Hoop Algebra Demonstration.

Conservative Division Algebras - Data, Procedures, Demonstrations, Physics.

R.H.Beresford. 2004-5, revised 12/10/2006.

".. we must be willing to question basic, seemingly self-evident, elements of reality." (Brian Greene, The Fabric of the Cosmos" 2004, p426.)

1. Foreword.

Symmetry-conserving partial-fraction-division "Hoop Algebras" are closely related to Groups and Moufang Loops. They multiply, divide, add and split vectors with generalized signs and employ many new or neglected mathematical concepts to re-define physical mathematics.

A shorter and simpler account of Hoop Algebras (using a much smaller package) is available at http://library.wolfram.com/infocenter/MathSource/6198/; it should be studied first. This notebook goes into much greater detail.
The demonstrations can be followed by non-Mathematica users, by downloading the free MathReader from the same MathSource page.
Mathematica users can save (in addons/ExtraPackages) & initialize the GrouploopHoop.nb from http://library.wolfram.com/infocenter/MathSource/4894/. to set up the package and investigate the demonstrations.
HoopDemo is a simplified version of GroupLoopDemo, omitting material on the creation, identification, and manipulation of Groups & Loops. Activate the following cell if you have Mathematica and wish to test the examples with different data.
Please send any comments to me at rhberesford@btinternet.com

<< GroupLoopHoop`

2. Illustrated Introduction.

2.1 Hoops.

Hoops are "symmetry-conserving partial-fraction-division algebras" that unify many physically significant algebras (including ℝ, ℂ, Clifford, Davenport, Dirac-γ, graded, ℍ, Lie, noncommutative, non-associative, 𝕆, Pauli-σ, and Wedge etc., algebras). Hoops act on "vecs" (generalized vectors) - indexed lists of operators or coefficients {(Subscript[a, 1]),...(Subscript[a, m])}, equivalent to (∑_i) (Subscript[a, i]) (Subscript[d, i]) where (Subscript[d, i]) is a sign or a dimension - using a Cayley index multiplication table G:i j→ k, where i, j & k are indices though k may be signed (see Figure 1). Generalized procedures multiply, divide and (in some cases) provide cartesian/polar interconversions that lead to powers and roots. Hoops have the Moufang division property, ensuring that vectors have multiplicative inverses. Consequently, division employs the same procedure, hoopTimes, as multiplication; AB=(∑_k) (Subscript[a, i]) (Subscript[b, j]) (Subscript[g, k]). Warning - this multiplication MUST NOT be confused with matrix multiplication (transformations of coordinates for a single vector); it represents interactions between two vectors to give a product and two (possibly null) remainder vectors.
	The defining feature of a hoop is "Frobenius conservation", Det[AB]=±Det[A] Det[B]; each algebra has conserved "sizes" or symmetries that may (by Noether's theoren) define particles and forces. These sizes are the factors of the determinant of the multiplication table inverse mapped with the vectors. The sizes are also denominators of the partial-fraction formulation of the inverse. Hoops comprise all groups, a few non-associative Moufang Loops (including Octonions), and the "signed tables" obtained by "folding" conservative unsigned tables. Hoops combine the "conservative" property with the Moufang division property, so they describe conservative division algebras. Most tables lack both properties.

2.2 New Concepts.

New concepts, transcending the limitations of "real algebras without divisors of zero", are introduced. The most important of these are unsigned continuous "primal numbers", "generalized signs", "folding", conserved "sizes" (symmetries), conservative multiplication-division with "remainders", degenerate "monosized" algebras, "polar-duals" with "offsets", "plex-conjugates", and "unital hoop algebras" with continuous "orbits" that provide analogies with deBroglie waves and Planck areas. Real numbers are "folded" (equivalenced) from a pair of "primal" (unsigned continuous, half-line) numbers via an equivalence relationship (Subscript[r, i]) ~ {(Subscript[a, i]), (Subscript[a, j])} ~ {(Subscript[b, i]), (Subscript[b, j])} iff (Subscript[a, i])+(Subscript[b, j]) = (Subscript[b, i])+(Subscript[a, j]); reals do not exist in some algebras. Other types of number are folded from larger sets of primal numbers by the introduction of generalized signs.
	Hoop multiplication tables are, or are "folded" from, algebraic loops. An unsigned m×m loop with r-fold symmetry can be folded to an (m/r)×(m/r) table whose elements are "r-signed". The tables inherit the Moufang division properties, including zx.yz=(z.xy)z. This ensures that there are multiplicative inverses for each group element and for each vector, so division (multiplication by a multiplicative inverse) is always possible. Groups are associative Moufang loops, so all groups define hoop algebras. Only a few of the many non-associative Moufang loops [4] are conservative (i.e. hoops). (Those with up to 32 elements are in the database as D4M, Oct, OctC2, P4M, KiC4M, C8pC2M, C4iC4M.)

Fig.1 shows three small Moufang loops (the C4C2, quaternion, and C9 groups) as index tables; the boldface top left squares are the folded versions. These, because of a loop symmetry, occur with or without offsets in the other squares of the same size. The folded tables are interpreted as algebraic multiplication tables with signed products. The minus sign is a square root of +1, the sign 𝕁 is a cube root of +1, and i, i, j are fourth roots of +1. The first and folded second tables define the complex and quaternion algebras; the third is a new algebra "C9J" developed to investigate ternary symmetry and "3-folding". 
	The last table is the symbolic form of the C9j table; {a,b,c} has been mapped onto the signed elements. This mapping allows the evaluation of the table determinant, Det. (Hoops are defined as having the Frobenius property, conservation of conjugate factors of the determinant Det[AB].) The conserved factors (omitting the voluminous C9 case) are shown. In all but the C9J case (where the determinant is -(a^3)(𝕁^2)-(b^3)𝕁+3abc𝕁-(c^3)) conjugation has had no effect as the determinants are real.

Figure 1. Two-fold and three-fold Folding of Moufang Loops, the
resulting signed algebras, and their conserved shapes..

C4C2 group          Quaternion group       C9 Group

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

  C4 & Complex    Quaternion        C9J      Symbolic
    algebra        algebra         algebra     C9J  
({{1, i, -1, -i}, {i, -1, -i, 1}, {-1, -i, 1, i}, {-i, 1, i, -1}}), ({{1, i, j, ij}, {i, -1, -ij, j}, {j, ij, -1, -i}, {ij, -j, i, -1}}),     ({{1, 2, 3}, {2, 3 , 𝕁1}, {3, 𝕁1, 𝕁2}}),({{a, b, c}, {b, c, 𝕁a }, {c, 𝕁a, 𝕁b}})

shape[{a, b, c, d, e, f, g, h}, "C4C2"]

shape[{a, b, c, d}, "C4"]

shape[{a, b, c, d, e, f, g, 
        h}, "Q8"]

shape[{a, b, c, d}, "Qr"]

shape[{a, b, c}, "C9J"]

Loops can be expressed as Cayley tables in many isomorphic ways. The Hoops database provides a preferred isomorph or "protoloop" for each table. The basic properties are unchanged if an isomorph is used - the elements are re-labelled in isomorphisms. However, different isomorphs can fold to different hoops (see Figures 2 and 24).
	Fig. 2 folds a different C4C2 isomorph to the Davenport algebra, by 5→-1, 6→-2, 7→-3. The symbolic table ({a,b,c,d} mapped onto {1,2,3,4}),  is also shown. The determinant of this table factorises into ((((b+c))^2)+(((a - d))^2) )*  ( (((b - c))^2)+(((a + d))^2)), and these two factors are conserved on multiplication.

Figure 2. Davenport algebra is folded from a C4C2 Isomorph.

C4C2 Isomorph                                 Davenport Algebra
({{({{1, 2, 3, 4, 5, 6, 7, 8}, {2, 5, 4, 7, 6, 1, 8, 3}, {3, 4, 5, 6, 7, 8, 1, 2}, {4, 7, 6, 1, 8, 3, 2, 5}, {5, 6, 7, 8, 1, 2, 3, 4}, {6, 1, 8, 3, 2, 5, 4, 7}, {7, 8, 1, 2, 3, 4, 5, 6}, {8, 3, 2, 5, 4, 7, 6, 1}}), ({{     1, 2, 3, 4}, {2, -1, 4, -3}, {3, 4, -1, -2}, {4, -3, -2, 1}})     , ({{a,    b,   c,   d}, {b, -a,   d, -c}, {c,   d, -a, -b}, {d, -c, -b,   a}})}})

Most hoop determinants have more than one factor. I call the determinant factors "sizes" and the list of symbolic sizes the "shape" for the hoop. Sizes are conserved properties or symmetries for the algebra, size[AB]=±size[A] size[B]. Hoops are so-named because they conserve their shape and come between groups and loops.

2.3 Primal numbers, generalized signs.

Loops are defined without introducing negation, so they multiply sets of essentially unsigned elements (numbers or operators). Unsigned continuous "Primal" numbers (the half-line) can be defined from set theory, via Natural numbers, unsigned rational numbers, and an axiom of continuity. Other numbers can then be defined via equivalence relations that "folds" a set of r primal numbers to a single r-signed number. m×m Moufang loops with r-fold symmetry can be folded to (m/r)×(m/r) Hoop tables; at the same time the sets of m unsigned elements are folded to vectors with m/r "r-signed" elements. Signs introduce the second operation that converts a loop into an algebra. The standard cases are r=2 and 4, creating numbers in real or complex fields and the signs "+", "𝕚", "-", and "-𝕚".

Most mathematical texts wrongly assume that negative numbers are fundamental; only a few define them as equivalence relations on pairs of unsigned numbers; even fewer include "Minus 1 exists" and "Minus 1 is unique" as axioms. The effect is that much of mathematics has been restricted to "fields" where "ordinary negation" and "subtraction" are conflated, the only "signs" are "+", "i", "-", & "-i", and other possibilities are excluded. Hoops transcend this artificial limitation.

Generalized signs (𝕤^j) (with (𝕤^r) = 1) are introduced as distinct j'th powers of r'th roots of unity. They include, when r = 4, (𝕤^0)="+", (𝕤^1)="i", (𝕤^2)="-", & (𝕤^3)="-i". Projecting other roots of unity onto the complex plane as "cyclotomic numbers" is a distortion and restriction of their properties. In many hoops, "Minus 1" is not unique; the standard proof that it is unique is based on the usually unstated axiom (assumption) that integers are equivalence relations on pairs rather than indexed sets of natural numbers. "Study numbers" have (k^2)=1,  k≠-1. "Minus 1" need not exist; r=3 gives "Terplex algebra", with signs "+", "𝕁", "(𝕁^2)", where 𝕁𝕁𝕁=1, and there is no entity that squares to 1. (The standard proof that all 3-element algebras are equivalent to complex algebra is only valid for fields with a single -1.)
	Complex functions are special cases of functions involving generalized signs. In particular, (Subscript[e, r]^((𝕤^j) x))defines multiphase symmetric sinusoids that fold to the asymmetric (e^(-i x))= Cos[x] +i Sin[x] when r=2.

2.4 Vector Division, Remainders.

The Moufang division property ensures that a multiplicative inverse Ainv exists for all hoop vectors A, with Aiinv.A = {1,0,..} (the unit vector). Vector division is multiplication by the inverse. (In non-abelian algebras Ai is a left inverse; a right inverse can be created.)  Inverses can be found via Cramer's rule, and so have the conjugate determinant as a denominator. In most hoops (excluding degenerate monosized ℝ, ℂ, ℍ, 𝕆, Pauli-σ, C9j, C9J, some Clifford algebras, etc.) the determinant has more than one distinct factor, and the inverse splits into partial fractions with each distinct real factor providing a partial-fraction denominator (repeated factors occur in non-abelian hoops, but only provide one size.)

Vectors give a zero determinant if any sizes are zero, even though some sizes are still finite. Such "zero-vectors" (with one or more size evaluating to zero) can still have conservative hoop multiplication and division. Division projects the results onto sub-algebras (constrained to have these sizes zero) and is defined so that it ejects left & right "remainders"; A/B=C+Rl+Rr, with C*B+Rl=A & A/C+Rr=B. The remainders Rl & Rr have the sizes of A or B that would be infinite in 1/B & 1/A respectively; i.e. 0⩽size[Rl]<size[1/B], just as integer division has I/J=K+R, 0≤R<J. Conservation is maintained by the creation of remainders wherever a size would otherwise be lost by becoming zero or infinity. Division-by-zero is eliminated by "projection" into sub-algebras. In effect, the zeroes and infinities are factored out of the result.
	Multiplication behaves in the same way (multiplication and division are one operation, using inverse vectors). In Fig. 3, multiplying A={2,1,-1,2} by B={1,3,5,2} in the Davenport algebra gives AB={8,-1,1,8} and a right remainder {-.5,4,4,.5} because the shape of A is {0,20}. The shape of B is {65,13}, so the shapes of AB and Rr are {0,13} and {65,0}.
	 (Most figures are the output from a set of Mathematica input instructions, which are usually hidden in a "closed cell" because they are probably meaningless to non-users of Mathematica. If desired, the instructions can be inspected by clicking on the tiny cell bracket, then on Cell, Cell Properties and Cell Open.)

Figure 3. Remainders in Dav multiplication and division.

