2)3, 5→ J 2:-

(({{1, 2, 3, 4, 5, 6, 7, 8, 9}, {2, 9, 1, 5, 3, 4, 8, 6, 7}, {3, 1, 5, 6, 4, 8, 9, 7, 2}, {4, 5, 6, 7, 8, 9, 1, 2, 3}, {5, 3, 4, 8, 6, 7, 2, 9, 1}, {6, 4, 8, 9, 7, 2, 3, 1, 5}, {7, 8, 9, 1, 2, 3, 4, 5, 6}, {8, 6, 7, 2, 9, 1, 5, 3, 4}, {9, 7, 2, 3, 1, 5, 6, 4, 8}}))

Results need extensive simplification, using the rule js, i.e. J^any_→J^Mod[any,3]:-

\(Use["C9j"]; {"Example 10. C9j division.
", "The table is ", hoopTbl // tf, "
 A= ", A = {3, 4, 6}, "
 B= ", 
      B = {x, y, z}, "
AB =", 
      AB = hoopTimes[A, B], "
Ainv = ", 
hoopInverse[
        A], "
Ainv.AB unsimplified ", 
      hoopTimes[hoopInverse[A], AB], "
Simplified Ainv.AB = B ", 

      Simplify[Simplify[hoopTimes[hoopInverse[A], AB]] //. js]}\)

J and (J^2) are the only "generalized signs" used in this document; the Hoops' package includes instructions that make J behave as a sign. Other generalized signs are developed in a similar way in [4], as powers of primitive roots of unity.

2.6 Non-Abelian Hoops, Left & Right Inverses.

So far all the examples (except g2401c) have used Abelian (commutative) hoops. Example 11 demonstrates the non-Abelian (AB differs from BA) D3 algebra and Clifford(2,1) algebra. 
	hoopInverse calculates a left inverse, so Ai.AB divides AB by A, recovering B. Similarly Bi.BA recovers A, whilst BA.Ai recovers B because A is the left inverse of Ai. (Examples 16 & 18 also use D3.)

\({"Example 11. D3 & CL21 Left and Right division
", 
      Use["D3"]; "The D3 table is", hoopTbl // tf, "
A= ", 
      A = {3, 4, 6, 1, 2, 7}, "
B= ", 
      B = {4, \(-1\), 2, 2, 5, 3}, "
AB = ", 
      AB = hoopTimes[A, B], "
Ainv = ", 
Ainv = 
        hoopInverse[A], "
Ainv.AB = B ", 

      Simplify[hoopTimes[Ainv, AB]], 
"
BA = ", 
      BA = hoopTimes[B, A], "
Binv.BA = A ", 

      Simplify[hoopTimes[hoopInverse[B], BA]], "
BA.Ainv = B ", 

      Simplify[hoopTimes[BA, Ainv, 
          AB]], 
Use["CL21"]; "The CL21 table is", 
      hoopTbl // tf, "
A= ", A = {3, 4, 6, 1, 2, 7, 2, 1}, "
B= ", 
      B = {4, \(-1\), 2, 2, 5, 3, 2, \(-3\)}, "
AB = ", 
      AB = hoopTimes[A, B], "
Ainv = ", 
Ainv = 
        hoopInverse[A], "
Ainv.AB = B ", 

      Simplify[hoopTimes[Ainv, AB]], 
"
BA = ", 
      BA = hoopTimes[B, A], "
Binv.BA = A ", 

      Simplify[hoopTimes[hoopInverse[B], BA]], "
BA.Ainv = B ", 

      Simplify[hoopTimes[BA, Ainv, AB]]}\)

2.7 Many Algebras are Hoops.

Example 12 shows that traditional vector dot & cross products are hoop multiplications where only the "univectors" are multiplied i.e. the scalar and multivector elements of the multiplicands are kept at zero. (I use "univectors" where many authors confusingly use "basis vectors", "vectors" or "scalars" for single element generators. Their products are bivectors, trivectors, etc.) The symbols are chosen to match those in [3, p11] for quaternions (Qr) and CL2, and to match [3, p37] for octonions (Octr) and CL3.

\(TraditionalForm\`{Use["Qr"]; \*""Example 12. Dot & Cross products with Qr & Octr.\n{0,(Subscript[e, 1]),(Subscript[e, 2]),0}×{0,(Subscript[u, 1]),(Subscript[u, 2]),0} gives\n   Qr Scalar       & Dot products\n"", 
      hoopTimes[{0, e\_1, e\_2, 0}, {0, u\_1, u\_2, 
          0}], 
Use["Octr"]; \*""\n\nOctr  {0,(Subscript[a, 1]),(Subscript[a, 2]),0,(Subscript[a, 3]),0,0,0}×{0,(Subscript[b, 1]),(Subscript[b, 2]),0,(Subscript[b, 3]),0,0,0}="", \ 
      hoopTimes[{0, a\_1, a\_2, 0, a\_3, 0, 0, 0}, {0, b\_1, b\_2, 0, b\_3, 
          0, 0, 0}]}\)

In both cases the first element of the result is the scalar product (up to a sign), whilst the univectors (and octonion trivector) in the product are zero. The Qr trivector element (Subscript[e, 2]) (Subscript[u, 1]) - (Subscript[e, 1]) (Subscript[u, 2]) is the 2D cross product, and Octc gives the (negated) 3D scalar product (Subscript[a, 1]) (Subscript[b, 1]) + (Subscript[a, 2]) (Subscript[b, 2]) + (Subscript[a, 3]) (Subscript[b, 3]) together with elements that add to give the vector cross product  {(-(Subscript[a, 2]) (Subscript[b, 1]) + (Subscript[a, 1]) (Subscript[b, 2]))(Subscript[e, 1]) (Subscript[e, 2]), (-(Subscript[a, 3]) (Subscript[b, 1]) + (Subscript[a, 1]) (Subscript[b, 3]))(Subscript[e, 1]) (Subscript[e, 3]), (-(Subscript[a, 3]) (Subscript[b, 2]) + (Subscript[a, 2]) (Subscript[b, 3]))(Subscript[e, 2]) (Subscript[e, 3])}. 
(The bivector (Subscript[e, 1]) (Subscript[e, 2]) is the 4th direction, having coefficient 0 in the multiplicands).

All Clifford algebras are hoops. Their tables are created by using the GroupLoopHoop procedure cl or by folding a specific group isomorph. Wedge (exterior) algebras are the bivector part of the Clifford product [3 p.10]. They multiply univectors using Clifford algebra rules; the scalars and multivectors are kept at zero in the multiplicands so various products are zero (i.e. these are Grassmann algebras). Multiplying univectors (Subscript[u, 1]), (Subscript[u, 2]) and (Subscript[v, 1]), (Subscript[v, 2]) in CL2 gives a scalar (Subscript[u, 1]) (Subscript[v, 1]) - (Subscript[u, 2]) (Subscript[v, 2]) and a bivector (Subscript[u, 1]) (Subscript[v, 2]) - (Subscript[u, 2]) (Subscript[v, 1]); the univectors in the product are zero. CL3 gives the wedge product of three univectors when the scalar and multivector terms are zero in the multiplicands. CL2 and CL3 univector products only differ from the quaternion and octonion univector products in the sign of the first term.
	Different 16-element Clifford algebras multiply 4 univectors to give scalars and three bivectors with assorted signs:-

{Use["CL2"]; "Example 13. Wedge Products, Exterior \
Algebras. Clifford Algebras.\nCL2",hoopTimes[{0, (Subscript[u, 1]), (Subscript[u, 2]), 0}, {0, (Subscript[v, 1]), (Subscript[v, 2]), 0}],
Use["CL3"]; "\nCL3",hoopTimes[{0, (Subscript[u, 1]), (Subscript[u, 2]), 0, (Subscript[u, 3]), 0, 0, 0}, {0, (Subscript[v, 1]), (Subscript[v, 2]), 
              0, (Subscript[v, 3]), 0, 0, 0}],
Use["CL4"]; "\nCL4",hoopTimes[{0, (Subscript[e, 1]), (Subscript[e, 2]), 0, (Subscript[e, 3]), 0, 0, (Subscript[e, 4])}, {0, (Subscript[u, 1]), 
                (Subscript[u, 2]), 0, (Subscript[u, 3]), 0, 0, (Subscript[u, 4])}],
Use["CL04"]; "\nCL04",hoopTimes[{0, (Subscript[e, 1]), (Subscript[e, 2]), 0, (Subscript[e, 3]), 0, 0, (Subscript[e, 4])}, {0, (Subscript[u, 1]), 
                (Subscript[u, 2]), 0, (Subscript[u, 3]), 0, 0, (Subscript[u, 4])}],
Use["CL31"]; "\nCL31",hoopTimes[{0, (Subscript[e, 1]), (Subscript[e, 2]), 0, (Subscript[e, 3]), 0, 0, (Subscript[e, 4])}, {0, (Subscript[u, 1]), 
                (Subscript[u, 2]), 0, (Subscript[u, 3]), 0, 0, (Subscript[u, 4])}]}

Finite Lie algebras appear to be Clifford hoops for which the real (scalar) elements have been zeroed. This has yet to be explored.
	Many algebras, some of which appear to be new, are defined by other hoops

2.8 Most loops are not conservative.

The majority of loops (quasigroups used as multiplication tables) are not conservative. The data contains a few non-conservative loops as examples, identified by names ending in n. The determinants of "Q4n" and "C6n" factorise nicely, but their quadratic factors are not conserved:-

Use["Q4n"]; {"Example 14. Q4n is not conservative.\n", "Q4n \
Table", hoopTbl // tf, "\nDet factors", 
      sh, 
"\nA =   ", 
      A = {4, 2, 3, 5}, 
"\nB =   ", 
      B = {1, 6, 2, 2}, 
"\nAB=   ", 
      AB = hoopTimes[A, B], "\nDet factors of A    sa =", 
      sa = sh /.  as[A], "\nDet factors of B    sb =", 
      sb = sh /.  as[B], "\nDet factors of AB   sab=",  
      sh /.  as[AB], "\nThis is not sa*sb      =", sa sb}

Use["C6n"];
{"Example 14a. C6n only conserves the linear factors.\n", "C6n \
Table", hoopTbl // tf, "\nDet factors", sh, "\n A =", 
      A = {4, 2, 3, 5, 1, 6}, "\n B =", 
      B = {0, 6, 2, 4, 3, 2}, "\nAB =              ", 
      AB = hoopTimes[A, B], "\nDet factors of A  sa =  ", 
      sa = sh /.  as[A], "\nDet factors of B  sb =  ", 
      sb = sh /.  as[B], "\nDet factors of AB sab=  ", 
      sab = sh /.  as[AB], "\nThis is not sa*sb       ", 
      sa sb}

Many non-conservative loops, with various properties, have been tested by multiplying random vectors and comparing the determinants. (No necessary and sufficient loop property has been found for conservation.)  The only conservative tables appear to be groups, a few non-commutative Moufang Loops [5] (including octonions and split-octonions), their direct composition with abelian groups, and tables folded from conservative loops. All known conservative tables have the Moufang property.

2.9 Quadratic sizes, Abelian Polar-Vector Duals, Roots & Powers.

Real quadratic hoop determinant factors are conserved functions (though their complex-conjugate-pair factors are not conserved). They are symmetric real polynomials that can often be "fragmented", i.e. split into signed sums of smaller squared terms or "fragments". A generalisation (the "PolyHelix Identity") of (Cos^2)[θ]+(Sin^2)[θ]=1 then leads to "polar-dual" formulations. E.g. C3 conserves (r^2) =(a^2)+(b^2)+(c^2)- a b - b c - c a which fragments into ((((a - b))^2)+(((b - c))^2)+(((c - a))^2))/2. The associated angle is ArcTan[2a-b-c,-(Sqrt[3])(b-c)]. (see Example 1.) Note that, in sizes, the "-" sign always occurs in squared terms so it corresponds to a "symmetric difference" that does not imply negation or subtraction, which only arise in hoops after 2-folding.
	In effect, each conserved quadratic size releases a degree of freedom, which is taken up by an angle. Angles are "hidden variables" for cartesian vectors. Polar-duals are the generalization of the {r,θ} formulation of the complex plane (or Argand-Wessel diagram) to other algebras. Linear sizes provide "offsets" for polar forms; these are extra degrees of freedom that are missing from complex algebra.
	As angles add on abelian hoop multiplication, powers and roots can be obtained by raising sizes to the exponent and multiplying angles by the exponent. This is implemented by hoopPower, which converts a cartesian vector to the polar form, applies the exponent, and then reverts to cartesian form. It has been demonstrated in previous examples. Example 16 demonstrates a "Hexal" algebra, C3C4c, which is "supersymmetric" (see section 3) to one of the "Dozal" algebras, C3C4. These two algebras are unusual - they have sizes that involve (Sqrt[3]). Note the "If" statements that handle specific cases in the angle calculations. These are essential when dealing with orbits (section 2.13); they are omitted (for simplicity) in most topol definitions in the data. ([4] supplies the complete versions.)

\(Use["C3C4c"]; {"Example 16. C3C4c Polar form and square root
The table is", hoopTbl // tf, "
The polar form is", 
      topol, "
A simplified cartesian form, in terms of radii ϵ,η,κ and angles σ,χ,ψ is
", \(tovec /. \ 
          as[{ϵ\^2, σ, η\^2, χ - π/
                  6, κ\^2, ψ - π/6}]\) /. \@any_\^2 -> 
          6  any, "
When A  = ", 
      A = {6. , 2. , 4. , 3. , 1. , 1. }, "
the shape of A is ", 
      sh /. \ as[A], "
the polar form is ", 
      pA = topol /. \ as[A], 
"tovec recovers  A ", 
      tovec /. \ as[pA], 
"
The square root is", 
      rootA = hoopPower[A], "
with polar form   ", 
      topol /. \ as[rootA], "
rootA.rootA = A   ", 
      hoopTimes[rootA, rootA]}\)

2.10 Non-Abelian Polar-Vector Duals, Rotated-roots, Uncertainty?

Non-abelian hoops have repeated determinant factors. They do not have ordinary polar forms because the repeated sizes release extra degrees of freedom. (As the tables are non-abelian, the determinant should possibly be found and factorised by noncommutative procedures. No progress has been made with this topic.) Non-abelian quadratic sizes often have more than three fragments, and do not fit the patterns that give abelian polar forms. Splitting them into sets of two or three fragments allows the formulation of invertible "pseudo-polar-powers" with ((((A^(1/p))))^p)=A, but as the angles are not additive this gives "pseudo-roots" that do not multiply to recover the original vector i.e. (A^(1/2)).(A^(1/2))≠A; the product has the correct sizes but the other elements of the polar form are modified in a non-obvious way. The angles are rotated. Conjecture - powers and roots may be associated with the introduction of uncertainty on making an observation of a quantum system.
	Pseudo-roots are demonstrated for D3:-

Example 16. Using D3, Division and Polar-Vector interconversions work, but repeated products AA differ from pseudo-powers (A^2)

Use["D3"]; {"The D3 table is", 
hoopTbl // 
        tf}
\(Chop[{Flatten[{"If  A=", 
              A = {2. , 3. , \(-3. \), 0. , \(-2. \), 1. }, "polarA=", 
              topol /. as[A]}], 
          Flatten[{"Vec(pol(A))=", 
              tovec /. as[
                  topol /. as[
                      A]], "OK, =A", "", "", "", "", "", ""}], 
Flatten[{"and  B=", 
              B = {1. , \(-5. \), 3. , \(-1. \), 2. , 2. }, "polarB=", 
              topol /. as[B]}], 
Flatten[{"AB=", 
              AB = hoopTimes[A, B], "polarAB=", 
              topol /. as[AB]}], 
Flatten[{"Ai.AB=", 
              ABA = hoopTimes[hoopInverse[A], 
                  AB], "OK, =B", "", "", "", "", "", ""}], 
Flatten[{"BA=", 
              BA = hoopTimes[B, A], "polarBA=", 
              topol /. as[BA]}], 
Flatten[{"Bi.BA=", 
              BAB = hoopTimes[hoopInverse[B], 
                  BA], "OK, =A", "", "", "", "", "", ""}], 
          Flatten[{"AA=", AA = hoopTimes[A, A], "polarAA=", 
              topol /. as[AA]}], 
          Flatten[{\*""(A^2)="", 
              A2 = hoopPower[A, 2], \*""(polarA^2)="", 
              topol /. as[A2]}], 
          Flatten[{\*""((A^2))^(1/2)="", 
              RA2 = hoopPower[A2, 1/2], "polar=", topol /. as[RA2]}]}] // 
      tf\)

The repeated quadratic factor of D3, ((a^2) - (b^2) + (c^2) - (d^2) + (e^2) - (f^2) - c a - 
        e a+b d  -c  e + b f + d f)/2, has been handled as two sets of sums of three squares \(TraditionalForm\`\((\((e - a)\)\^2 + \((a - c)\)\^2 + \((c - e)\)\^2)\)/2\) and \(TraditionalForm\`\((\((f - b)\)\^2 + \((b - d)\)\^2 + \((d - f)\)\^2)\)/2\) to allow the formulation of a pseudo-polar dual. The angles do not add, nor are the squared radii conserved. (The conserved length is their difference.) I have sought (without success) a reformulation with additive angles, using the distinction between left and right multiplication.