{"A= ", A = {2, 1, -1, 2}, "B= ", 
      B = {1, 3, 5, 2}, "\nUsing Davenport Algebra, AB=", 
      AB = hoopTimes[A, B, "Dav"], "\nRl is", 
      Rl, " with shape[Rl]=", shape[Rl], "\nRr is", 
      rr = Rr, " with shape[Rr]=", shape[Rr], "\nAinverse=", 
      Ai = hoopInverse[A], "\nAinverse.AB = ", 
      B0 = hoopTimes[ Ai, 
          AB], "which only recovers B when Rr is added to give ", 
      B0 + rr}
A =. ; AB =. ; B0 =. ; rr =.

2.5 Polar Duals, Orbits.

Many hoops have quadratic sizes that can be expressed as sums or differences of squares. These lead to polar-dual forms (generalizations of the polar description of the complex plane) that involve multiphase sinusoids with angles; the quadratic sizes are squared radii. Because angles add on abelian multiplication, polar-duals provide abelian hoops with roots, powers, logarithms, and continuous multi-phase sinusoidal "orbits" that are are related to the discrete orbits of loops. Whilst complex-conjugate factor-pairs are not conserved, their real products are conserved sizes that define the squared radii of orbits.

Figure 4. The polar form of "Terplex" C3 is given by "toPol[{a,b,c}]". The Orbit has an offset d=a+b+c, a squared radius r2, and an angle σ. The function "toVec[{d,r2,σ}]" recovers the Cartesian form.

{"The Terplex Polar form of {a,b,c} is", 
      toPol[{a, b, 
          c}, "C3"], "\nThe general toVec[{d,r2,σ},C3] \
expression is", 
      toVec[{d, 
          r2, σ}, "C3"], "\nThe Polar form of {1.,2.,5.} is", 
      p3 = toPol[{1. , 2. , 
            5. }, "C3"], 
"\ntoVec recovers  ", 
      toVec[p3, "C3"]}
p3 =.

Some unitary algebras, in which the determinant is constrained to be 1 (apart from any zero sizes), have 3, 4, 6, & 12-phase de-Broglie-wave-like orbits, stable Planck-area-like sizes, and reciprocal-radius pairs. 12-element hoops provide "half-spin" quantum operators related to 6-element "unit-spin" operators; there are fewer 6-element "supersymmetric" elements because they result from a fold that frees them from the Pauli exclusion principle. Note that, whilst algebras are usually taken to have vector elements that are "coefficients", elements can also be operators.
	Sizes are symmetries that may (via Noether's theorem) account for forces and particles. They may determine the different families and the limited number of fundamental particles. They may also lead to the "law of large numbers" and three size regimes, R = the radius of universe, ρ = a Planck-size , and r = ρ/R = the radius of "pseudopoint" particles. Hoops may provide a new basis for symmetry in mathematical physics, based on finite hoops rather than infinite groups.

2.6 The GroupLoopHoop.m package.

GroupLoopHoop.m is a package that includes data on over 90 hoop algebras (including some isomorphs). Most of these have under 16 elements; large symbolic determinant factors are difficult to find. The package provides the generalized procedures hoopTimes, hoopInverse, hoopPower. In addition to the hoops, it provides a database of all groups with up to 73 elements, and some other tables, and includes procedures to generate, manipulate, and identify tables. GroupLoopTest.nb validates the package.
	HoopGlossary.nb gives brief explanations of the relevant concepts. GroupLoopDemo.nb demonstrates the database and procedures. This notebook, HoopDemo.nb, is a shortened version concerned only with hoops. MathSource /6198 gives a simplified account of hoops, with a database restricted to hoops.

3. Hoop Tutorial.

3.1. "Hoops", Hoop Names.

Summary. Conservative division algebras (Hoops) are created by "folding" Moufang loops (groups and some alternative loops such as octonions) by the introduction of generalized signs. Hoops multiply and divide vectors whilst conserving "symmetries" or "sizes" (factors of the symbolic inverse table determinant). A database provides named tables and sizes for over 90 hoops, together with much more material on small loops and groups.

Hoops are "conservative division algebras" that unify most algebras (including the Cayley-Dickson algebras {ℝ, ℂ, ℍ, 𝕆}, Clifford, Davenport, Dirac-γ, Lie, non-commutative, non-associative, Pauli-σ, Terplex & Wedge algebras, etc) by employing generalized procedures for multiplication, division, and (where appropriate) conversion to and from the polar-dual forms that provide powers and roots. They act on generalized vectors - indexed lists of coefficients {(Subscript[a, 1]),...(Subscript[a, m])}, equivalent to (∑_i) (Subscript[a, i]) (Subscript[d, i]) where (Subscript[d, i]) is a sign, a direction, or a dimension.
	Hoops are division algebras because their multiplication tables have the Moufang division property, zx.yz=(z.xy)z. This ensures that each element in the table, and each vector, has a multiplicative inverse. Division is multiplication by the inverse vector.
	Hoops are conservative algebras because (by definition) they have the Frobenius determinant conservation property, Det[AB] = ±Det[A]Det[B] where Det is evaluated from the inverted multiplication table mapped with the vector. The factors of the determinant are conserved properties ("symmetries" or "sizes") and the list of distinct factors is the "shape" for a vector in a specified algebra.
	Hoops are algebras because they have another operation (apart from multiplication/division), addition with generalized negation and "generalized signs". This operation is obtained by an equivalence relationship on sets of coefficients,  (Subscript[h, i])~ {(Subscript[a, i]),(Subscript[a, i + m/r]),(Subscript[a, i + 2  m/r]),..} ~ {(Subscript[b, i]),(Subscript[b, i + m/r]),(Subscript[b, i + 2  m/r])...}. The commonest case is r=2, giving half as many real coefficients, (Subscript[r, i])~ {(Subscript[a, i]) ,(Subscript[a, i + m/2])} ~ {(Subscript[b, i]), (Subscript[b, i + m/2])} iff (Subscript[a, i])+(Subscript[b, i + m/2])= (Subscript[b, i])+(Subscript[a, i + m/2]). Equivalencing an m×m table with r-fold symmetry "folds" it to an (m/r)×(m/r) table. This may be a smaller unsigned Moufang loop (e.g. in Fig.1 the C4C2 isomorph folds to C4) but in many cases it is a signed table in which some products are signed; e.g. C4 2-folds to C4c (complex algebra) with elements {1,i} and the signed product i.i is -1; the C4C2 isomorph in Fig.2 2-folds to the signed Davenport multiplication table.

The database in GroupLoopHoop.m includes multiplication tables and properties for over 90 loops with known determinant factors. Each has a string as a name. The names without the quotation marks define Cayley tables, providing the "preferred isomorph" for hoop multiplication tables. The procedure hoopShape[name] selects a "target algebra" for further operations. The names, in order of increasing vector size, (apart from 4 non-conservative loops, moved to the end) are:-

hoopList

The names are based on the names of loops used in GroupLoopHoop.m. Thus C2, C3, etc. are cyclic groups, D3, D4, etc. are dihedral groups, Q8, Q12 are quaternionic groups (Q12 is called T in Mathworld Groups, (Subscript[Dic, 3])in Wikipedia). C3C2 is the direct composition of C3 with C2; C4iC4 is the indirect composition of C4 with C4; K=C2C2 is the Klein 4-group. P4, P8, P16 have Pauli-σ matrices as generators. CLmn is Clifford(m,n). Names ending in r, j, J, or i are signed tables that compose with C2, C3, or C4 to Moufang loops. Oct is the 16-element Octonion loop, which folds to the 8-element octonion hoop Octr. O4, O4a, O4b are hoops with polar-duals having four {angle,radius} pairs. Dav is the Davenport (hypercomplex) algebra. An M indicates a non-associative Moufang loop, obtained by doubling a non-abelian group, so Q8M2 is the non-associative but conservative Moufang loop obtained from Q8. The last four, with names ending in n, are not conservative; they are included as non-hoop counter-examples with known determinant factors that do not define inverses.

All these hoops have been tested, using a routine in GroupLoopTest.nb. This shows them to be conservative division algebras by tests with random real or complex vectors. Polar-duals, positive powers, roots, and quadratic equation solution have also been tested successfully wherever duals are known.

Conservative Moufang loops are a small subset of quasi-groups and semi-groups. Most tables have neither the Moufang nor the Frobenius property, so they lack inverses and conserved sizes.

3.2. Multiplication, Conserved "sizes".

Summary. hoopShape selects an algebra & hoopTimes defines multiplication in it. Conservation of determinant factors is demonstrated for C3 algebra.

The GroupLoopHoop.m database contains a multiplication table and a "shape" for each hoop algebra. A "target hoop" is selected by a hoopShape[name] instruction (or by specifying the loop in a multiplication operation). hoopTimes uses the selected multiplication table to multiply two vectors. The result is a vector (Subscript[c, k]) = Σ(Subscript[a, i])(Subscript[b, j]) where i× j = k in the table. Do not confuse hoop multiplication (which creates a vector and remainders from two vectors) with matrix multiplication (which transforms a single vector to new coordinates) or the Dot product, which creates a scalar from two vectors.

Figure 6. Multiplication. The C3 algebra is selected by hoopShape["C3"]; and two 3-element vectors are set up and multiplied:-

hoopShape["C3"]; {"If A=", 
      A = {3. , 1. , 4. }, "& B=", 
      B = {1. , 2. , -4. }, "AB in C3 algebra =", 
      AB = hoopTimes[A, B], "\nThis is not the Dot product, which is" , 
      A . B, "\nbut a vector. The general product {a,b,c}×{d,e,f} is \
", hoopTimes[{a, b, c}, {d, e, f}]}

Three variables, the hoop name gmn, the multiplication table glo, and the length mm, are automatically set when an algebra is selected by hoopShape. 
	The multiplication index-table can be shown in "TraditionalForm" by appending //tf , the symbolic form by gmap[glo,{a,b,c}]//tf, (which puts a,b... in place of 1,2.., etc.) and the shape is given by sh[gmn]
	
Figure 7. C3 Index table, Symbolic table, and Shape.

{gmn, "Index Table", 
      glo // tf, "Symbolic Form", 
      gmap[glo, {a, b, c}] // tf, "\nhas shape=", 
      shape[{a, b, c}, "C3"]}

Figure 8. Size Conservation in Multiplication. The shapes of A, B, and AB (from Fig. 6.) are shown; conservation is demonstrated in the last line by subtracting to give {0,0}:-

{{"", sh1, sh2}, Flatten[Join[{shA }, sA = shape[A]]], 
        Flatten[Join[{shB }, sB = shape[B]]], 
        Flatten[Join[{shAB}, sAB = shape[AB]]], 
        Flatten[Join[{shA*shB - shAB}, sA*sB - sAB]]} // tf

3.3. Multiplicative Inverses, Division.

Summary. Division is multiplication by a multiplicative inverse that can be found by Cramer's method and so has the determinant as denominator.  hoopInverse creates the left inverse Ainv. This is demonstrated for abelian C3 and non-abelian D3, where Ainv.AB recovers B and AB.Binv recovers A (because B is the left inverse of Binv), provided that A & B have the same zero-sizes.

hoopInverse[A] gives the multiplicative left inverse Ainv (which is also a right inverse for abelian hoops). The inverses have been found by Cramer's method, solving Ainv.A= I, where I is the hoop unit vector, {1,0....0}. (Again, do not confuse this with matrix inversion, where I is the unit diagonal matrix.) Cramer's method makes the table determinant the denominator for every element of the inverse.
Figure 9. C3 Division using hoopInverse. Ainv.AB recovers B:-

{"Ainv", 
      Ainv = hoopInverse[A], "\nDeterminant ", 
      Det[gmap[C3, A]], "\nShape Ainv", 
      shape[Ainv], "\nAinv.AB recovers B", 
      hoopTimes[Ainv, AB]}

Non-abelian (noncommutative) algebra determinants have repeated roots, as shown by the third line of Figure 10., where {a,b,c,d,e,f} is mapped onto the index table of D3 (the smallest non-abelian group), and the determinant is factorised. (The repeated factor "fragments" into ((((a - c))^2)+(((c - e))^2)+(((e - a))^2))/2-((((b - d))^2)+(((d - f))^2)+(((f - b))^2))/2 ). The D3 products AB and BA differ. Inverses are "left inverses", so both Ainv.AB & BA.Ainv recover B, but AB.Ainv gives rubbish.

Figure 10. Noncommutative Multiplication & Division using D3.

hoopShape["D3"]; {"A=", 
      A = {3, 1, 4, 2, 6, 5}, "B=", 
      B = {1, 2, 1, 3, 3, 4}, "\nD3 Determinant ", 
      Factor[Det[gmap[D3, {a, b, c, d, e, f}]]], "\nAB in D3 algebra =", 
      AB = hoopTimes[A, B], "\nBA in D3 algebra =", 
      BA = hoopTimes[B, A], "\nAinv", 
      Ainv = hoopInverse[A], "\nShapes of A & Ainv", shape[A], 
      shape[Ainv], "\nAinv.AB recovers B", 
      hoopTimes[Ainv, AB], "\nBA.Ainv recovers B", 
      hoopTimes[BA, Ainv], "\nAB.Ainv is rubbish", 
      hoopTimes[AB, Ainv], "\nBinv", 
      Binv = hoopInverse[B], "\nShapes of B & Binv", shape[B], 
      shape[Binv], "\nAB.Binv recovers A", 
      hoopTimes[AB, Binv], "\nBinv.BA recovers A", 
      hoopTimes[Binv, BA]}

The database contains a few hoops that are non-associative, with Octonions being best known. "Octr" is the 8-element octonion algebra, and "Oct" is its parent 16-element unsigned table, the third nonassociative Moufang loop with 16 elements. "Q8M2" is the second, and is parent to another 8-element signed table "Octi" (split-octonions in Wikipedia); both have a size consisting of four positive and four negated squares, giving the signature (4,4). Left and right multiplication of two 16-element vectors is demonstrated for "Q8M2" in Figure 10a:-

Figure 10a. Noncommutative Nonassociative Multiplication & Division Q8M2.

{"A=", 
      A = {3, 1, 4, 2, 6, 5, 0, 7, 1, 0, 4, 2, 3, 
          0, -1, -4}, "B=", 
      B = {1, 2, 1, 3, 3, 4, 5, 0, 2, 6, 5, 0, 7, 1, 0, 
          4}, "\nQ8M2 Determinant Factors\n ", 
      hoopShape["Q8M2"], "\nAB in Q8M2 algebra =", 
      AB = hoopTimes[A, B], "\nBA in Q8M2 algebra =", 
      BA = hoopTimes[B, A], "\nAinv", 
      Ainv = hoopInverse[A], "\nShapes of A & Ainv", shape[A], 
      shape[Ainv], "\nAinv.AB recovers B", 
      hoopTimes[Ainv, AB], "\nBA.Ainv recovers B", 
      hoopTimes[BA, Ainv], "\nAB.Ainv is rubbish", 
      hoopTimes[AB, Ainv], "\nBinv", 
      Binv = hoopInverse[B], "\nShapes of B & Binv", shape[B], 
      shape[Binv], "\nAB.Binv recovers A", 
      hoopTimes[AB, Binv], "\nBinv.BA recovers A", 
      hoopTimes[Binv, BA]}

Recovery fails unless A & B have the same zero-sizes; disparate zeroes are handled in the next sections.

3.4. "Zero sizes" & Remainders.

Summary. Vectors B with zero-sizes are given an inverse Binv with the same zeroes, so these sizes are lost in the dividend. A/B (= Binv*A) is "projected" onto a sub-algebra. Conservation is maintained by "ejecting" a remainder. The analogy with integer division, A/B = C, C*B+R=A with R<B, is that the ejected remainder contains those sizes of A that are less than the (infinite) sizes omitted from Binv.

Most hoop determinants have more than one factor, so most inverses split into partial fractions with the factors (sizes) as denominators. (A mathematical quirk makes repeated factors give a single partial fraction, weighted by the multiplicity, rather than the general partial fractions.)
	hoopInverse creates the required partial fractions, but omits any in which the denominator is effectively zero (to within an arbitrary hmin). Consequently, Ainv has the same zeroes as A and dividends have the same zeroed sizes. The dividends are "projected" onto a sub-algebra in which these sizes are constrained to be zero. By analogy with natural number division, where A/B = C with remainder R (with 0⩽R<B) such that C*B+R=A, there is a dividend C and a remainder R, into which any unconserved sizes are "ejected". This is implemented by hoopTimes[A,Binv]; it is demonstrated here and discussed in the next section.
	
Figure 11.  B is altered to have a zero D3 size. Dividing A by B splits A into the dividend A0, with the same size zero, and the (left) remainder Rl, with the other sizes zero:-

hoopShape["D3"]; {"A=", 
      A = {3, 1, 4, 2, 6, 5}, "B=", 
      B = {1, 2, 1, 3, 7, 4}, "\nAB in D3 algebra =", 
      AB = hoopTimes[A, B], "\nBinv", 
      Binv = hoopInverse[B], "\nwith remainders Rl, Rr", R1 = Rl, 
      Rr, "\nShapes A,B    ", shape[A], shape[B], "\nShapes Rl,Binv", 
      shape[Rl], shape[Binv], "\nAB/B+Rl=A", 
      hoopTimes[AB, Binv] + R1}

3.5. Multiplication and Remainders.

Summary. Multiplication by a vector Z with zero-sizes gives a product with the same zeroes, losing any corresponding sizes in the other multiplicand, so that ZB/Z does not recover B. The product ZB has been "projected" onto a sub-algebra, just as in division. Conservation is maintained by "ejecting" left and right remainders so that ZB/B+Rl = Z & ZB/Z + Rr = B. Abelian, non-abelian, and monosized algebra remainders are demonstrated.

Division needs remainders to maintain conservation. As division is multiplication by an inverse, using hoopTimes, remainders must also be involved in multiplication. If different sizes of A & B are zero, these sizes will be zero in the product AB. To maintain conservation, the "missing sizes" have to be "ejected" as remainders when AB is "projected" onto a sub-algebra. Rl is the remainder for the left multiplicand A, and Rr for the right multiplicand B. They ensure that AB/B+Rl=A and AB/A+Rr=B. The procedure hoopTimes creates the product and both remainders, which are {0,0..} if different zeroes are not involved.
A general function muldiv is set up to demonstrate remainders. (The details are irrelevant to non-users of Mathematica):-

muldiv[Ao_, Bo_, H_] := 
        Module[{s = Length[sh[H]], z = Table[0, {i, 32}]}, gmn = H; 
          glo = ToExpression[H]; mm = Length[glo]; 
          s = Take[Flatten[{"shape no.", Range[s]}]]; 
          A = Take[Flatten[Append[Ao, z]], mm]; 
          B = Take[Flatten[Append[Bo, z]], mm]; 
          Print[{Flatten[Append[{"A  ="}, A]], 
                Flatten[Append[{"B  ="}, B]], 
Flatten[
                  Append[{"AB ="}, 
                    AB = hoopTimes[A, B]]], 
Flatten[
                  Append[{"Rl ="}, R1 = Rl]], 
Flatten[
                  Append[{"Rr ="}, R2 = Rr]]} // 
              tf]; 
Print[{s, 
                Flatten[Append[{"shA  ="}, shape[A]]], 
                Flatten[Append[{"shB  ="}, shape[B]]], 
                Flatten[Append[{"shAB ="}, shape[AB]]], 
                Flatten[Append[{"shRl ="}, shape[Rl]]], 
                Flatten[Append[{"shRr ="}, shape[Rr]]]} // 
              tf]; {Print["AB/B+Rl=A  ", 
              hoopTimes[AB, hoopInverse[B]] + R1, "\nAB/A+Rr=B  ", 
              hoopTimes[hoopInverse[A], AB] + R2]};];

Figure 12. The C3C2 algebra is used. A is chosen to have the first size zero, B to have the second and fourth zero, so AB has only the third size non-zero. Rl & Rr have the shapes that are missing from AB:-

muldiv[{2, 1, -2, -2, 3, -2}, {3, 
          1, 1, 3, 1, 1}, "C3C2"];

Figure 13. Non-abelian D3 is used; two of the sizes of B are zero and Rl contains the sizes ejected from A:-

muldiv[{2, 1, -2, 2, 3, -2}, {3, 1, 
          1, 3, 1, 1}, "D3"];

Figure 14. P4 is a non-abelian group with one size; the Pauli-σ matrices provide a representation. It gives a complex product from real vectors. As the size is (a^2)-(b^2)-(c^2)-(d^2), real vectors can have a size of zero (the "light-cone" of relativity.) A is chosen to have size 0, so the remainder for B is all of B:-

muldiv[{5, 4, 0, 3}, {2, 1, 4, 
          3}, "P4"];

3.6. Polar and Hyperbolic Duals.

Summary. Some hoops have quadratic sizes that can be written as sums or differences of squared terms. In many cases, their cartesian forms can be converted to polar-duals or hyperbolic-duals in which each quadratic size is a squared radius (or "ulna" in the hyperbolic case) associated with an angle that can be calculated by an ArcTan (or ArcTanh) function. These duals generalize the complex plane; angles add on abelian multiplication.

Many hoops have Polar, as well as Cartesian, forms because they have quadratic sizes that "fragment" into sums of squares. The polar form (here called pA) is given by toPolar[A], and the cartesian form is recovered by toVector[pA]. The polar forms have squared radii and angles, and often involve linear "offsets" such as a+b+c, a+b+c+d, a-b+c-d that are conserved on addition as well as on multiplication The forms for the radii and angles are exemplified by those of C3 & C4:-

Figure 15. Typical Polar-duals.

toPolar[{a, b, c}, "C3"]

toPolar[{a, b, c, d}, "C4"]

In Figure 16, the C4 algebra is used to multiply arbitrary real A & B; the polar form pA=toPolar[A] is found; toVector[pA] is shown to recover A. The two linear offsets, the squared radii and the angles are tabulated for A, B, and AB. The zeroes in the last two lines show that sizes multiply and angles add:-

Figure 16. Interconversion between C4 vector & polar

hoopShape["C4"]; {Flatten[
            Append[{"A ="}, A = {2. , 0.5, 1. , 0.1}]], 
          Flatten[Append[{"B ="}, B = {3. ,  .7, 2.5,  .5}]], 
          Flatten[Append[{"AB="}, AB = hoopTimes[A, B]]]} // tf

Chop[{{"" ,"a+b+c+d","a-b+c-d","(a-c)^2+(b-d)^2","angle or d"},Flatten[Append[{"pA="}, pA = toPolar[A]]],Flatten[Append[{"toVector[pA]"}, toVector[pA]]],Flatten[Append[{"pB="}, pB = toPolar[B]]],Flatten[Append[{"pAB="}, pAB = toPolar[AB]]],Flatten[Append[{"pA pB-pAB"}, pA pB - pAB]],Flatten[Append[{"pA+pB-pAB"}, pA + pB - pAB]]}]//tf

Some algebras, with sizes expressible as differences of squares, have hyperbolic dual formulations with hyperbolic angles and squared "ulnae" (named by analogy with radii). Thus the (a+b+c+d) and (a-b+c-d) sizes of the C4 algebra lead to conservation of (((a + c))^2)-(((b + d))^2) and a hyperbolic angle ArcTanh[a+c, b+d]. This has to be written in the logarithmic form for full generality, and the infinite case has to be allowed for in the calculation.
	Example 17 shows the hyperbolic form of Example 16. (It has to use "local variables" b$, x$ and y$ to handle special cases.) The polar components are unchanged, and the two linear sizes are replaced by an {ulna, hyperbolic angle} pair. The hyperbolic angles add on multiplication.

Figure 17. Interconversion between vector & "hypolar". Hyperbolic angle addition.

{"tohyPolar[{a,b,c,d},C4] =", 
        tohyPolar[{a, b, c, d}, "C4"]}

Chop[{{"" ,"(a+c)^2-(b+d)^2","hyperangle","(a-c)^2+(b-d)^2","angle"},Flatten[Append[{"hA="}, hA = tohyPolar[A]]],Flatten[Append[{"Reverts to A"}, tohyVector[hA]]],Flatten[Append[{"hB="}, hB = tohyPolar[B]]],Flatten[Append[{"hAB="}, hAB = tohyPolar[AB]]],Flatten[Append[{"hA hB-hAB"}, hA hB - hAB]],Flatten[Append[{"hA+hB-hAB"}, hA + hB - hAB]]}]//tf

Many non-abelian hoops have "pseudo-polar-duals" that revert correctly to the cartesian form, but do not have additive angles. This is because they have repeated determinant factors, and the pseudo-polar-duals are created by splitting a repeated quadratic into two sections that are not conserved independently. Figure 18 shows this for D3, the smallest non-abelian loop. The third size contains six squared fragments, 3 positive and 3 negative. These are used to create two {squared radius, angle} pairs that allow reversion but give neither conserved radii nor additive angles. (Nor do the angles subtract, as I once thought.) Further analysis might resolve this problem - I have failed to do so. Non-abelian multiplication appears to introduce uncertainty into the polar angles.

Figure 18. D3 has polar reversion without {radius,angle} conservation.

\(\(\( (*Figure\ 17. *) \)\({"Shape of D3 is", 
      hoopShape["D3"], "

toPolar D3 is", 
toPolar[{a, b, c, d, e, f}]}

    \(({Flatten[Append[{"A="}, A = {3. , 2. ,  .1, 2. , 1. ,  .4}]], 
          Flatten[Append[{"B="}, 
              B = {1. , 3. , \(- .1\),  .2, \(-1. \), 1. }]], 
          Flatten[Append[{"AB="}, 
              AB = hoopTimes[A, B]]], 
Flatten[
            Append[{"pA"}, pA = toPolar[A]]], 
Flatten[
            Append[{"Revert pA"}, toVector[pA]]], 
          Flatten[Append[{"pB"}, 
              pB = toPolar[B]]], 
Flatten[
            Append[{"Revert pB"}, toVector[pB]]], 
          Flatten[Append[{"pAB"}, pAB = toPolar[AB]]], 
          Flatten[Append[{"pA+pB-pAB"}, pA + pB - pAB]], 
          Flatten[Append[{"pA*pB-pAB"}, Chop[pA*pB - pAB]]], 
          Flatten[Append[{"pA+pB+pAB"}, Chop[pA + pB + pAB]]]} // 
        tf)\)\)\)\)

Figure 19. Clifford(2) Algebra has pseudo-polar reversion but angles do not add.