2.11 Vectors with some zero sizes, Remainders.

Sizes provide the denominators of the partial-fraction formulation of the multiplicative inverse (because it is found by Cramer's method). Vectors can have coefficients that make one or more size zero in a particular algebra; these algebras have "divisors of zero". Most mathematicians choose to restrict their work to the "algebras without real divisors of zero", i.e. ℝ, ℂ, ℍ (Quaternions), and 𝕆 (Octonions). A determinant is zero if it has one or more zero factor, but it may still have non-zero factors. Hoop algebras overcome division-by-zero via two innovations, projection into sub-algebras and ejection of remainders. Zeroed sizes correspond to operations in sub-algebras with these sizes constrained to be zero; inverses are in the same sub-algebras, and also have these zeroes. In effect, zeroes are "factored out". Conservation is maintained by ejecting left and right remainders A/B = C +Rl +Rr; sizes that are zero in C but not in A go into the left remainder; sizes that are zero in C but not in B go into the right remainder. This is a generalization of integer division. As hoopTimes implements both division and multiplication, remainders also extend to multiplication A*B=P+Rl+Rr so that P/A+Rr=B and P/B+Rl=A. Example 17 sets up two vectors with C4 shapes {6,2,0} & {0,10,10}; their product has shape {0,20,0}; Rl conserves the 6 from A0; Rr conserves the second 10 from B0. Dividing AB by A does not recover B until Rr is added; dividing AB by B does not recover A until Rl is added.

Use["C4"];Chop[{"Example 17. C4 multiplication with remainders","\n         A0 = ",A0 = {2, 1, 2, 1},
"\nshape of A0 = ",sA0 = sh /.  as[A0],
"\n         B0 = ",B0 = {3, -1, 2, -4},
"\nshape of B0 = ",sh /.  as[B0],
"\n         AB = ",AB = hoopTimes[A0, B0],
"\nshape of AB = ",sh /.  as[AB],
"\nNote that sizes  ↑     ↑ have become 0",
"\nRemainder Rl =",RA = Rl,"\nhas a shape      ",sh /.  as[Rl],"\nRemainder Rr =",RB = Rr,"\nhas a shape   ",sh /.  as[Rr],"\nAinverse    = ",Ainv = hoopInverse[A0],"\nwith shape    ",sh /.  as[Ainv],"\nAB/A0       = ",ABoA = hoopTimes[Ainv, AB],"\nwith shape    ",sh /.  as[ABoA],"\nAdding Rr recovers B0",ABoA + RB,
"\nBinverse    = ",Binv = hoopInverse[B0],"\nwith shape    ",sh /.  as[Binv],"\nAB/B0       = ",ABoB = hoopTimes[AB, Binv],"\nwith shape    ",sh /.  as[ABoB],"\nAdding Rl recovers A0  ",ABoB + RA}]

ℂ, ℍ, and 𝕆 can have divisors of zero if their coefficients are complex.The hoop algebras defined by the Pauli-σ matrices (P4, P8, P16) conserve variations on the Minkowski metric (t^2)-(x^2)-(y^2)-(z^2) and can have real divisors of zero. The "light cone" is a zero sub-space.
	Some algebras can have "annihilators", real vectors with all sizes zero. Operations with them give results with all sizes zero (but with non-zero coefficients), so one remainder will be the other operand, unchanged. Example 18 shows this for left and right multiplication by a D3 annihilator A0.

Use["D3"]; 
    Chop[{"Example 18. D3 annihilator", "\nD3 has shape", 
        sh, "\nA0 =  ", 
        A0 = {2, 0, -2, -2, 0, 
            2}, 
"\nA0 shape", 
        sh /.  as[A0], "\nB  =  ", 
        B = {2, 3, -4, -1, 5, 
            2}, 
"\nB shape ", 
        sh /.  as[B], "\nAB  = ", 
        AB = hoopTimes[A0, B], "\nAB shape", 
        sh /.  as[AB], "\nRl =  ", Rl, "\nRr = B", 
        Rr, "\nBA  = ", BA = hoopTimes[B, A0], "\nBA shape", 
        sh /.  as[BA], "\nRl = B", Rl, "\nRr =  ", Rr}]

2.12 Subtraction is Additive Elimination.

Negation "-a" and subtraction "b-a" are conflated and treated as a single fundamental process by most mathematicians, but [5] requires over a hundred pages of "Post-Modern Algebra" before subtraction can be tackled. In hoops, negation is the result of a fold with r=2. This equivalences {(Subscript[a, i]),(Subscript[a, i + m/2])} to a real number (Subscript[a, i]), with {(Subscript[a, i + m/2]),(Subscript[a, i])} ~ -(Subscript[a, i]) and {(Subscript[a, i]),(Subscript[a, i])} ~ 0. The first size, ∑(Subscript[a, i]), is destroyed, as it equivalences to zero. Subtraction can now be seen as "additive elimination", adding a set of coefficients that make all the terms of an un-folded vector identical, so that they equivalence to zero: {a,b}+{b,a} ={a+b,a+b} ~ 0. (The commutative property of addition is inherited from the natural numbers.)
	When r is 3, the equivalence relationship is (Subscript[t, j])~{(Subscript[a, j]),(Subscript[a, j + m/3]),(Subscript[a, j + 2  m/3])} ~{(Subscript[b, j]),(Subscript[b, j + m/3]),(Subscript[b, j + 2  m/3])} iff ((Subscript[a, j])+(Subscript[b, j + m/3])) = ((Subscript[a, j + m/3])+(Subscript[b, j + 2  m/3])) = ((Subscript[a, j + 2  m/3])+(Subscript[b, j])). The "terplex number" {a,b,c} has a left negation (or rotation), {b,c,a} and a right negation {c,a,b}, with {a,b,c} +{b,c,a} +{c,a,b} ={a+b+c, a+b+c, a+b+c} ~ 0. A number is eliminated by adding the two rotations of its vector form. This extends to other r values; a number is eliminated by adding the r-1 rotations of a primal form. Simultaneous equations can be solved in this way in hoop algebras, but this is not demonstrated here.

2.13. Orbits.

I define a "unital" sub-algebra by the restriction that "the product of non-zero sizes is unity". The name "orbit" is appropriate to polar forms of unital sub-algebras because the coefficients are restricted to circles in m-dimensional space, and the group orbits are points on these circles. Multiplying one orbit by another (using the same hoop) gives another orbit, and any remainders will also be orbits. If some sizes are zero, operations are restricted to a sub-algebra in which these sizes remain at zero, by conservation. This generates sub-orbits that have analogies with particles - discussed in section 3.
 	I use the convention that orbit polar forms have names OHijk, with vector forms VHijk, where H is the hoop name and ijk... gives the sizes. Linear sizes provide "offsets", displacing the orbit origin from zero. This may be a new concept; it introduces additive properties to orbits. The squared radius is a multiplicative property for abelian radius/angle formulations.