{"Clifford(2) Shape is ", 
      hoopShape["CL2"]}

    {Flatten[Append[{"A="}, A = {3. , 2. ,  .1,  .3}]], 
        Flatten[Append[{"B="}, B = {1. , 3. , - .1,  .7}]], 
        Flatten[Append[{"AB="}, 
            AB = hoopTimes[A, B]]], 
Flatten[
          Append[{"pA"}, pA = toPolar[A]]], 
Flatten[
          Append[{"Revert pA"}, Chop[toVector[pA]]]], 
        Flatten[Append[{"pB"}, 
            pB = toPolar[B]]], 
Flatten[
          Append[{"Revert pB"}, Chop[toVector[pB]]]], 
        Flatten[Append[{"pAB"}, pAB = toPolar[AB]]], 
        Flatten[Append[{"pA+pB-pAB"}, pA + pB - pAB]], 
        Flatten[Append[{"pA+pB+pAB"}, Chop[pA + pB + pAB]]], 
        Flatten[Append[{"BA="}, BA = hoopTimes[B, A]]], 
        Flatten[Append[{"pBA"}, pBA = toPolar[BA]]], 
        Flatten[Append[{"pA+pB-pBA"}, pA + pB - pBA]], 
        Flatten[Append[{"pA+pB+pBA"}, Chop[pA + pB + pBA]]]} // 
      tf

Section 12 returns to duals, showing them to create "orbits".

3.7. Powers, Roots, Quadratic equations.

Summary. Multiplication of polar-duals multiplies sizes and (if abelian) adds angles. This generalizes to powers and roots, in which the sizes are raised to the exponent and angles are multiplied by the exponent. Quadratic equations can be solved via square roots. If angles exceed 2π the results are rotated vectors.

Abelian Hoops with polar-duals or hyperbolic-duals have powers and roots. (The others may have higher analogues using pseudo-polar-duals, but I have not found them.)  hoopPower[A,p] raises A to the power p, giving the square root if no power is specified. A dual of the algebra must be known, because powers involve the multiplication of the angle(s) by the exponent. If the result exceeds 2π, recovering a root will give a rotated value with the same sizes.
Figure 20. Squaring the C3 square root recovers the original. :-

\(hoopShape["C3"]; {"A=", 
      A = {3, 1, 4. }, \*""\nIn C3, \!\(\@A\) ="", 
      RootA = hoopPower[A], \*""\n(\!\(\@A\)\!\(\()\^2\)\) ="", 
      hoopPower[RootA, 2]}\)

Figure 21.  High angle Powers give rotated roots,

hoopShape["C4"];{"A=",A = {1.6, 1, 1.4, 0},"\nIn C4, shape[A] is ",shape[A],"\n     and polar A is", toPolar[A],"\n     with angle             1.3734 ↑\n(A^2) is ",SquareA = hoopPower[A, 2],"\nwith Polar-dual",toPolar[SquareA],"\nwith squared sizes & doubled angle ↑.\nThe square root recovers A",hoopPower[SquareA],"\n(A^3) takes the angle beyond 2π \n(A^3) is ",CubeA = hoopPower[A, 3],"\nWith Polar-dual",toPolar[CubeA],"\nNow the cube root of (A^3)",A3 = hoopPower[CubeA, 1/3],"has the same sizes but a rotated angle",Chop[toPolar[A3]]}

QPower calculates powers in the quaternion algebra (28/1/05), and handles complex (biquaternion) elements. It multiplies the angle t by the power n. It may lead to better topolar & toVector formulations for quaternion & other non-abelian hoops; the current formulations do not have angle addition.

Figure 22. QPower works with biquaternions (quaternions over the complex field).

\({"A =       ", 
      A = {\  .6 -  .1  I,  .5,  .3, \(- .2\) I}, "
rootA = ", 
      rA = QPower[A, 
          1/2], 
\*""\n(rootA)^2 = "", 
      Chop[QPower[rA, 
          2]], 
\*""\n(A^2)     =   "", 
      AA = Chop[QPower[A, 2]], \*""\n(Sqrt[(A^2)])   =   "", 
      Chop[QPower[AA, 1/2]]}\)

Quadratics can be solved in algebras that have powers. Figure 23 sets up three random C4 vectors A, B, C. Then genQuadratic is used to solve A (X^2)+B X == C. Substituting the two results into A (X^2)+B X shows that they are both solutions. Calculating C from  A (Xo^2)+B Xo = C and solving for X often does not give Xo as one of the solutions because negative sizes and large angles can result in many different solutions. The next example uses random coefficients in the range [0.,1.] for A, B, & C. Two solutions are found, and validated by recovering C.

Figure 23. Solving a random quadratic equation in C4.

\(genShape["C4"]; 
    Print[{"A=", 
        A = Table[Random[Real, {\(- .1\), 1. }], {i, mm}], "
 B=", 
        B = Table[
            Random[Real, {\(- .1\), 1. }], {i, 
              mm}], 
"
 C=", 
        Cc = Table[
            Random[Real, {\(- .1\), 1. }], {i, 
              mm}], 
\*""\nSolutions for x if A (x^2)+B x=C:-\n"", 
        result = 
          genQuadratic[A, B, 
            Cc]}]; \*""A (x^2)+B x recovers C in both cases:-""\)
Chop[
      hoopTimes[result[[1]], 
        B + hoopTimes[A, result[[1]]]]]
Chop[
      hoopTimes[result[[2]], 
        B + hoopTimes[A, result[[2]]]]]

3.8. Many Moufang Loops "Collapse"  to Hoops.

Summary. Hoops are obtained from Moufang loops. These act on sets of unsigned coefficients; Algebras operate on signed vectors, usually sets of real or complex numbers. Real numbers are obtained from pairs of unsigned numbers via a "folding" equivalence relationship, (Subscript[r, i])~ {(Subscript[a, i]),(Subscript[a, i + m/2])}; this halves the number of coefficients, and halves the number of elements in Moufang loops with 2-fold symmetry - it "folds" the loop to an algebra. The result may be either smaller Moufang loops, or signed multiplication tables such as those for Complex & Quaternion algebras. Most of the non-associative Moufang loops [4] are not conservative; only a few describe hoops (all associative loops describe hoops).

Hoops are obtained from Moufang Loops, inheriting vector multiplication and division from them. The definition of Moufang loops does not involve signs, so they multiply (and divide) sets of unsigned coefficients, but they are not algebras because they lack a second operation. They only become algebras (hoops) if generalized signs are introduced, as equivalence relations on sets of coefficients,  (Subscript[h, i])~ {(Subscript[a, i]), (Subscript[a, i + m/r]), (Subscript[a, i + 2  m/r]), ..} ~ {(Subscript[b, i]), (Subscript[b, i + m/r]), (Subscript[b, i + 2  m/r]), ...}. The general case is discussed in Section 9, but the commonest cases, r=2 or 4, are developed here. r=2 gives half the number of real coefficients, (Subscript[r, i]) ~ {(Subscript[a, i]),(Subscript[a, i + m/2])} ~ {(Subscript[b, i]),(Subscript[b, i + m/2])} iff  (Subscript[a, i])+(Subscript[b, i + m/2])=(Subscript[b, i])+(Subscript[a, i + m/2]). The m×m table "folds" to an (m/r×m/r) table, which may a smaller (unsigned) Moufang loop or a signed table in which some products are signed. Thus C4 is a hoop over the real field because it can be regarded as the folded version of C4C2; the highlights in the illustration of C4C2 (figure 24) give the C4 table. C4 is also a loop that fold to complex algebra, ℂ. This is shown by mapping {1,i,-1,-i} onto the C4 table, and highlighting the complex multiplication table that results from folding. This matches the signed C4c table, which is complex multiplication.

Figure 24. Collapsing Moufang Tables to Hoops.

({{C4C2, 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}})

({{C4, 1, I, -1, -I,  , C4c, 1, 2,  , ℂ, 1, I}, { , I, -1, -I, 1,  ,  , 2, -1,  ,  , I, -1}, { , -1, -I, 1, I,  ,  ,  ,   ,  ,  ,  ,  }, { , -I, 1, I, -1,  ,  ,  ,  ,  ,  ,  ,  }})

The C4 multiplication of {2., 0.5, 1., 0.1} and {3., .7, 2.5, .5} (in Fig. 16) gave {8.82, 3.65, 8.4, 3.25}. These vectors fold to complex numbers (2-1)+(.5-.1)i and (3-2.5)+(.7-.5)i, with product (8.82-8.4)+(3.65-3.25)i; i.e. (1+.4i)(.5+.2i) = .42+.4i. 
This is confirmed in Figure 25:-

(*Figure 25. C4 example folds to C4c*){" {(2-1),(.5-.1)}*{(3-2.5),(.7-.5)}\n\
~.42+.4\[ImaginaryI]", "\nC4c product folds to (1+.4\[ImaginaryI])(.5+.2\
\[ImaginaryI]) & gives", 
        hoopTimes[{(2 - 1), ( .5 -  .1)}, {(3 - 
                2.5), ( .7 -  .5)}, "C4c"]}

The properties of complex algebra result from the Moufang & Frobenius properties of C4. The rules for quaternion and octonion algebras arise similarly, from the Q8 and Oct tables. As their determinant factors are (repeated) sums of the squares of their elements, they are "real algebras without divisors of zero".
	After finding several conservative tables related to Octonions, I conjectured that all non-associative Moufang Loops might be conservative. This was comprehensively disproved by the examples in [4]; on testing the 88 with up to 36 elements, only 7 were hoops. Figure 24a shows these, together with the non-hoop 12 & 20 element cases, and 4 larger nonassociative hoops composed from Oct.

Figure 24a. A few nonassociative Moufang Tables are conservative.

{conservativeQ[mp["D3Mn"]], conservativeQ[mp["Q8M2"]], 
      conservativeQ[Oct], conservativeQ[mp["D5Mn"]], 
      conservativeQ[mp["P16M"]], conservativeQ[mp["OctC2"]], 
      conservativeQ[mp["C8pC2M"]], conservativeQ[mp["KiC4M"]], 
      conservativeQ[C4iC4M], conservativeQ[co[Oct, C3]], 
      conservativeQ[co[Oct, C4]], conservativeQ[co[Oct, K]], 
      conservativeQ[co[Oct, C5]]}

Oct and Q8M2 are the two 16 element non-associative Moufang loops. They each have 8 linear and one quadratic size, and fold to the signed tables for the "Octr" Octonion and the "Octi" ("Split-Octonion" in Wikipedia) algebras. Q8M2 operations were demonstrated in Figure 10a. Figure 24b gives the shapes of these algebras
	Octi is an algebra with divisors of zero and signature {4,4} as it conserves {(a^2)-(b^2)+(c^2)-(d^2)+(e^2)-(f^2)+(g^2)-(h^2)}. Oct and Q8M2 have one quadratic and 8 linear sizes and the linear sizes produce divisors of zero.

Figure 24b. Conservative nonassociative Octonion and "Split-octonion" Algebras.

{"Q8M2 shape", sh["Q8M2"], "\nOct shape", 
      sh["Oct"], "\nOcti shape", sh["Octi"], "\nOctr shape", 
      sh["Octr"]}

Of the 88 non-associative Moufang loops with up to 36 elements in [4], only M1602 (Q8M2), M1603 (Oct), M3202 (Q8C2M), M3216 (OctC2), M3221 (P16M), M3235 (C8pC2M), M3254 (KiC4M), and M3265 (C4iC4M) are conservative. They are all hoops, becase their compositions with C2, C3, C4, K, etc. are conservative. Figure 24c gives some examples. I have not found (folded) signed tables for P16M, C8pC2M, KiC4M or C4iC4M; these are related to the corresponding groups (without M) via a "Moufang modification". Unlike D3Mn, their diagonals do not preclude a folded version.

Figure 24c. Composition of conservative nonassociative Moufang Loops with C2, C3, C4 & K gives conservative hoops.

{conservativeQ[co[mp["P16M"], C2]], 
      conservativeQ[co[mp["C8pC2M"], C3]], 
      conservativeQ[co[mp["KiC4M"], K]], 
      conservativeQ[co[mp["C4iC4M"], C4]]}

3.9. "Generalized signs" fold unsigned numbers and Moufang loops.

Summary. The signs {1,i,-1,-i} or {(i^( 0)),(i^( 1)),(i^( 2)),(i^( 3))} are the r=4 (complex) case of generalised signs (𝕤^j)where (𝕤^r)=+1 & 0≤j<r. Other algebras use other r values, e.g. r=3 gives Terplex algebra. m× m Moufang loops operate on sets of m unsigned coefficients, but if they have r-fold symmetry they can be folded to (m/r)×(m/r) hoop tables that  multiply vectors with m/r "r-signed" coefficients. These tables inherit the Moufang and Frobenius properties, but some sizes are destroyed by folding.

Section 8 demonstrated the folding of Moufang loops with 2-fold symmetry to introduce the traditional "-" sign. Many hoops in the database involve "generalized signs", ("-", 𝕁, i), that are primitive roots of unity - e.g. ("-"^2)=+1, (𝕁^3)=+1, (i^4)=+1. (They should not be confused with the cyclotomic numbers, which are points in the complex plane). The package defines 𝕁 and some other signs (e.g. (𝕜^2)=1, 𝕜 distinct from "-") as extensions to Mathematica. (Only i & 𝕁 are used in this notebook, but the generalization is explained for completeness). These numbers resemble i but are not real, not (apart from i) complex, not positive, not negative; they are their own sign and have an absolute value (magnitude) of 1. The general case 𝕤 is set up as the r'th root of unity, with (𝕤^r)=1. A minSign function is provided to reduce the powers of any signs in an expression to the minimum range 0≤j<r. 
	Each signed table has been obtained from a larger Moufang table that (because negation is not required in the definition of Moufang loops) consists of unsigned indices. Moufang coefficients may be operators, but if they are numbers they are therefore fundamentally "unsigned", i.e. natural numbers, unsigned rational numbers, or unsigned (non-negative) continuous numbers (for which I propose the name Primal numbers.) Signed numbers are equivalence relations on sets of unsigned numbers. Thus, Real coefficients are equivalence relations on pairs of primal numbers, (Subscript[r, i])~{(Subscript[a, i]),(Subscript[a, i + m/2])}~ {(Subscript[b, i]),(Subscript[b, i + m/2])}, -(Subscript[r, i])~{(Subscript[a, i + m/2]),(Subscript[a, i])}, 0~{(Subscript[a, i]),(Subscript[a, i])} with the relationship (Subscript[a, i])+(Subscript[b, i + m/2])=(Subscript[a, i + m/2])+(Subscript[b, i]). Complex coefficients are equivalence relations on quads of real numbers, (Subscript[c, i])~{(Subscript[a, i]), (Subscript[a, i + m/4]), (Subscript[a, i + m/2]), (Subscript[a, i + 3  m/4])}, -(Subscript[c, i])~{(Subscript[a, i + m/2]), (Subscript[a, i + 3  m/4]), (Subscript[a, i]), (Subscript[a, i + m/4])} with the relationships (Subscript[a, i])+(Subscript[b, i + m/2])= (Subscript[a, i + m/2])+ (Subscript[b, i]), (Subscript[a, i + m/4])+ (Subscript[b, i + 3  m/4])= (Subscript[a, i + 3  m/4])+(Subscript[b, i + m/4]). Terplex coefficients, with signs {1,𝕁, (𝕁^2)} are equivalence relations on triples of primal numbers, (Subscript[t, i]) ~ {(Subscript[a, i]), (Subscript[a, i + m/3]), (Subscript[a, i + 2  m/3])}, 𝕁(Subscript[t, i]) ~ {(Subscript[a, i + m/3]), (Subscript[a, i + 2  m/3]), (Subscript[a, i])}, (𝕁^2)(Subscript[t, i]) ~ {(Subscript[a, i + m/3]), (Subscript[a, i]), (Subscript[a, i + 2  m/3])} with the relationship (Subscript[a, i])+(Subscript[b, i + m/3])= (Subscript[a, i + m/3])+(Subscript[b, i])=(Subscript[a, i + 2  m/2])+(Subscript[b, i]). I call the equivalencing of 2, 3, 4 or more primal coefficients to a single real, terplex, complex or hypercomplex (etc) number "folding". 
	Signed tables are produced by folding Moufang Loops that have suitable symmetry. A Moufang table of size m×m can be folded to a signed table of size (m/r)×(m/r) if it has r-fold symmetry. The unsigned tables consist of (r^2) blocks (of size (m/r)×(m/r)) that are repeated with offsets. Twofold folding gives the top left (m/2)×(m/2) table; indices greater than r become negated smaller indices. Threefold folding gives a (m/3)×(m/3) table that may involve 𝕁.

Figure 26. Collapsing the unsigned C6, Quaternion (Q8) and C9 tables to signed Hoop tables.

({{C6=(({{1, 2, 3,  , 4, 5, 6}, {2, 3, 4,  , 5, 6, 1}, {3, 4, 5,  , 6, 1, 2}, { ,  ,  ,  ,  ,  ,  }, {4, 5, 6,  , 1, 2, 3}, {5, 6, 1,  , 2, 3, 4}, {6, 1, 2,  , 3, 4, 5}}))
, C3C2c=(({{1, 2, 3}, {2, 3, -1}, {3, -1, -2}}))
4  & 5  become  - 1  &  - 
              2}, {Q8=(({{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}})), Qr=(({{1, 2, 3, 4}, {2, -1, -4, 3}, {3, 4, -1, -2}, {4, -3, 2, -1}}))
{5, 6, 7, 8}  becomes
{-1, -2, -3, -4}
}, {C9=(({{1, 2, 3,   , 4, 5, 6,  , 7, 8, 9}, {2, 3, 4,  , 5, 6, 7,  , 8, 9, 1}, {3, 4, 5,  , 6, 7, 8,  , 9, 1, 2}, { ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  }, {4, 5, 6,  , 7, 8, 9,  , 1, 2, 3}, {5, 6, 7,  , 8, 9, 1,  , 2, 3, 4}, {6, 7, 8,  , 9, 1, 2,  , 3, 4, 5}, { ,  ,  ,  ,  ,  ,  ,   ,  ,  ,  }, {7, 8, 9,  , 1, 2, 3,  , 4, 5, 6}, {8, 9, 1,  , 2, 3, 4,  , 5, 6, 7}, {9, 1, 2,  , 3, 4, 5,  , 6, 7, 8}})), C9J=(({{1, 2, 3}, {2, 3, 𝕁}, {3, 𝕁, 2 𝕁}}))
4  & 5 become 𝕁 &2𝕁}})

Figure 27. shows that C9J is a division algebra. It was developed to show that hoops were not restricted to "traditional" signs. After finding and applying the appropriate minsign, the shapes are seen to be conserved. Then multiplying Ainverse by AB recovers B, showing that C9J is a division algebra over the real field, though its products involve 𝕁. The C9 conjugate determinant has one linear, one quadratic, and one 6th order factor; folding reduces these to a single cubic factor, the size for C9J.

Figure 27. Using the C9J multiplication table to multiply and divide:-

{"hoopShape[C9J] ={(a^3)+(b^3) 𝕁-3 a b c 𝕁+(c^3) (𝕁^2)}\nA=",A = {2, 1, 3},"\nB=",B = {4, -2, 5},"\nAB=",AB = hoopTimes[A, B, "C9J"],"\nshape[A] ",sa = shape[A] ,"\nshape[B]",sb = shape[B],"\nshape[AB]",sab = Expand[shape[AB]],"\nApplying minsign= {(𝕁^3)→1,(𝕁^any)_"→(𝕁^mod)[any, 3],mins = minsign[AB]; "\nshAB gives",Expand[shape[AB]] /. mins,"\nshA shB = ",Expand[sa sb] /. mins,"\nAi=",Ai = hoopInverse[A],
"\nAiAB=",AiAB = hoopTimes[Ai, AB],"\nminsign simplifies this to B ",Simplify[Expand[AiAB] /. mins]}

3.10. Exponent of Generalized Signs, Multiphase Sinusoids.

Summary. An important property of generalized signs is that their exponents give generalized symmetric multiphase sinusoids, \(TraditionalForm\`e\_r\%\(\(\(\(𝕤\)\(\ \)\)\^j\) x\)\)=(∑_j) 2(𝕤^j)/r Sin[p(x+2π j/r)], with period p= Sin[2π /r]. (Subscript[e, r]) is a normalizing extrapolation of e  to r phases, (Subscript[e, r])=Re[(e^( Im[(((-1))^(2/r))]))]. The traditional (e^(-ix))=Cos[x]+i Sin[x] is an asymmetric folded case with r=4.

Extended signs 𝕤 are a significant generalization of complex mathematics. One important property is that their exponents generalize the sinusoids Cos[x] = Re[(e^(i x))], Sin[x] = Im[(e^(i x))] to multi-phase sinusoids genCos[x,r,j] = Re[j \(TraditionalForm\`e\_r\%\(\(\(\(𝕤\)\(\ \)\)\^j\) x\)\)] where j is a phase & (Subscript[e, r]) is a (normalizing) modification of e that depends on r, the number of phases, with (Subscript[e, 3])=(Subscript[e, 6])=(e^((Sqrt[3])/2))≃2.3774, (Subscript[e, 12])=(Sqrt[e])≃1.6487. The Mathematica implementation of the normalising element is (Subscript[e, r])= Re[(e^Im)[(((-1))^(2/r))]]. The generation of these sinusoids as solutions to banded differential equations is described in GroupLoopDemo Appendix E [1].

Figure 28 & Figure 29 demonstrate genCos & genSin by plotting three phases of the 3-phase & 12-phase sinusoids over their periods 2π/Sin[2π/3]=7.2552 & 4π=12.5664.

\(\(\( (*\(Figs\ 28\  &\)\ 9. \ Multiphase\ sinusoids*) \)\(Unprotect[m]; 
    m = 3; Plot[{2/3 
           Re[\((\(-1\))\)\^\(\(2\ \((m - 0)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)], 
        2/3  Re[\((\(-1\))\)\^\(\(2\ \((m - 1)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)], 
        2/3  Re[\((\(-1\))\)\^\(\(2\ \((m - 2)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]}, {x, 0, 
        2  π/Sin[2  π/3]}, 
      PlotLabel -> "Fig.28. genCos. Phases 0,1,2 for m = 3"]; m = 12; 
    Plot[{Im[\((\(-1\))\)\^\(\(2\ \((m - 0)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]/6, 
        Im[\((\(-1\))\)\^\(\(2\ \((m - 1)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]/6, 
        Im[\((\(-1\))\)\^\(\(2\ \((m - 2)\)\)\/m\)\ E\^\(I\ Im[\((\(-1\))\)\^\(2/m\)]\ x\)]/6}, {x, 0, 
        2  π/Sin[2  π/12]}, 
      PlotLabel -> "Fig.29. genSin. Phases 0,1,2 for m = 12"];\)\)\)

Multi-phase sinusoids have an exponential series formulation that simplifies, when m = 4, to half the standard Cos and Sin series, with alternate terms becoming zero. (e^(i x)) = {Cos[x]/2, Sin[x]/2, -Cos[x]/2, -Sin[x]/2}. This is traditionally folded and mis-interpreted as two asymmetrical phases, {Cos[x], Sin[x]} instead of four symmetrical phases.
	It is convenient to include a phase parameter "j" in the generalised sinusoid procedures genCos[x,m,j] & genSin[x,m,j], and to normalize the amplitude to 1. Cosh and Sinh functions are included via m=0 and m=2, which do not give sinusoids.

Figure 30.  genCos, genSin.

?genCos

genCos[x_, m_:  4, j_:  0] := 
      Which[m == 0, 
        j ((E^( x)) + (E^(-x)))/2, 
        m == 2, Re[(E^x) + (((-1))^j) (E^( -x))]/2, 
True, 
        Re[(((-1))^(2 (m - j)/
                    m)) (E^(I Im[(((-1))^(2/m))] x))]]

?genSin

genSin[x_, m_:  4, j_:  0] := 
      Which[m == 
          0, ((E^( x)) - (E^(-x)))/
          2, m == 2, 
        Re[(((-1))^j) (E^( x)) - (E^(-x))]/2, 
True, 
        Im[(((-1))^(2 (m - j)/
                    m)) (E^(I Im[(((-1))^(2/m))] x))]]

Note that neither the phasality m nor the phase j need be an integer. Also genCos[x,m,m/4] ≡ genSin[x,m], so one of the series is redundant. Both are included for compatibility with normal mathematics, with genSin as the antisymmetric sinusoid. genSin is preferable to genCos in some situations such as Direction Sines, as genSin[x,m,0]→ x for small x and the value tends to zero with x. When x = 0, terms involving genSin[x,m,0] drop out of expressions, making them compatible with renormalization.

3.11. Generalizing Complex maths.

Summary. Equations and functions containing i  are often special cases of  more generalized functions containing the generalized sign 𝕤.

Just as (e^(i x)) is a special case of \(TraditionalForm\`e\_r\%\(\(\(\(𝕤\)\(\ \)\)\^j\) x\)\)and Cos[x]+i Sin[x] generalizes to multi-phase sinusoids, many other complex functions appear to be special cases of more general functions involving 𝕤. Others may be unique to the r=4 case, giving complex maths a unique central position in mathematics. The author has neither the time nor the ability to explore this vast topic, but makes the following observations.
(1). Generalized signs act as relators for most groups. Only a subset have i as a relator, and many of this subset have additional relators that are different fourth roots of unity. E.g. C4C4, C4iC4, Pauli-σ algebra.
(2). A lot of group theory is concerned with unitary complex matrices that dot-multiply to provide the elements of particular classes of groups. The Pauli-σ algebra is archetypical, as it has a representation as σ-matrices, which involve i. Many other groups have representations via small matrices involving other signs. These are widely used in GroupLoopHoop.m
(3). i is an intrinsic feature of both Schroedinger's equation and Fourier analysis. Together [3] they lead to dispersive Gaussian wave packets that describe "information about particles". But particles do not disperse, and it is an article of the physicist's creed that mathematics is the language in which existence is written. Therefore there should be a mathematical formulation for particles, possibly as a version of Schroedinger's equation involving 𝕤. Hoop algebras provide tantalizing glimpses of possibilities for physical mathematics using generalized signs.