Use["C3"]; {"Example 19. C3 Orbits", "\nOC301 is {0,1,a} with \
vector form", 
      OC301 = tovec /.  
          as[{0, 1, a}], "\nOC311 is {1,1,b} with vector form", 
      OC311 = tovec /.  
          as[{1, 1, 
              b}], "\nProducts are orbits with added angles; the polar form \
of the product of OC301 & OC311 is", 
      Simplify[topol /.  
          as[hoopTimes[OC301, OC311]]], "\nand the remainders are", Rl, 
      Rr}
\(Use["C4"]; {"Example 20b. C4 Orbits", "
OC4001 is ", 
      OC4001 = {0, 0, 1, a}, "
 with vector form", 
      VC4001 = tovec /. \ as[OC4001], "
OC4011 is", 
      OC4011 = {0, 1, 1, b}, "
 with vector form"\ , 
      VC4011 = tovec /. \ as[OC4011], "
OC4101 =", 
      OC4101\  = \ {1, 0, 1, c}, "
 with vector form", 
      VC4101 = tovec /. \ as[OC4101], "
OC4111 is", 
      OC4111\  = \ {1, 1, 1, d}, "
with vector form", 
      VC4111 = tovec /. \ 
          as[OC4111], "

", 
Simplify[{"OC4001×OC4001 has the 001 shape", 
          sh /. \ as[
              hoopTimes[VC4001, 
                VC4001]], "
OC4101×OC4011 also has the 001 shape
as the two offsets disappear   ", 
          sh /. \ as[hoopTimes[VC4101, VC4011]], "
with remainders ", 
          Rl, Rr, "
OC4101×OC4111 has the 101 shape", 
sh /. \ 
            as[hoopTimes[VC4101, VC4111]], "
with remainders ", Rl, 
          Rr, "
OC4111×OC4111 has the 111 shape", 
          sh /. \ as[hoopTimes[VC4111, VC4111]]}], "
with remainders ", 
      Rl, Rr}\)

3. Analogies with Particles.

Summary
	The "Dozal" algebras based on 12-element groups have elements that act as half-spin (fermionic) and unit-spin (bosonic) quantum operators. Their orbits have analogies with fundamental particles. This leads to the possibility that hoop algebras may provide a new paradigm for physics. The following conjectures, which the author cannot develop further, are outlined:- 
	Dozal interactions, with remainders conserving different properties, resemble particle interactions and decay. Four hoops {C3K, C4C4, D3C2, Q12} possibly correspond to four forces - they conserve different but overlapping sets of properties. The non-Abelian hoops may introduce uncertainty. Linear sizes are additive properties.
	Orbits resemble multi-phase deBroglie waves, and quadratic sizes resemble Planck areas. Wave packets in stable orbits with 3, 6 or 12 phases may relate to quarks whilst 4-phase orbits may relate to leptons. Resonant orbits, at integer multiples of the fundamental frequency, may relate to particles resonances.
	Bosonic "hexal" elements are supersymmetric to (but fewer than) fermionic dozal elements. Bosons are not subject to the Pauli exlusion principle because they are equivalence relations.

3.1. Dozal and Hexal.

Dozal and Hexal algebras, with 12 and 6 elements, have many analogies with fundamental particles. They may lead to a new paradigm for physics, based on finite Moufang Loop symmetries (together withLie group symmetries). Sporadic discoveries, spread over fifteen years of research, have repeatedly related different physical phenomena to newly discovered aspects of hoop algebra, encouraging the conjecture that these algebras are "physical mathematics".

3.2. Dozal elements act as Quantum Operators.

J. J. Hamilton [6] developed a "hypercomplex arithmetic", with half-spin "non-real, non-complex" quantum operators {a, \(TraditionalForm\`\(a\^*\)\),b ,\(TraditionalForm\`\(b\^*\)\)} as well as integer-spin quantum operators, having the following multiplication table:-

Example 21. J.J.Hamilton's Table II.

({{ , (Subscript[E, 0]), (Subscript[E, 1]), (Subscript[E, 2]), (Subscript[E, 3]), (Subscript[E, 4]), (Subscript[E, 5]), (Subscript[E, 6]), (Subscript[E, 7])}, {(Subscript[E, 0]), 1, i, h, k, a, \(b\^*\), \(a\^*\), b}, {(Subscript[E, 1]), i, -1, k, -h, \(b\^*\), -a, b, \(-\(a\^*\)\)}, {(Subscript[E, 2]), h, k, 1, i, \(a\^*\), b, a, \(b\^*\)}, {(Subscript[E, 3]), k, -h, i, -1, b, \(-\(a\^*\)\), \(b\^*\), -a}, {(Subscript[E, 4]), a, \(b\^*\), \(a\^*\), b, 1, i, h, k}, {(Subscript[E, 5]), \(b\^*\), -a, b, \(-\(a\^*\)\), i, -1, k, -h}, {(Subscript[E, 6]), \(a\^*\), b, a, \(b\^*\), h, k, 1, i}, {(Subscript[E, 7]), b, \(-\(a\^*\)\), \(b\^*\), -a, k, -h, i, -1}})

His a* and b* are "biconjugates" (my term) with a**=-a, b**=-b. The table has four elements that square to -1; Hamilton also introduces several other elements and many aliases. Replacing the ambiguous "-1" by ii and including extra rows and columns that match his multiplication rules gives a 12×12 table with his table embedded (in boldface, with ii in place of -1):-

Example 22. C3K contains Hamilton's table.

({{1, i, ii, h, k, -h, a, \(b\^*\), -a, \(a\^*\), b, \(-\(a\^*\)\)}, {i, ii, 1, k, -h, h, \(b\^*\), -a, a, b, \(-\(a\^*\)\), \(a\^*\)}, {ii, 1, i, -h, h, k, -a, a, \(b\^*\), \(-\(a\^*\)\), \(a\^*\), b}, {h, k, -h, 1, i, ii, \(a\^*\), b, \(-\(a\^*\)\), a, \(b\^*\), -a}, {k, -h, h, i, ii, 1, b, \(-\(a\^*\)\), \(a\^*\), \(b\^*\), -a, a}, {-h, h, k, ii, 1, i, \(-\(a\^*\)\), \(a\^*\), b, -a, a, \(b\^*\)}, {a, \(b\^*\), -a, \(a\^*\), b, \(-\(a\^*\)\), 1, i, ii, h, k, -h}, {\(b\^*\), -a, a, b, \(-\(a\^*\)\), \(a\^*\), i, ii, 1, k, -h, h}, {-a, a, \(b\^*\), \(-\(a\^*\)\), \(a\^*\), b, ii, 1, i, -h, h, k}, {\(a\^*\), b, \(-\(a\^*\)\), a, \(b\^*\), -a, h, k, -h, 1, i, ii}, {b, \(-\(a\^*\)\), \(a\^*\), \(b\^*\), -a, a, k, -h, h, i, ii, 1}, {\(-\(a\^*\)\), \(a\^*\), b, -a, a, \(b\^*\), -h, h, k, ii, 1, i}})

The extended table is isomorphic to C3K, so i and ii are interpreted as a cube roots of 1. His half-spin operators {a, a*, -a≡  a**,b, b*, -b≡ b**≡ -a*} are {a, a h, a i i, a h i, a i , a h i i}; conjugation a* is multiplication by h (with h h =1), but biconjugation a** becomes multiplication by i or ii. His half-spin operators are still valid, but they are now defined in terms of two distinct square roots of 1 (h & a) and the cube roots (i, ii) of 1. Complex numbers are not involved.

3.2. Half-spin and Dozal algebra.

Banded sets of differential equations generate multi-phase sinusoids (just as \(TraditionalForm\`\(\(\ \)\(x\_1\)\)\& . \) = a (Subscript[x, 2,])\(TraditionalForm\`\(\(\ \)\(\(\(\ \)\(x\_2\)\)\& . \)\)\)= -a (Subscript[x, 1]), equivalent to (Overscript[x, ¨]) = -(a^2)x, generates ordinary sinusoids). A relevant simple set is the m equations (with m-1 being independent):-
 Example 23. Banded Differential Equations and Multi-phase Sinusoids
     D[(Subscript[x, j]), t] = c'[t] + 
            a ((Subscript[x, Mod[j + k, m]]) - (Subscript[x, Mod[j - k, m]])), 
        j = 0.  ..  m - 1, k < m/2. 
Substituting the trial function (Subscript[x, j]) = c[t] + (Subscript[c, 1]) Sin[(Subscript[c, 2]) t + ϕ + 2  j k π/m] leads to                  (Subscript[x, j]) = 
        c[t] + (∑_p) (Subscript[c, p]) 
              Sin[2  a Sin[2  k π (Subscript[f, p])/m] t + (Subscript[ϕ, p]) + 
                    2  j k π/m]/(Subscript[f, p]).

	j is one of the m phases; c[t] gives freedom from the origin;  (Subscript[c, p]), (Subscript[f, p]) and (Subscript[ϕ, p]) are phase amplitude, resonant multiplier and phase offset constants which have to be subscripted because independent sets can arise; k is a phase step. Sin[2k π/m] is a constant that determines the frequency & period. When m=4, k must be 1; folding then removes the c[t] term and reproduces the ordinary sinusoid equation-pair with doubled amplitude. Other values of m and k give a frequency reduction.
	The period is 2π(Subscript[f, p])/Sin[2k π/m], and the function is non-sinusoidal (constant) if 2/m is integral.
	The Dozal C3C4 loop gives half-spin with a period of 4π when m=12, k=1, (Subscript[f, p])=1. The twelve components are elements of the C12 hoop.However, C3C4 is the preferred isomorph because it has sizes in common with the other Dozal hoops C3K, Q12, D3C2, and A4.
	 A lot more research is needed here. The differential equations need to be related to hoops; the resonances (orbits with frequencies that are integer multiples of the fundamental frequency) need study; what is the relationship between hoops and the theory of strings and branes, etc.? Before folding, the sinusoids are “free”; they do not have a zero mean because they include constants of integration c[t]. This may relate to position in space.

3.3. Sizes and Orbits as Planck Areas & deBroglie Waves with polarization & chirality.

C3K has a polar dual that can be expressed as {α, β, γ, δ, ζζ, τ, λλ, ϕ, ηη, χ, κκ, ψ} and reverted to a vector as follows:-

Example 24. C3K Polar form and reversion to Cartesian form.

C3KO = {
α + β + γ + δ + ζ Cos[τ]       + λ Cos[ϕ]       + η Cos[χ]       + κ Cos[ψ], α + β + γ + δ + ζ Cos[τ + ω] + λ Cos[ϕ + ω] + η Cos[χ + ω] + κ Cos[ψ + ω],  
α + β + γ + δ + ζ Cos[τ - ω] + λ Cos[ϕ - ω] + η Cos[χ - ω] + κ Cos[ψ - ω], 
α - β + γ - δ + ζ Cos[τ]       - λ Cos[ϕ]       + η Cos[χ]       - κ Cos[ψ],  α - β + γ - δ + ζ Cos[τ + ω] - λ Cos[ϕ + ω] + η Cos[χ + ω] - κ Cos[ψ + ω],  α - β + γ - δ + ζ Cos[τ - ω] - λ Cos[ϕ - ω] + η Cos[χ - ω] - κ Cos[ψ - ω],  α + β - γ - δ + ζ Cos[τ]       + λ Cos[ϕ]       - η Cos[χ]       - κ Cos[ψ],  α + β - γ - δ + ζ Cos[τ + ω] + λ Cos[ϕ + ω] - η Cos[χ + ω] - κ Cos[ψ + ω],  α + β - γ - δ + ζ Cos[τ - ω] + λ Cos[ϕ - ω] - η Cos[χ - ω] - κ Cos[ψ - ω],  α - β - γ + δ + ζ Cos[τ]       - λ Cos[ϕ]       - η Cos[χ]       + κ Cos[ψ],  α - β - γ + δ + ζ Cos[τ + ω] - λ Cos[ϕ + ω] - η Cos[χ + ω] + κ Cos[ψ + ω],  α - β - γ + δ + ζ Cos[τ - ω] - λ Cos[ϕ - ω] - η Cos[χ - ω] + κ Cos[ψ - ω]}/12;

Use["C3K"]; topol

Here the substitutions ζ=2(Sqrt[ζ])ζ, λ=2(Sqrt[λ])λ, η=2(Sqrt[η])η, κ=2(Sqrt[κ])κ have been made, to give a compact expression. Note that ζζ, λλ, ηη, & κκ represent quadratic sizes that are expressible as a sums of squares, so they are positive for real elements. I conjecture that they correspond to Planck areas, and provide a fundamental scale; their square roots provide radii (2(Sqrt[ζ])ζ/12 etc) for multiphase deBroglie-like sinusoids. The angles τ, ϕ, χ, & ψ are "hidden variables" in the vector form. PROBLEM - how do these relate to space-time?

C3C4 has a polar dual that can be expressed as {α, β, ϵϵ, σ, ζζ, τ, λλ, ϕ, ηη, χ, κκ, ψ} (with {α, β, ζζ, τ, λλ, ϕ} in common with C3K) and reverted as follows:-

Example 25. C3C4 Full Polar form and Orbits.

Use["C3C4"]; sh

C3C4O = {
α + β + ϵ  Cos[σ] + ζ Cos[τ]       + λ Cos[ϕ]           + η Cos[x] + η r3 Sin[
                  x] + κ Cos[y] + κ r3 Sin[
                  y], 
α + β + ϵ  Cos[σ] + ζ Cos[τ + ω] + λ Cos[ϕ + ω]      - η Cos[x] + η r3 Sin[
                  x] - κ Cos[y] + κ r3 Sin[
                  y],  
α + β + ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω] - η r6 Sin[
                  x]                                           - κ r6 Sin[
                  y], 
α - β + ϵ  Sin[σ] + ζ Cos[τ]      - λ Cos[ϕ]       + η r3 Cos[x]      - η Sin[
                  x] - κ r3 Cos[y] + κ Sin[
                  y], 
α - β + ϵ  Sin[σ] + ζ Cos[τ + ω] - λ Cos[ϕ + ω] + η r3 Cos[x]      + η Sin[
                  x] - κ r3 Cos[y] - κ Sin[
                  y], 
α - β + ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] - η r6 Cos[
                  x]                                           + κ r6 Cos[
                  y], 
α + β - ϵ  Cos[σ] + ζ Cos[τ]      + λ Cos[ϕ]            - η Cos[x] - η r3 Sin[
                  x] - κ Cos[y] - κ r3 Sin[
                  y] , 
α + β - ϵ  Cos[σ] + ζ Cos[τ + ω] + λ Cos[ϕ + ω]      + η Cos[x] - η r3 Sin[
                  x] + κ Cos[y] - κ r3 Sin[
                  y],  
α + β - ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω] + η r6 Sin[
                  x]                                           - κ r6 Sin[
                  y], 
α - β - ϵ  Sin[σ] + ζ Cos[τ]      - λ Cos[ϕ]       - η r3 Cos[x]      + η Sin[
                  x] + κ r3 Cos[y] - κ Sin[
                  y], 
α - β - ϵ  Sin[σ] + ζ Cos[τ + ω] - λ Cos[ϕ + ω] - η r3 Cos[x]      - η Sin[
                  x] + κ r3 Cos[y] + κ Sin[
                  y], 
α - β - ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] + η r6 Cos[
                  x]                        + κ r6 Cos[y]}/12;

Here ϵ=2(Sqrt[ϵ])ϵ, ζ=2(Sqrt[ζ])ζ, λ=2(Sqrt[λ])λ, η=(Sqrt[3  ηη]), κ=(Sqrt[3  κκ]),  x=χ+π/6, y=ψ+π/6, ω=2π/3, r3=(Sqrt[3]), r6=2(Sqrt[3])to give a compact expression.
	Orbits are obtained by setting some of the parameters {α, β, γ, δ, ϵϵ, ζζ ,λλ , ηη, κκ} to zero or 1 and constraining the product of the remainder to be 1. Each set of angles involving ω=2π/3 gives a chiral orbit; the chirality is reversed if the angle is negated. 
	Similar, but simpler, expressions give Q12 & D3C2 orbits. Their parameters are related to those for C3K and C3C4. A 12-phase system arises if ψ=χ+π/6. Problem - non-commutativity prevents angle additivity in these orbits. Does this introduce uncertainty?

Q12O = 
{  (α + β + ϵ Cos[σ] + ζ Cos[τ]       + η Cos[χ]), 
(α + β + ϵ Cos[σ] + ζ Cos[τ + ω] + η Cos[χ + ω]), 
(α + β + ϵ Cos[σ] + ζ Cos[τ - ω] + η Cos[χ - ω]), 
(α - β + ϵ Sin[σ]      + λ Cos[ϕ]       + ϰ Cos[ψ]), 
(α - β + ϵ Sin[σ]      + λ Cos[ϕ + ω] + ϰ Cos[ψ + ω]), 
(α - β + ϵ Sin[σ]      + λ Cos[ϕ - ω] + ϰ Cos[ψ - ω]), 
(α + β - ϵ Cos[σ] + ζ Cos[τ]       - η Cos[χ]), 
(α + β - ϵ Cos[σ] + ζ Cos[τ + ω] - η Cos[χ + ω]), 
(α + β - ϵ Cos[σ] + ζ Cos[τ - ω] - η Cos[χ - ω]), 
(α - β - ϵ Sin[σ]       + λ Cos[ϕ]      - ϰ Cos[ψ]), 
(α - β - ϵ Sin[σ]       + λ Cos[ϕ - ω] - ϰ Cos[ψ + ω]), 
(α - β - ϵ Sin[σ]       + λ Cos[ϕ - ω] - ϰ Cos[ψ - ω])}/12;

This (and D3C2O, below) gives a 12-phase system if ψ=χ±π/6 & χ=η. A polarised 3-phase system (with polarisation angle φ) is given by ϕ=τ & λ=±ζ.