3.12. Unital "Orbits" & sub-algebras. Orbits with "Offsets".

Summary. Polar-duals become continuous "Orbits" when the non-zero offsets and squared radii (or their product) are constrained to be one, defining "unital" sub-algebras. Many orbits appear to be unstable and have zero size. The implications of the few cases with finite sizes are discussed later.

C3, K & C4 Orbits have fixed amplitudes.

Finite groups have finite "orbits", sets of points. These become continuous orbits (continuous multi-phase sinusoids) in algebras with polar-duals - the duals form an equivalence class if offsets and squared radii (or ulnae) are constrained to values of 0 or 1 whilst angles are allowed to vary. They constitute continuous closed unitary sub-algebras; group multiplication of two orbits gives another orbit.
	Once a hoop is specified, Orbit vector values are given by toVector[P] using arbitrary or symbolic angles, all other parameters in P being 0 or 1. The three C3 and four C4 orbits are set up in Figure 31; the names correspond to the sizes. Note that shape["C3"] has one L1 (linear) and one L2 (quadratic) size whilst a C3 vector has 3 degrees of freedom. Similarly, shape["C4"] has three size parameters to describe a 4-element vector. Each squared radius releases a degree of freedom. This is taken up by the angle parameter, which is a "hidden variable" for the cartesian form. Each polar-dual includes an angle for each quadratic size in the shape. The linear sizes become offsets from zero for the centre of rotation, providing conserved properties that are not present in the "standard model" of particles.
	Figure 31 also sets up and multiplies the C3 orbits, showing the product to be an orbit with the angles added. Then four C4 orbit pairs are set up and multiplied; the products have the correct sizes for orbits:-

Figure 31. C3 & C4 Orbit Multiplication.

\({"C3 Orbits", 
      hoopShape["C3"]; "
OC301 is {0,1,a} with vector form", 
      OC301 = toVector[{0, 1, a}], "
OC311 is {1,1,b} with vector form", 
      OC311 = toVector[{1, 1, 
            b}], "
Products are orbits with added angles; the polar form of the product is", 
      Simplify[toPolar[
          hoopTimes[OC301, OC311, 
            C3]]], "

C4 Orbits", 
\(hoopShape["C4"];\); "
OC4001 is {0,0,1,a} with vector form", 
      OC4001 = toVector[{0, 0, 1, 
            a}], "
OC4011 is {0,1,1,b} with vector form", 
      OC4011 = toVector[{0, 1, 1, 
            b}], "
OC4101 = {1,0,1,c} with vector form", 
      OC4101 = toVector[{1, 0, 1, 
            c}], "
OC4111 is {1,1,1,d} with vector form", 
      OC4111 = toVector[{1, 1, 1, 
            d}], "

", 
Simplify[{"OC4001×OC4001 has the 001 shape", 
          shape[hoopTimes[OC4001, OC4001, 
              C4]], "
OC4101×OC4011 also has the 001 shape
as the two offsets disappear   ", 
          shape[hoopTimes[OC4101, 
              OC4011]], "
OC4101×OC4111 has the 101 shape", 
          shape[hoopTimes[OC4101, 
              OC4111]], "
OC4111×OC4111 has the 111 shape", 
          shape[hoopTimes[OC4111, OC4111]]}]}\)

The group K (C2C2) cannot have a polar orbit as it has no L2 size, but the four L1 sizes can be combined in six ways to give paired difference-of-squares formulations. Hyperbolic orbits can be defined using any of these as (adjacent\(TraditionalForm\`\^2\), opposite\(TraditionalForm\`\^2\)) pairs. Two orbits are set up in Figure 32 and multiplied. The product is shown to be an orbit with unit sizes; its "angles" appear in complex form but numeric substitutions show that they are the sums of those in the multiplicands.

Figure 32. Klein Group Hyperbolic Orbit Multiplication.

\({hoopShape["K"]; "OKh11a is {1,α,1,a} with vector form
", OKh11a = 
        tohyVector[{1, α, 1, 
            a}], "
OKh11b is {1,β,1,b} with vector form", 
      OKh11b = tohyVector[{1, β, 1, 
            b}], "
Their product Polar form is
", 
      Simplify[tohyPolar[
          hoopTimes[OKh11a, 
            OKh11b]]], "
Substituting α->.1, β->.3, a->.22, b->.28 shows that the angles have added; the product angles are {α+β, a+b}:-
{Log[(Cosh[α+β]+Sinh[α+β])],Log[(Cosh[a+b]+Sinh[a+b])]} = ", {Log[\((Cosh[α + β] + Sinh[α + β])\)], 
          Log[\((Cosh[a + b] + Sinh[a + b])\)]} /. {α ->  .1, 
          a ->  .22, β ->  .3, b ->  .28}}\)

C5, C7 & C8 Orbits have unstable amplitudes.

The orbits of C3 are of fixed squared radius 2/3 and offsets {0,0,0} or {1/3, 1/3, 1/3).  Those of C4 are of fixed squared radii or ulnae of magnitude 1/2, and offsets {0,..} or {±1/4,..}. For C2 there is only one, hyperbolic, orbit with a unit ulna. 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. Examples in [1] shows that OC51r, a five-phase orbit with 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 in C8 has a zero size. These orbits also appear to be unstable - numerical integration of the relevant differential-equation sets is unstable.

Figure 33.  Zero-sized C5 orbit.

{genShape["C5"];"OC51r is",OC51r = 
    toVector[{d, r, σ}], 
  "\nThe radius disappears from the shape", 
  Simplify[shape[OC51r]]}

Stable, finite amplitude, orbits with 3, 4, 6, 9, and 12 phases have been found, some of which have polarization and chirality (Section 13).

3.13. Dozal Orbits, Analogies with Particles & Forces.

Summary. Dozal (the algebra of vectors with 12 elements) has elements that act as half-spin and unit-spin quantum operators, and stable orbits (some with polarization and chirality) that match the families of fundamental particles. Dozal is in a real subsymmetric relationship with Hexal (6 elements), and a ternal subsymmetric relationship with Tetral (4 elements). Multiplication with remainders may correspond to particle interactions, with fission and fusion. Instead of being "point particles", quarks may be multi-phase wave packets with ternary symmetry, and leptons may be wave packets with 4-fold symmetry.

The Appendix shows that the C3K algebra can be interpreted as having elements that act as half-spin quantum operators. They cannot be expressed in real or complex terms; they combine C3 (terplex numbers) and K (Study numbers, or the hyperbolic plane). Section 10. showed that 12-phase systems have a natural period of 4π - i.e. they have "half-spin".

Stable, finite amplitude, orbits with 3, 4, 6, and 12 phases have been found for "Dozal", the unitary algebra of 12 elements with several multiplication rules. Some of these orbits have polarization and chirality. In the following examples, the vector form is expressed as {a,b,c,d,e,f,g,h,i,j,k,l} and its polar-dual as {α, β, γ, δ, ζζ, τ, λλ, ϕ, ηη, χ, ϰϰ, ψ} for C3K & D3C2, or {α, β, ϵϵ, σ, ζζ, τ, λλ, ϕ, ηη, χ, ϰϰ, ψ} for C3C4 or Q12. The latter have a 4-phase {ϵϵ, σ} pair in place of the two linear sizes {γ ,δ} of C3K and D3C2. Note that {α, β} and {γ, δ} can be replaced by hyperbolic orbits. The following expressions are edited to emphasise the inter-relationships between the vector and polar forms. In particular, square roots of quadratic polar elements such as ζζ become radii such as ζ, and 2π/3 is written as ω:-

Figure 34. C3K Cartesian & Polar-Dual interconversions.

toPolar[{a, b, c, d, e, f, g, h, i, j, k, 
          l}, "C3K"];

{α = a + b + c + d + e + f + g + h + i + j + k + l, β = 
        a + b + c - d - e - f + g + h + i - j - k - 
          l, 
γ = 
        a + b + c + d + e + f - g - h - i - j - k - l, δ = 
        a + b + c - d - e - f - g - h - i + j + k + 
          l, 
ζζ = ((((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, 
τ = 
        ArcTan[2 a - b - c + 2 d - e - f + 2 g - h - i + 2 j - k - 
            l, -(Sqrt[3]) (b - c + e - f + h - i + k - 
                l)], 
λλ = ((((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, 
ϕ = 
        ArcTan[2 a - b - c - 2 d + e + f + 2 g - h - i - 2 j + k + 
            l, -(Sqrt[3]) (b - c - e + f + h - i - k + 
                l)], 
ηη = ((((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, 
χ = 
        ArcTan[2 a - b - c + 2 d - e - f - 2 g + h + i - 2 j + k + 
            l, -(Sqrt[3]) (b - c + e - f - h + i - k + 
                l)], 
ϰϰ = ((((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, 
ψ = 
        ArcTan[2 a - b - c - 2 d + e + f - 2 g + h + i + 2 j - k - 
            l, -(Sqrt[3]) (b - c - e + f - h + i + k - l)]}

\(C3Kv = 
      toVector[{α, β, γ, δ, ζζ, τ, λλ, ϕ, ηη, χ, ϰϰ, ψ}, "C3K"] /. {2  π/3 -> ω, \ 
          4  π/3 -> 
            2  ω, \@ζζ -> ζ, \@λλ -> λ, \@ηη -> η, \@ϰϰ -> ϰ}\)

The above expression is not shown directly, but it tidies up to give
Figure 35:-

C3Kαβγδζτλϕηχκψ  . 
{α + β + γ + δ + 
            2 ζ Cos[τ]       + 
            2 λ Cos[ϕ]       + 
            2 η Cos[χ]      + 
            2 ϰ Cos[ψ],  α + β + γ + δ + 2 ζ Cos[τ + ω] + 
            2 λ Cos[ϕ + ω] + 
            2 η Cos[χ + ω] + 
            2 ϰ Cos[ψ + ω],  α + β + γ + δ + 2 ζ Cos[τ - ω] + 
            2 λ Cos[ϕ - ω] + 
            2 η Cos[χ - ω] + 
            2 ϰ Cos[ψ - ω],  α - β + γ - δ + 2 ζ Cos[τ]       - 
            2 λ Cos[ϕ]      + 
            2 η Cos[χ]       - 
            2 ϰ Cos[ψ],  α - β + γ - δ + 2 ζ Cos[τ + ω] - 
            2 λ Cos[ϕ + ω] + 
            2 η Cos[χ + ω] - 
            2 ϰ Cos[ψ + ω],  α - β + γ - δ + 2 ζ Cos[τ - ω] - 
            2 λ Cos[ϕ - ω] + 
            2 η Cos[χ - ω] - 
            2 ϰ Cos[ψ - ω],  α + β - γ - δ + 2 ζ Cos[τ]       + 
            2 λ Cos[ϕ]      - 
            2 η Cos[χ]       - 
            2 ϰ Cos[ψ],  α + β - γ - δ + 2 ζ Cos[τ + ω] + 
            2 λ Cos[ϕ + ω] - 
            2 η Cos[χ + ω] - 
            2 ϰ Cos[ψ + ω],  α + β - γ - δ + 2 ζ Cos[τ - ω] + 
            2 λ Cos[ϕ - ω] - 
            2 η Cos[χ - ω] - 
            
            2 ϰ Cos[ψ - ω],  α - β - γ + δ + 2 ζ Cos[τ]       - 
            2 λ Cos[ϕ]      - 
            2 η Cos[χ]       + 
            2 ϰ Cos[ψ],  α - β - γ + δ + 2 ζ Cos[τ + ω] - 
            2 λ Cos[ϕ + ω] - 
            2 η Cos[χ + ω] + 
            2 ϰ Cos[ψ + ω],  α - β - γ + δ + 2 ζ Cos[τ - ω] - 
            2 λ Cos[ϕ - ω] - 
            2 η Cos[χ - ω] + 
            2 ϰ Cos[ψ - ω]}/12

Here, τ occurs in four copies of a 3-phase system, λ, η, & ψ each occur in two copies of different 6-phase systems. 12-phase systems can be obtained by making two of these systems have the same radii and period, but with a phase shift of π/6. I conjecture that these are related to the three families of quarks.
	The C3C4 orbits duplicate the {τ, λ} orbits and replace the {γ, δ} linear sizes by a 4-phase {ϵ, σ} sinusoid. I conjecture that this is related to the leptons. The {η, κ} orbits are more complicated in C3C4 than in C3K:-

Figure 36.   General C3C4 orbit.

toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, 
            l_}, "C3C4"] := 
Module[{r = -Sqrt[3], 
            e1 = a + b + c - g - h - i, 
            e2 = d + e + f - j - k - l, 
a1 = 
              a - b + d - e + g - h + j - k, 
            a2 = a - b - d + e + g - h - j + k, 
b1 = 
              b - c + e - f + h - i + k - l, 
            b2 = b - c - e + f + h - i - k + l, 
c1 = 
              c - a + f - d + i - g + l - j, 
            c2 = c - a - f + d + i - g - l + j, 
            f1 = a - b - g + 
                h + (d + e - 2 f - j - k + 
                      2  l)/(Sqrt[3]), 
f2 = (a + b - 2  c - 
                      g - h + 2  i)/(Sqrt[3]) - d + e + j - 
                k, 
f3 = 
              a - b - g + 
                h - (d + e - 2 f - j - k + 
                      2  l)/(Sqrt[3]), 
f4 = (a + b - 2  c - 
                      g - h + 2  i)/(Sqrt[3]) + d - e - j + 
                k}, 
{  a + b + c + d + e + f + 
              g + h + i + j + k + l,   a + b + c - d - e - f + g + 
              h + i - j - k - l, 
  (e1^2) + 
              (e2^2),   ArcTan[e1, 
              e2], 
  ((a1^2) + (b1^2) + 
                  (c1^2))/2,   ArcTan[a1 - c1, 
              r b1], 
  ((a2^2) + (b2^2) + 
                  (c2^2))/2,   ArcTan[a2 - c2, 
              r b2], 
  ((f1^2) + (f2^2)) 
              3/4,   
              ArcTan[f1, f2] - π/
                6, 
  ((f3^2) + (f4^2)) 
              3/4,   ArcTan[f3, f4] - π/6}];

{  a = (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]), 
b = (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]),  
c = (α + β + ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω]                         - η r6 Sin[x]                   - κ r6 Sin[
                  y]), 
d = (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]), 
e = (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]), 
f = (α - β + ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] - η r6 Cos[
                  x]                        + κ r6 Cos[
                  y]                   ), 
g = (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]), 
h = (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]),  
i = (α + β - ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω]                         + η r6 Sin[x]                    + κ r6 Sin[
                  y]), 
j = (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]), 
k = (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]), 
l = (α - β - ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] + η r6 Cos[
                  x]                        + κ r6 Cos[y]           )}/12

The non-abelian groups D3C2 and Q12 are very closely related to C3K and C3C4 respectively - their shapes include some shared sizes.