D3C2O = 
{  (α + β + γ + δ + ζ Cos[τ]       + η Cos[χ]), 
(α + β + γ + δ + ζ Cos[τ + ω] + η Cos[χ + ω]), 
(α + β + γ + δ + ζ Cos[τ - ω] + η Cos[χ - ω]), 
(α - β + γ - δ       - λ Cos[ϕ]       - ϰ Cos[ψ]), 
(α - β + γ - δ       - λ Cos[ϕ + ω] - ϰ Cos[ψ + ω]), 
(α - β + γ - δ       - λ Cos[ϕ - ω] - ϰ Cos[ψ - ω]), 
(α + β - γ - δ + ζ Cos[τ]       - η Cos[χ]), 
(α + β - γ - δ + ζ Cos[τ + ω] - η Cos[χ + ω]), 
(α + β - γ - δ + ζ Cos[τ - ω] - η Cos[χ - ω]), 
(α - β - γ + δ       - λ Cos[ϕ]       + ϰ Cos[ψ]), 
(α - β - γ + δ       - λ Cos[ϕ + ω] + ϰ Cos[ψ + ω]), 
(α - β - γ + δ       - λ Cos[ϕ - ω] + ϰ Cos[ψ - ω])}/12;

For these orbits to represent particles, the product of the non-zero quadratic sizes should define a fundamental size - a Planck Area. Section 3.7 explores this topic. The product of the non-zero linear sizes {α, β, γ, δ} should be 1; some quantization process may constrain them to be ±1. The orbits may represent multi-phase deBroglie waves.

3.4. C5, C7 & C8 Orbits have unstable amplitudes.

The orbits of C3 have a squared radius of magnitude 2/3, with offsets {0,0,0} or {1/3, 1/3, 1/3).  Those of C4 have squared radii (or ulnae) of magnitude 1/2, and offsets {0,..} or {±1/4,..}. Orbits of C5, C7, Cp (p prime>3) etc can be developed, with p sinusoidal phases, but their radii are unconstrained, though their offsets are 0 or 1/p. Example 21 shows that OC5dr, a five-phase orbit with a displacement d and radius r, has a quartic size of zero and so is unconstrained. The angle, linear L1 and quartic parameters only use 3 of the 5 degrees of freedom. Similarly, a seven-phase orbit with radius r has a zero size in C7 (I have not found a polar expression for this group) and an eight-phase orbit with radius r has a zero size in C8. These orbits also appear to be unstable - numerical integration of the relevant differential-equation sets is unstable. Orbit instability is a vague concept. It may be related to the occurrence of determinant factors that are larger than quadratic.

{Use["C5"];"Example 26.  Zero-sized C5 orbit.\nOC5dr is",OC5dr = 
    tovec/. as[{d, r, σ}], 
  "\nThe radius disappears from the shape", 
  Simplify[sh/. as[OC5dr]]}

3.5. The limited number of stable orbits may limit the number of stable particles.

I surmise that stable orbits represent stable particles, and that the number of stable particles is determined by the limited number of stable particles. 
	Stable, finite amplitude, orbits with 3, 4, 6 & 12 phases have been found, some with polarization and chirality. The following examples are restrictions (with various radii and offsets set to zero) of the tables in section 3.3.

{"Example 27. 3-phase C3K Orbit (Omitting the 1/12 factor).\n",  
      12  C3KO /. {λ -> 0, η -> 0, κ -> 
              0, ϰ -> 0} // tf}

The linear terms {α, β, γ, δ} define offsets (α/12 etc) from zero. Scaled square roots of the quadratic terms appear as radii. Each phase occurs four times in a 3-phase dozal orbit, with different linear offsets.

topol shows that the squared amplitude (((ζ/12))^2)is the sum of three squares:- 
((a-b-d+e+g-h-j+k)\(TraditionalForm\`\^2\)+(-a+c+d-f-g+i+j-l)\(TraditionalForm\`\^2\)+(b-c-e+f+h-i-k+l)\(TraditionalForm\`\^2\))/72

{"Example 28. 4-phase C3C4 Orbit \n(omitting ϵ/12 \
factor).",  
      12  C3C4O/ϵ /. {α -> 0, β -> 
              0, ζ -> 0, λ -> 0, η -> 
              0, κ -> 0} // tf}

Each of the four phases occur three times. topol shows that the squared amplitude (((ϵ/12))^2) is the sum of two squares  \(\((\((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2)\)/144\).

{"Example 29. 6-phase C3K Orbit.\n",  
      Simplify[C3KO /. {ψ -> 0, ζ -> 
                0, ϰ -> 0}] // tf}

Example 29.  6-phase C3K Orbit.
{α + β + γ + δ + κ       + λ cos(ϕ)        + η cos(χ), α + β + γ + δ - κ/
                        2 + λ cos(ϕ + ω) + η cos(χ + ω), α + β + γ + δ - κ /
                        2 + λ cos(ϕ - ω) + η cos(χ - ω), α - β + γ - δ - κ        - λ cos(ϕ)       + η cos(χ), α - β + γ - δ + κ /
                        2 - λ cos(ϕ + ω) + η cos(χ + ω), α - β + γ - δ + κ /
                        2 - λ cos(ϕ - ω) + η cos(χ - ω), α + β - γ - δ - κ        + λ cos(ϕ)       - η cos(χ), α + β - γ - δ + κ /
                        2 + λ cos(ϕ + ω) - η cos(χ + ω), α + β - γ - δ + κ /
                        2 + λ cos(ϕ - ω) - η cos(χ - ω), α - β - γ + δ + κ        - λ cos(ϕ)       - η cos(χ), α - β - γ + δ - κ/
                        2 - λ cos(ϕ + ω) - η cos(χ + ω), α - β - γ + δ - κ/2 - λ cos(ϕ - ω) - η cos(χ - ω)}/12

The linear terms have been omitted. Each phase occurs twice, with different offsets.

3.6. Particle Interactions.

As Dozal elements resemble quantum operators, the addition/splitting and multiplication/division of Dozal vectors should resemble particle decays and interactions. C3K and D3C2 conserve the linear sizes {α, β, γ, δ} and  C3C4 and Q12 conserve {α, β} on addition and subtraction, corresponding to additive particle properies. Multiplication, with remainders, conserves overlapping sets of sizes for each group. Conjecture - this may correspond to interactions subject to the four forces. (What is the role of A4 here? It appears to discriminate between different classes of orbit.) Many other functions are conserved on multiplication of orbits, rather than of general vectors. Example 30 lists the known conserved functions, together with two {L4Q,L4σ} conserved by two signed tables related to quaternions and Pauli-σ hoops.
	I have not been able to correlate these functions with conserved particle properties. Resonances are an outstanding problem; if hoops are related to differential equations (as the multi-phase sinusoidal orbits suggest) there should be solutions with multiple frequencies, corresponding to the plethora of short-lived particles.

Example 30. Functions conserved in Dozal interactions.