Figure 37. General D3C2 orbit..

D3C2αβγδζτλϕηχϰψ

    {(α + β + γ + δ + 
          4 ζ Cos[τ]       + 
          4 η Cos[χ]), 
(α + β + γ + δ + 4 ζ Cos[τ + ω] + 
          4 η Cos[χ + ω]), 
(α + β + γ + δ + 
          4 ζ Cos[τ - ω] + 
          4 η Cos[χ - ω]), 
(α - β + γ - δ       - 
          4 λ Cos[ϕ]       - 
          4 ϰ Cos[ψ]), 
(α - β + γ - δ       - 
          4 λ Cos[ϕ + ω] - 
          4 ϰ Cos[ψ + ω]), 
(α - β + γ - δ       - 
          4 λ Cos[ϕ - ω] - 
          4 ϰ Cos[ψ - ω]), 
(α + β - γ - δ + 
          4 ζ Cos[τ]       - 
          4 η Cos[χ]), 
(α + β - γ - δ + 4 ζ Cos[τ + ω] - 
          4 η Cos[χ + ω]), 
(α + β - γ - δ + 
          4 ζ Cos[τ - ω] - 
          4 η Cos[χ + 
                2 ω]), 
(α - β - γ + δ       - 4 λ Cos[ϕ]       + 
          4 ϰ Cos[ψ]), 
(α - β - γ + δ       - 
          4 λ Cos[ϕ + ω] + 
          4 ϰ Cos[ψ + ω]), 
(α - β - γ + δ       - 
          4 λ Cos[ϕ - ω] + 
          4 ϰ Cos[ψ - ω])}

\(toVec[{α_, β_, γ_, δ_, ζζ_, τ_, λλ_, ϕ_, ηη_, χ_, κκ_, ψ_}, "D3C2"] := 
Module[{ζ = 
            4 \@ ζζ, λ = 
            4 \@ λλ, η = 4 \@ ηη, κ = 
            4 \@ κκ}, {\ \ \ \ \ \ 
α + β + γ + δ + ζ\ Cos[τ]\ \ \ \ \ \  + η\ Cos[χ], 
α + β + γ + δ + ζ\ Cos[τ + ω] + η\ Cos[χ + ω], \ 
α + β + γ + δ + ζ\ Cos[τ - ω] + η\ Cos[χ - ω], 
α - β + γ - δ - λ\ Cos[ϕ]\ \ \ \ \ \  - κ\ Cos[ψ]\ , 
α - β + γ - δ - λ\ Cos[ϕ + ω] - κ\ Cos[ψ + ω], 
α - β + γ - δ - λ\ Cos[ϕ - ω] - κ\ Cos[ψ - ω], 
α + β - γ - δ + ζ\ Cos[τ]\ \ \ \ \ \  - η\ Cos[χ]\ , 
α + β - γ - δ + ζ\ Cos[τ + ω] - η\ Cos[χ + ω], 
α + β - γ - δ + ζ\ Cos[τ - ω] - η\ Cos[χ - ω], 
α - β - γ + δ - λ\ Cos[ϕ]\ \ \ \ \ \  + κ\ Cos[ψ]\ , 
α - β - γ + δ - λ\ Cos[ϕ + ω] + κ\ Cos[ψ + ω], 
α - β - γ + δ - λ\ Cos[ϕ - ω] + κ\ Cos[ψ - ω]}/12]\)

Figure 38. Q12 Cartesian & Polar-Dual interconversions.

Q12p[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_}] := {α = 
            a + b + c + d + e + f + g + h + i + j + k + 
              l, 
β = 
            a + b + c - d - e - f + g + h + i - j - k - 
              l, 
ϵϵ = (((a + b + c - g - h - i))^2) + (((d + e + f - j - k - l))^2), 
σ = 
            ArcTan[a + b + c - g - h - i, 
              d + e + f - j - k - 
                l], 
ζζ = ((((a - b + g - h))^2) + (((b - c + h - i))^2) + (((-a + c - g + i))^2))/
              2, 
τ = 
            ArcTan[2 a - b - c + 2 g - h - 
                i, -(Sqrt[3]) (b - c + h - 
                    i)], 
λλ = ((((d - e + j - k))^2) + (((e - f + k - l))^2) + (((-d + f - j + l))^2))/
              2, 
ϕ = 
            ArcTan[2 d - e - f + 2 j - k - 
                l, -(Sqrt[3]) (e - f + k - 
                    l)], 
ηη = ((((a - b - g + h))^2) + (((-a + c + g - i))^2) + (((b - c - h + i))^2))/
              2, 
χ = 
            ArcTan[2 a - b - c - 2 g + h + 
                i, -(Sqrt[3]) (b - c - h + 
                    i)], 
ϰϰ = ((((d - e - j + k))^2) + (((-d + f + j - l))^2) + (((e - f - k + l))^2))/2, 
ψ = 
            ArcTan[2 d - e - f - 2 j + k + 
                l, -(Sqrt[3]) (e - f - k + l)]};

The Q12 polar reversion has been tidied up by the substitutions
ϵ -> (Sqrt[ϵ])ϵ, ζ -> (Sqrt[ζ])ζ,  λ -> (Sqrt[λ])λ, η  -> (Sqrt[ηη ]), ϰ   -> (Sqrt[ϰ])ϰ, ω -> 
      2  π/3 : -
{α + β + 
                2 ϵ Cos[σ] + 
                4 ζ Cos[τ]       + 
                4 η Cos[χ], 
α + β + 2 ϵ Cos[σ] + 4 ζ Cos[τ + ω] + 
                4 η Cos[χ + ω], 
α + β + 2 ϵ Cos[σ] + 
                4 ζ Cos[τ - ω] + 
                4 η Cos[χ - ω], 
α - β + 
                2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ]       + 
                4 ϰ Cos[ψ], 
α - β + 2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ + ω] + 
                4 ϰ Cos[ψ + ω], 
α - β + 2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ - ω] + 
                4 ϰ Cos[ψ - ω], 
α + β - 2 ϵ Cos[σ] + 
                4 ζ Cos[τ]       - 
                4 η Cos[χ], 
α + β - 2 ϵ Cos[σ] + 4 ζ Cos[τ + ω] - 
                4 η Cos[χ + ω], 
α + β - 2 ϵ Cos[σ] + 
                4 ζ Cos[τ - ω] - 
                4 η Cos[χ - ω], 
α - β - 
                2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ]       - 
                4 ϰ Cos[ψ], 
α - β - 2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ + ω] - 
                4 ϰ Cos[ψ + ω], 
α - β - 2 ϵ Sin[σ]      + 
                4 λ Cos[ϕ - ω] - 
                4 ϰ Cos[ψ - ω]}/12

In all cases, orbits are created by conversion to unital sub-algebras, with arbitrary angles and with the other components (α,β,γ,δ,ϵ,ζ,η,λ,ξ,ϰ) all being either 0 or 1. They can be identified by appending names of the non-zero parameteres to that of the algebra, as in Figure 39:-

Figure 39.  Some C3K orbits.

({{     C3Kτϕ

          {  (Cos[τ]          + 
                  Cos[ϕ]),        
(Cos[τ +    ω] + 
                  Cos[ϕ + ω]), 
(Cos[τ + 2 ω] + 
                  Cos[ϕ + 
                      2  ω]), 
(Cos[τ ]         - 
                  Cos[ϕ]), 
(Cos[τ +    ω] - 
                  Cos[ϕ +    ω]), 
(Cos[τ + 2 ω] - 
                  Cos[ϕ + 
                      2 ω]), 
(Cos[τ]          + 
                  Cos[ϕ]), 
(Cos[τ +    ω] + Cos[ϕ + ω]), 
(Cos[τ + 
                      2 ω] + 
                  Cos[ϕ + 
                      2 ω]), 
(Cos[τ]           - 
                  Cos[ϕ]), 
(Cos[τ +    ω]  - Cos[ϕ + ω]), 
(Cos[τ + 
                      2 ω] - Cos[ϕ + 2 ω])}/
            6,               C3Kατχ

          {  (1/2 + Cos[τ]          + 
                  Cos[χ]),        
(1/2 + 
                  Cos[τ +    ω] + 
                  Cos[χ +  ω]), 
(1/2 + 
                  Cos[τ + 2 ω] + 
                  Cos[χ + 2 ω]), 
(1/2 + 
                  Cos[τ ]         + 
                  Cos[χ]), 
(1/2 + 
                  Cos[τ +    ω] + 
                  Cos[χ +    ω]), 
(1/
                    2 + Cos[τ + 2 ω] + 
                  Cos[χ + 2  ω]), 
(1/2 + 
                  Cos[τ]          - 
                  Cos[χ]), 
(1/2 + 
                  Cos[τ +    ω] - 
                  Cos[χ +   ω]), 
(1/2 + 
                  Cos[τ + 2 ω] - 
                  Cos[χ + 2 ω]), 
(1/2 + 
                  Cos[τ]          - 
                  Cos[χ]), 
(1/2 + 
                  Cos[τ + ω]    - 
                  Cos[χ +    ω]), 
(1/
                    2 + Cos[τ + 2  ω] - 
                  Cos[χ + 2 ω])}/6}, {  
      C3Kαδχψ 

          {  (1 + Cos[χ]          + 
                  Cos[ψ]), 
(1 +  
                  Cos[χ +   ω] + 
                  Cos[ψ +    ω]), 
(1 +  Cos[χ + 2 ω] + 
                  Cos[ψ + 
                      2 ω]), 
(       Cos[χ]          - 
                  Cos[ψ]), 
(       Cos[χ +    ω] - 
                  Cos[ψ +    ω]), 
(       Cos[χ + 2 ω] - 
                  Cos[ψ + 
                      2 ω]), 
(     -Cos[χ]          - 
                  Cos[ψ]), 
(     -Cos[χ +    ω] - 
                  Cos[ψ +    ω]), 
(     -Cos[χ + 2 ω] - 
                  Cos[ψ + 
                      2 ω]), 
(-1 - 
                  Cos[χ]         + 
                  Cos[ψ]), 
(-1 - 
                  Cos[χ +    ω] + 
                  Cos[ψ +    ω]), 
(-1 - Cos[χ + 2  ω] + Cos[ψ + 2 ω])}/
            6, 
           C3Kβγδϕχψ

          {   (1.5 + Cos[ϕ]          + 
                  Cos[χ]           + 
                  Cos[ψ]), 
 (1.5 + 
                  Cos[ϕ +    ω] + 
                  Cos[χ +    ω] + 
                  Cos[ψ +   ω]), 
 (1.5 + Cos[ϕ + 2 ω] + Cos[χ + 2 ω] + 
                  Cos[ψ + 
                      2 ω]), 
 (- .5 - 
                  Cos[ϕ]          + 
                  Cos[χ]          - 
                  Cos[ψ]), 
 (- .5 - 
                  Cos[ϕ +    ω] + 
                  Cos[χ +    ω] - 
                  Cos[ψ +    ω]), 
 (- .5 - Cos[ϕ + 2 ω] + Cos[χ + 2 ω] - 
                  Cos[ψ + 
                      2 ω]), 
 (- .5 + 
                  Cos[ϕ]          - 
                  Cos[χ]           - 
                  Cos[ψ]), 
 (- .5 + 
                  Cos[ϕ +    ω] - 
                  Cos[χ +     ω] - 
                  Cos[ψ +    ω]), 
 (- .5 + Cos[ϕ + 2  ω] - Cos[χ + 2 ω] - 
                  Cos[ψ + 
                      2  ω]), 
 (- .5 - 
                  Cos[ϕ]          - 
                  Cos[χ]           + 
                  Cos[ψ]), 
 (- .5 - 
                  Cos[ϕ +    ω] - 
                  Cos[χ +     ω] + 
                  Cos[ψ +   ω]), 
 (- .5 - Cos[ϕ + 2  ω] - Cos[χ + 2  ω] + 
                  Cos[ψ + 2 ω])}/6}})