\(\(sh12[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}] := 
        Module[{ab = a - b - g + h, ag = a - b + g - h, bc = b - c - h + i, 
            bh = b - c + h - i, 
ca = a - c - g + i, 
            ci = a - c + g - i, de = d - e - j + k, 
            dj = d - e + j - k, 
ef = e - f - k + l, 
            ek = e - f + k - l, fd = d - f - j + l, fl = d - f + j - l, 
            f1 = a - b - g + 
                h + \((d + e - 2\ f - j - k + 
                      2  l)\)/\@3, 
f2 = \((a + b - 2  c - 
                      g - h + 2  i)\)/\@3 - d + e + j - 
                k, 
f3 = 
              a - b - g + 
                h - \((d + e - 2\ f - j - k + 
                      2  l)\)/\@3, 
f4 = \((a + b - 2  c - 
                      g - h + 2  i)\)/\@3 + d - e - j + k, abef, 
            agek}, 
abef = 3 \((ab\ ef - bc\ de)\)\^2; 
          agek = 3 \((ag\ ek - bh\ dj)\)\^2; 
{o1 = 
              a + b + c + d + e + f + g + h + i + j + k + l, 
            o2 = a + b + c - d - e - f + g + h + i - j - k - 
                l, 
o3 = 
              a + b + c + d + e + f - g - h - i - j - k - l, 
            o4 = a + b + c - d - e - f - g - h - i + j + k + 
                l, 
 (*5*) l22a = 
              3 \((f1\^2 + f2\^2)\)/4,  (*6*) l22b = 
              3 \((f3\^2 + f4\^2)\)/4, 
 (*7*) 
              p22 = \((a + b + c + g + h + i)\)\^2 + \((d + e + f + j + k + l)\)\^2,  (*8*) 
              q22 = \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2,  (*9*) 
              r22 = \((a + b + c + g + h + i)\)\^2 - \((d + e + f + j + k + l)\)\^2,  (*10*) 
              s22 = \((a + b + c - g - h - i)\)\^2 - \((d + e + f - j - k - l)\)\^2, 
 (*11*) 
              p23 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + fl)\)\^2)\)/2,  (*12*) 
              q23 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - fl)\)\^2)\)/2, 
 (*13*) 
              r23 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + fd)\)\^2)\)/2,  (*14*) 
              s23 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - fd)\)\^2)\)/2, 
 (*15*) 
              q24 = \((a + b + c)\)\^2 + \((d + e + f)\)\^2 + \((g + h + i)\)\^2 + \((j + k + l)\)\^2, 
 (*16*) 
              s24 = \((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + h + i)\)\^2 - \((j + k + l)\)\^2, 
 (*17*) 
              p26 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)/
                2,  (*18*) 
              q26 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)/
                2, 
 (*19*) 
              r26 = \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)/
                2,  (*20*) 
              s26 = \((ab\^2 + bc\^2 + ca\^2 + de\^2 + ef\^2 + fd\^2)\)/
                2, 
\ \ \ \ \ \ \ \  (*21*) \ 
            l3 = \((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + 
                      j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - 
                      e - h + k)\) + 
\((c + f - \ i\  - 
                      l)\)\ \((c - f + \ i\  - l)\)\ \((c - f - \ i\  + 
                      l)\) - \((a - d + g - j)\)\ \((b - e - h + k)\)\ \((c + 
                      f - \ i\  - l)\) - 
\((a - d - g + 
                      j)\)\ \((b + e - h - k)\)\ \((c - f + i - l)\) - \((a + 
                      d - g - j)\)\ \((b - e + h - k)\)\ \((c - f - \ i\  + 
                      l)\), 
 (*22*) l4p = 
              p26\^2 + agek, 
 (*23*) l4q = 
              q26\^2 + abef, 
 (*24*) l4r = 
              r26\^2 - 3  agek, 
 (*25*) l4s = 
              l22a\ l22b (*\ \(=\)\(\ \)\(s26\^2 - abef\)*) ,  (*26*) 
              L4Q = \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((a - c)\)\^2 + \((d - e)\)\^2 + \((e - f)\)\^2 + \((d - f)\)\^2 + \((g - h)\)\^2 + 
\((h - i)\)\^2 + \((g - i)\)\^2 + \((j - k)\)\^2 + \((k - l)\)\^2 + \((j - l)\)\^2)\)^2/4 - 
                3\ \((\((a\ \((f - e)\) + b\ \((d - f)\) + c\ \((e - d)\))\)\^2 + \((a\ \((i - h)\) + b\ \((g - i)\) + c\ \((h - g)\))\)\^2 + \((d\ \((h - i)\) + e\ \((i - g)\) + f\ \((g - h)\))\)\^2 + \((g\ \((k - l)\) + h\ \((l - j)\) + i\ \((j - k)\))\)\^2 + 
\((\((b - c)\)\^2 + \((e - f)\)\^2)\)\ j\^2 + \((\((a - c)\)\^2 + \((d - f)\)\^2)\)\ k\^2 + \((\((a - b)\)\^2 + \((d - e)\)\^2)\)\ l\^2)\) + 
6\ \((\((\((c - a)\)\ \((c - b)\) + \((f - d)\)\ \((f - e)\))\)\ j\ k + \((\((b - 
                        
                                a)\)\ \((b - c)\) + \((e - d)\)\ \((e - 
                                  f)\))\)\ j\ l + 
\((\((a - b)\)\ \((a - c)\) + \((d - e)\)\ \((d - 
                                  f)\))\)\ k\ l)\), 
 (*27*) 
              L4σ = \ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((c - a)\)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((f - d)\)\^2 - \((g - h)\)\^2 - \((h - i)\)\^2 - \((i - g)\)\^2 - \((j - k)\)\^2 - \((k - l)\)\^2 - \((l - j)\)\^2)\)^2/4 + 
                3\ \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((b\ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 - \((e\ g - f\ g - d\ h + f\ h + d\ i - e\ i)\)\^2 + \((b\ j - c\ j - a\ k + c\ k + a\ l - b\ l)\)\^2 - \((\(-e\)\ j + f\ j + d\ k - f\ k - d\ l + e\ l)\)\^2 - \((h\ j - i\ j - g\ k + i\ k + g\ l - h\ l)\)\^2)\), 
 (*28*) 
              L4a3 = \((\((ag + dj)\)\^2 + \((bh + ek)\)\^2 + \((ci + fl)\)\^2)\)\^2/4 + agek,  (*29*) 
              L4b3 = \((\((ag - dj)\)\^2 + \((bh - ek)\)\^2 + \((ci - fl)\)\^2)\)\^2/4 + agek,  (*30*) 
              L4c3 = \((\((ab + de)\)\^2 + \((bc + ef)\)\^2 + \((ca + fd)\)\^2)\)\^2/4 - abef,  (*31*) 
              L4d3 = \((\((ab - de)\)\^2 + \((bc - ef)\)\^2 + \((ca - fd)\)\^2)\)\^2/4 - abef,  (*32*) 
              L4a6 = \((ag\^2 + bh\^2 + ci\^2 - dj\^2 - fl\^2 - ek\^2)\)\^2/
                  4 + agek,  (*33*) 
              L4b6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/
                  4 + abef,  (*34*) 
              L4c6 = \((ab\^2 + bc\^2 + ca\^2 - de\^2 - ef\^2 - fd\^2)\)\^2/
                  4 - abef, 
	 (*35*) 
              L4d6 = \ \((ag\^2 + bh\^2 + ci\^2 + dj\^2 + fl\^2 + ek\^2)\)\^2/
                  4 - agek, 
 (*36*) 
              L49 = \((\((ab\^2 - de\^2)\)\^2 + \((ab\^2 - ef\^2)\)\^2 + \((ab\^2 - fd\^2)\)\^2 + \((bc\^2 - de\^2)\)\^2 + \((bc\^2 - ef\^2)\)\^2 + \((bc\^2 - fd\^2)\)\^2 + \((ca\^2 - de\^2)\)\^2 + \((ca\^2 - ef\^2)\)\^2 + \((ca\^2 - fd\^2)\)\^2)\), 
 (*37*) 
              L49a = \((\((ag\^2 - dj\^2)\)\^2 + \((ag\^2 - ek\^2)\)\^2 + \((ag\^2 - fl\^2)\)\^2 + \((bh\^2 - dj\^2)\)\^2 + \((bh\^2 - ek\^2)\)\^2 + \((bh\^2 - fl\^2)\)\^2 + \((ci\^2 - dj\^2)\)\^2 + \((ci\^2 - ek\^2)\)\^2 + \((ci\^2 - fl\^2)\)\^2)\), 
 (*38*) 
              L49b = \((\((ab\^2 - dj\^2)\)\^2 + \((ab\^2 - ek\^2)\)\^2 + \((ab\^2 - fl\^2)\)\^2 + \((bc\^2 - dj\^2)\)\^2 + \((bc\^2 - ek\^2)\)\^2 + \((bc\^2 - fl\^2)\)\^2 + \((ca\^2 - dj\^2)\)\^2 + \((ca\^2 - ek\^2)\)\^2 + \((ca\^2 - fl\^2)\)\^2)\), 
 (*39*) 
              L49c = \((\((ag\^2 - de\^2)\)\^2 + \((ag\^2 - ef\^2)\)\^2 + \((ag\^2 - fd\^2)\)\^2 + \((bh\^2 - de\^2)\)\^2 + \((bh\^2 - ef\^2)\)\^2 + \((bh\^2 - fd\^2)\)\^2 + \((ci\^2 - de\^2)\)\^2 + \((ci\^2 - ef\^2)\)\^2 + \((ci\^2 - fd\^2)\)\^2)\)}];\)\)

These sh12 functions are conserved quantities in some Dozal interactions; different hoops conserve different selections, and some are only conserved for orbit interactions. This is a rich topic that has yet to be explored, but has promise as an explanation of particle properties.

3.7. Reciprocal Radii; Three size regimes? The "Large numbers hypothesis"?

The Dozal quadratic sizes,{ζ, λ, η, κ}, must multiply to 1 (excluding any zeroes) in any orbits. ζ has (leptonic?) 4-fold symmetry. The three with ternary (hadronic?) symmetry could have one (λ) corresponding to a Planck area (p^2) and the other two (η, κ) in a reciprocal relationship, two squared radii (r^2) and (R^2) corresponding to the dimensions of "pseudo-point" particles and the universe. p=r.R then gives three size regimes. There is an analogy here with the "T-duality" of M-theory, where equivalent theories are created by exchanging a large dimension and a small dimension. The "Dirac large numbers hypothesis" could be a result of this possible reciprocal relationship between two sizes, with size ratios being powers of (10^40) approximately.  More work is needed.

3.8.  Mass and the Unit Velocity Equation.

The basic wave equation in many directions, (d^2)G/d(t^2)-∑[(d^2)G/d(x^2)]=0, has an infinite number of planar wave solutions, each with a unit velocity. These project onto the different directions as velocity components. This is proved in Example 31, which sums the second differentials of a general travelling wave G[-t+x lx+y ly+z lz] with three space directions.

Example 31. Unit Velocity Equation with 3 space directions.

\(Simplify[∂\_{t, 2}G[\(-t\) + x\ lx + y\ ly + 
              z\ lz] - 
∂\_{x, 2}G[\(-t\) + 
              x\ lx + y\ ly + 
              z\ lz] - 
\ ∂\_{y, 2}G[\(-t\) + 
              x\ lx + y\ ly + z\ lz] - ∂\_{z, 2}G[\(-t\) + x\ lx + 
              y\ ly + z\ lz]]\)

The result is zero if the lx are direction sines so that their squares sum to 1. (Direction sines are used because the angles are measured towards the x direction and the lx contribution disappears in sub-algebras where this angle is zero.). This generalises to any number of space directions. Each additional space direction adds the square of its direction sine. The partial differential equation is therefore satisfied by any unit velocity function moving forward in any number of directions. This gives a light-like limiting velocity in two possible ways:-
1. If pairs of directions fold to single dimensions, the disturbance travels at a speed that is less than unity unless there is a zero velocity in one of each of the folded pairs.
2. If all but 3 space directions are curled-up (Kaluza-Klein) directions, KK velocities reduce the velocity in the spatial directions; light-like disturbances have no velocity in the KK directions. KK "orbital" velocities give mass to the disturbances. Massless particle waves would be restricted to space-like dimensions, whilst others would have velocities in Kaluza-Klein dimensions. This relates to the reciprocal radii conjecture. Constant orbital frequencies in small dimensions would contribute less mass than the same in larger dimensions.

An article [7] on Huygen's principle develops both the linear and the spherical wave versions for n spatial dimensions. n=3 gives sinusoidal waves with the square law of decay, amplitude=A/(r^(n - 2)). n=2 gives Bessel function solutions. [8] applies the spherical wave versions to spaces with more than 3 space dimensions. Odd n's  give expanding spherical waves that decay according to r^[n-2]; the square law decay is restricted to 3 spatial dimensions.

The  Schrodinger equation provides "information about particles" as wave packets which disperse with the passage of time. As dispersion arises when the velocity is frequency-dependent, the unit velocity equation describes non-dispersing wave packets. Generalizing the Schrodinger equation to 12 directions (instead of the 4 implied by using i) might lead to a more general description of non-dispersing particles.

3.9.  The 6-vectors and Dependent Time.

The C3C4 hoop has a conserved quartic l4s which factorizes into two sums of  squares l22a & l22b and can be expressed as  s26²-abef =(((((ab^2) + (bc^2) + (ca^2) + (de^2) + (ef^2) + (fd^2))/
                2))^2)-3(((ab ef - bc de))^2). This may correspond to a Kaluza-Klein 6-vector ((ab^2) + (bc^2) + (ca^2) + (de^2) + (ef^2) + (fd^2))/2 (with 3 spce dimensions and 3 KK dimensions) and a dependent "time" -3 (ab ef - bc de). Other conserved quartics provide a variety of conserved sums or differences of squares; l4r = r26²-3agek is also the difference between the squared sum of six squares and another squared term. These functions provide possible extensions of the Minkowski metric to Kaluza-Klein spaces where "time" is dependent on the particle velocity. Four functions consisting of the sum of nine squared terms are also defined in Example 30. Their significance is unknown.

3.10.  Hyperbolic orbits.

Hyperbolic orbits exist when hoops have sizes that can be expressed as the difference of two squared fragments, corresponding to Cosh[x]^2-Sinh[x]^2=1. They are not implemented in the hoops.m package, but are included in MathSource/4894. They occur in non-Abelian hoops, and can be created in Abelian hoops if the product of two linear sizes can be expressed as the difference of two squares. The GroupLoopHoop.m package includes nine examples. Example 29 in GroupLoopDemo.nb (reproduced below as Example 32) multiplies two complex vectors, using the Klein group K. This conserves {a+b+c+d, a-b+c-d, a-b-c+d, a+b-c-d}, so it also conserves (((a + c))^2)-(((b + d))^2) and (((a - c))^2)-(((b - d))^2). The example shows that sizes multiply, (complex) angles add, and hyperbolic forms revert correctly to Cartesian vectors. Figure 32 includes a skeleton implementation of the relevant procedures. The hyperbolic tangent function has to be extended to handle two complex arguments {x,y}. The results depend on the quadrants in which x & y occur (details omitted); arcTanh[-x,-y] differs from arcTanh[x,y] although arcTanh[-x/(-y)]=arcTanh[x/y].

Example 32.Implementation of the "K" Hyperbolic Dual;

arcTanh[x, y] := Log[(x + y)/(Sqrt[(x^2) - (y^2)])]
tohyPol[{a, b, c, 
              d}, "K"] := {u^2 = (a + c)^2 - (b + d)^2, ϕ = 
              arcTanh[a + c, 
                b + d], 
v^2 = (a - c)^2 - (b - 
                      d)^2, ψ = arcTanh[a - c, b - d]}
tohyVec[{u, ϕ, 
              v, ψ}, "K"] := 
  { 
              u*Cosh[ϕ]  + v*Cosh[ψ],   u*Sinh[ϕ] + 
                v*Sinh[ψ], 
-u*Sinh[ϕ] - 
                v*Cosh[ψ], -u*Sinh[ϕ] - v*Cosh[ψ]}/2
If A={3.  - 2  I, 5. , -2, 
                2.  + I} with hyperbolic form hA=
{-51 - 
              18 I,  .0919 + 
              1.2768 I, 
            13 - 
              14 I,  .67026 +  .0524 I}
and B={5.  + I, 
                3.  - 2  I, -2.  - 3  I, 4. } 
              with hyperbolic form hB=
{-40 + 
              16 I,  .49525 - 
              1.761 I, 
            36 + 
              52 I, - .2291 -  .1609 I}
AB={44 - 7 I, 
              21 - 15 I, 
              6 - 8 I, 
              5 - 12 I}with hyperbolic form hAB={2328 - 
              96 I,  .5872 -  .4843 I, 
            1196 + 
              172 I,  .4412 -  .1084 I}
sizes multiply, hA hB - hAB={0, 1.7068 +  .955  I, 0, - .586 -  .01  I}
hyperangles add,hA + hB - hAB={-2419 + 94  I, 0, -1147 - 134  I, 0}

4. Conclusions.	
	Hoop Algebras unify many algebras with applications in physics. Complex algebra is a hoop, but is degenerate because it only conserves one size. This reduces Hoop properties to special cases.

References.

[1] MathSource/6198/
[2] Hoops&Physics.doc
[3] P. Louenesto, Clifford Algebras & Spinors 2nd ed., Cambridge University Press 2001
[4] MathSource/4894/
[5] J.D.H.Smith, A.B.Romanowska, Post-Modern Algebra,Wiley Interscience 1999.
[6] J.J.Hamilton, Hypercomplex numbers and the description of spin states, J.Math.Phys.38(10) Oct. 1997. pp4914-4928
[7] http://www.mathpages.com/home/kmath242/kmath242.htm
[8] http://www.mathpages.com/home/kmath614/kmath614.htm