These orbits may be related to deBroglie waves. Elements of the generating algebras (e.g. C3K) act as unit-spin and half-spin quantum operators. This leads to the surmise that quarks are wave packets with ternary symmetry, and leptons are wave packets with 4-fold symmetry. The orbit magnitudes may be related to the Planck size of "extended" particles; the number of types of particles may be related to the number of stable orbits. The 12-element algebras are in a "subsymmetric" relation to the algebras with 6 elements, "Hexal". (This respects the convention that unit-spin is supersymmetric to half spin. It also suggest that supersymmetry involves fewer, rather than the same number of, partners.) The symmetry is shown in Figure 40, where the elements of C3K are folded by replacing a & g by a/2 & -a/2, etc. The polar form loses 6 elements whilst the other elements match those of C3C2:-

Figure 40.  C3K is Subsymmetric to  C3C2.

toPolar[{a, b, c, d, e, 
          f, -a, -b, -c, -d, -e, -f}/
        2, "C3K"]

{0, 0, a + b + c + d + e + f, a + b + c - d - e - f, 0, 0, 0, 
      0, 
((((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)]}

toPolar[{a, b, c, d, e, f}, "C3C2"]

{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)]}

3.14. Time as a parameter.

Inherent in my conjecture that orbits relate to particles is the concept that angles change with "time" in some way. C3C4 has the following angles:-
ArcTan[a+b+c-g-h-i,d+e+f-j-k-l]
ArcTan[2a-b-c+2d-e-f+ 2g-h-i+ 2j-k-l, -(Sqrt[3])(b-c+e-f+h-i+k-l)]
ArcTan[2a-b-c-2d+e+f+2g-h-i-2j+k+l, -(Sqrt[3])(b-c-e+f+h-i-k+l)]
ArcTan[2a-b-c+2d-e-f-2g+h+i-2j+k+l, -(Sqrt[3])(b-c+e-f-h+i-k+l)]
ArcTan[2a-b-c-2d+e+f-2g+h+i+2j-k-l, -(Sqrt[3])(b-c-e+f-h+i+k-l)].

C3K has the same last four angles It folds to C3C2 with
ArcTan[2a-b-c+2d-e-f, -(Sqrt[3])(b-c+e-f)]
ArcTan[2a-b-c-2d+e+f, -(Sqrt[3])(b-c-e+f)]

C3C4 3-folds to C4, with these and 
ArcTan[a-c,b-d]
All these angles increase with -a/b, so put t=-a/b, in the form b=-a t

{ArcTan[2  a - b - c + 2  d - e - f,(Sqrt[3])(c - e + f - b)],ArcTan[a - c, b - d]}/.b -> -a t

But there are the constant size constraints to maintain, such as (((a - c))^2) + (((b - d))^2)=1. More work (and inspiration!) is needed here

The l22a & l22b sizes of C3C4 are factors of the L4s size, which is a 6-Vector, the sum of six squared terms with a seventh squared term subtracted:-

L4s=l22a*l22b=(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).

l22a =. ; l22b =. ;

sh = sh12[{a, b, c, d, e, f, g, h, i, j, k, l}];

sh[[5]]
sh[[6]]

toPol[{a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, 
            l_}, "C3C4"] := 
Module[{r = -Sqrt[3], 
            e1 = a + b + c - g - h - i, 
            e2 = d + e + f - j - k - l, 
a1 = 
              a - b + d - e + g - h + j - k, 
            a2 = a - b - d + e + g - h - j + k, 
b1 = 
              b - c + e - f + h - i + k - l, 
            b2 = b - c - e + f + h - i - k + l, 
c1 = 
              c - a + f - d + i - g + l - j, 
            c2 = c - a - f + d + i - g - l + j, 
            f1 = a - b - g + 
                h + (d + e - 2 f - j - k + 
                      2  l)/(Sqrt[3]), 
f2 = (a + b - 2  c - 
                      g - h + 2  i)/(Sqrt[3]) - d + e + j - 
                k, 
f3 = 
              a - b - g + 
                h - (d + e - 2 f - j - k + 
                      2  l)/(Sqrt[3]), 
f4 = (a + b - 2  c - 
                      g - h + 2  i)/(Sqrt[3]) + d - e - j + 
                k}, 
{  a + b + c + d + e + f + 
              g + h + i + j + k + l,   a + b + c - d - e - f + g + 
              h + i - j - k - l, 
  (e1^2) + 
              (e2^2),   ArcTan[e1, 
              e2], 
  ((a1^2) + (b1^2) + 
                  (c1^2))/2,   ArcTan[a1 - c1, 
              r b1], 
  ((a2^2) + (b2^2) + 
                  (c2^2))/2,   ArcTan[a2 - c2, 
              r b2], 
  ((f1^2) + (f2^2)) 
              3/4,   
              ArcTan[f1, f2] - π/
                6, 
  ((f3^2) + (f4^2)) 
              3/4,   ArcTan[f3, f4] - π/6}];

{  a = (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]), 
b = (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]),  
c = (α + β + ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω]                         - η r6 Sin[x]                   - κ r6 Sin[
                  y]), 
d = (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]), 
e = (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]), 
f = (α - β + ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] - η r6 Cos[
                  x]                        + κ r6 Cos[
                  y]                   ), 
g = (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]), 
h = (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]),  
i = (α + β - ϵ  Cos[σ] + ζ Cos[τ - ω] + λ Cos[ϕ - ω]                         + η r6 Sin[x]                    + κ r6 Sin[
                  y]), 
j = (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]), 
k = (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]), 
l = (α - β - ϵ  Sin[σ] + ζ Cos[τ - ω] - λ Cos[ϕ - ω] + η r6 Cos[
                  x]                        + κ r6 Cos[y]           )}/12

3.15. Noncommutative Features. (Needs OPScalar. Inconclusive!)

Most loops (but only a few hoops, including octonions) are noncommutative (NC). The ge procedure employs the explicit NC features of the relators to create the protoloop as a table of lex-ordered-string elements, but converts this to an index-table for output. I conjecture that the repeated factors of NC loop determinants may split when the determinant is found by applying the NC rules.
	OPScalar` (from Mathsource/4271) extends the functionality of the operation NonCommutativeMultiply "**" as a "Diamond" operation "⋄". I modified det and ge to use NC multiplication when the option o is negative, which can be used after Mathsource/4271 has been placed in ExtraPackages. Unfortunately it has several awkward side effects, and this investigation has been suspended.
I cannot see where to go next with this approach, as non-commutative factorization is needed to test my speculation that NC hoop repeated factors can be split.

17. Particle interactions.

Summary. Basic Conjecture. Dozal describes the different particle families as Fermions, with Hexal describing Bosons as equivalence relations on Dozal elements, freed from the Pauli exclusion principle.

Lepton interactions. Incomplete.

Summary. Conjecture. The electron family is described by the C3C4 and Q12 α, β, ϵ, & σ  parameters; their lower analogs are photons.

Electrons have spin, charge, and mass; they are associated with neutrinos, muons and tauons; all have anti-particles. The Dozal 4-phase system {ϵ Cos[σ], ϵ Sin[σ], -ϵ Cos[σ], -ϵ Sin[σ]} wave system, together with the α and β parameters, provides a possible description.
If mass is related to velocity relative to an origin, perhaps the low-mass neutrinos have α=β=0; the electron has α=0, β=1 and the phases are displaced by ±1; α=1, β=1 displaces half of them by 2; α=1, β=0 displaces all of them by 1

{  a = (α + β + ϵ  Cos[σ]), 
        b = (α + β + ϵ  Cos[σ]), 
        c = (α + β + ϵ  Cos[σ]), 
d = (α - β + ϵ  Sin[σ]), 
        e = (α - β + ϵ  Sin[σ]), 
        f = (α - β + ϵ  Sin[σ]), 
g = (α + β - ϵ  Cos[σ]), 
        h = (α + β - ϵ  Cos[σ]), 
        i = (α + β - ϵ  Cos[σ]), 
j = (α - β - ϵ  Sin[σ]), 
        k = (α - β - ϵ  Sin[σ]), 
        l = (α - β - ϵ  Sin[σ])}/
      12

id[C3C2]; shape[{a, b, c, d, e, f}]
toPolar[{a, b, c, d, e, f}]

id[D3]; shape[{a, b, c, d, e, f}]
toPolar[{a, b, c, d, e, f}]

Hadron Interactions. Incomplete.

id[C3C4]; 
    shape[A12 = {a, b, c, d, e, f, g, h, i, j, k, 
          l}]
toPolar[A12]

V12 = 
      toVector[P12 = {1, 1, 0. , 0. , 1, τ, 1, ϕ, 1, χ, 
            1, ψ}]

Chop[Simplify[toPolar[V12]]]

Appendix. J.J.Hamilton's Half Spin operators.

J.J.Hamilton published [2] a "hypercomplex arithmetic" with half-spin "non-real, non-complex" quantum operators as well as integer-spin complex quantum operators. He generalised complex conjugation, in which the conjugate of a conjugate is the original, i:(i*)*=+i, to include elements {a,b,a*,b*} demonstrating "biconjugation" (my term) in which the conjugate of a conjugate is the negated original, a:a** =-a, (a*)** =-a*,  b:b** =-b, (b*)** =-b*.  He demonstrated that {a,b,a*,b*}  are fundamental half-integral spin quantum operators with properties that cannot be expressed as real or complex numbers. (This is because they exist in the hyperbolic plane rather than the complex plane. They are different square roots of unity whilst the full-spin operators i & k are different fourth roots.) His multiplication table (Table II, reproduced below) uses the eight elements (Subscript[E, n ])= {1, i, h, k, a, \(TraditionalForm\`\(b\^*\)\), \(TraditionalForm\`\(a\^*\)\), b}. However, he has several distinct "-1" elements ((h^2),  (a^2) and (a^(*2)) all equal +1); he also introduces several other elements and many aliases. Example 12a reproduces his Table II.

Example 12a. J.J.Hamiltons's Hypercomplex Multiplication Table.  (His 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}})

Example 12b shows that Table II is embedded in C3K. Completing an "extended table" using Hamilton's definitions for -1,-h,-a, and -a*, and for their products, gives a 12×12 table. Choosing generators  (i^3)=(h^2)=(a^2)=1 (to match Hamilton's nomenclature) gives the elements that are shown above the table in Example 12b. If the supplementary and ill-defined "-1" term is converted to an arbitrary variable ii (it is (i^2) or (Subscript[E, 1]^2) in Table II), the table becomes C3K. Deleting the rows and columns starting with negated elements reduces this table to the bold face elements in Example 12b; they match Table II, which is therefore embedded in C3K:-

Example 12b. J.J.Hamiltons's Table II is embedded in C3K.

jjh3=({{1, i, -1, h, k, -h, a, \(b\^*\), -a, \(a\^*\), b, \(-\(a\^*\)\)}, {i, -1, 1, k, -h, h, \(b\^*\), -a, a, b, \(-\(a\^*\)\), \(a\^*\)}, {-1, 1, i, -h, h, k, -a, a, \(b\^*\), \(-\(a\^*\)\), \(a\^*\), b}, {h, k, -h, 1, i, -1, \(a\^*\), b, \(-\(a\^*\)\), a, \(b\^*\), -a}, {k, -h, h, i, -1, 1, b, \(-\(a\^*\)\), \(a\^*\), \(b\^*\), -a, a}, {-h, h, k, -1, 1, i, \(-\(a\^*\)\), \(a\^*\), b, -a, a, \(b\^*\)}, {a, \(b\^*\), -a, \(a\^*\), b, \(-\(a\^*\)\), 1, i, -1, h, k, -h}, {\(b\^*\), -a, a, b, \(-\(a\^*\)\), \(a\^*\), i, -1, 1, k, -h, h}, {-a, a, \(b\^*\), \(-\(a\^*\)\), \(a\^*\), b, -1, 1, i, -h, h, k}, {\(a\^*\), b, \(-\(a\^*\)\), a, \(b\^*\), -a, h, k, -h, 1, i, -1}, {b, \(-\(a\^*\)\), \(a\^*\), \(b\^*\), -a, a, k, -h, h, i, -1, 1}, {\(-\(a\^*\)\), \(a\^*\), b, -a, a, \(b\^*\), -h, h, k, -1, 1, i}})/.-1 -> i i//tf

Hence Hamilton's hypercomplex algebra can be interpreted as the C3K algebra if his i is a cube root of 1, with a and h being different square 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 is multiplication by h (with h h =1), but biconjugation becomes multiplication by i or i i. This introduces ternary symmetry, and defies the prejudice that there must be (2^n) elements in an algebra - which is only true for real algebras without divisors of zero.

References.

[1] http://library.wolfram.com/infocenter/MathSource/4894/

[2] J.J.Hamilton, Hypercomplex numbers and the description of spin states, J.Math.Phys. 38(10) Oct. 1997. pp4914-4928

[3] Remoissenet, Waves called Solitons, 1997 p29,p33.

[4] G.P.Nagy & P.Vojtechovsky, Loops package in GAP,  www.gap-system.org