Hoop Algebras & Physics.

R.H.Beresford.  21/2/2007.

Abstract.
Symmetry-conserving Hoop Algebras* (*=new mathematical concept) add, multiply and divide* vecs* (generalised vectors), introducing unsigned continuous Primal* numbers, generalised signs* and multiplication tables folded* (equivalenced) from finite group Cayley tables. Their Frobenius symmetry-conservation* and Moufang partial-fraction-division* properties suggest a new physics paradigm. Vecs have partial-fraction multiplicative inverses* with conserved symbolic determinant factors* (symmetries or “sizes”*) as denominators. Multiplying or dividing two vecs gives a third vec and two (often null) remainder* vecs carrying different symmetries, emulating symmetry-conserving particle interactions and eliminating division-by-zero. Non-Abelian hoops* may introduce uncertainty. Multi-phase sinusoidal orbits* generalise Cartesian-polar duality*. Some hoops have half-spin quantum operators*, stable de-Broglie-like orbits* and Planck-like intrinsic areas*, possibly explaining particle stability, ternary quark symmetry, mass, M-theory T-duality, and the “large numbers hypothesis”. Hoop symmetries complement the use of continuous groups as coordinate transformation matrices. Single-symmetry real, complex, quaternion and octonion algebras are degenerate* cases of the more general hoop operations that describe physics.

1. Introduction

1.1 Overview.

General relativity and quantum physics are both correct, but are incompatible in 20th century mathematics; is the mathematics wrong? A search for errors led to the conjecture that physics is over-reliant on real and complex numbers. A new approach was found. Sir W.R.Hamilton observed that quaternion and octonion algebra multiplication tables had four and eight signed “dimensions”, and were obtained from tables with eight and sixteen unsigned “directions”. This generalizes; many algebras are obtained by equivalence relationships folding* (*=new mathematical concept) Cayley tables with r-fold symmetry, creating signed algebras. Real, complex, quaternion and Clifford algebras are folded from the C2, C4, quaternion, etc. groups. Folding is usually two-fold with signs {+,-}, or four-fold, with signs {+,i,-,-i}; three-fold symmetry gives terplex* algebras with signs {+, J, JJ}*. Generalised signs* arise as powers of orthogonal (NOT cyclotomic) primitive roots of unity. Frobenius [1] proved that associative tables “conserve” the determinant of the inverse symbolic table. All groups, “signed table” algebras folded from them, and a few “alternative” Moufang loops [2] (e.g. octonions and split-octonions) conserve the determinant factors as “symmetries” or “sizes”; most loops do not. Hoops* are defined as symmetry-conserving algebras based on unsigned continuous Primal Numbers*, "generalised signs" and multiplication tables folded (equivalenced to introduce signed elements*) from conservative Cayley tables. Hoops add, multiply and divide* generalised* vectors (abbreviated to "vecs"). They are implemented and demonstrated [3,4] in notebooks that Mathematica users can test with different data. [5] provides edited Word document transcripts of these demonstrations.  
	Frobenius symmetry-conservation* and Moufang partial-fraction-division* properties make hoops relevant to, and hint at a possible new paradigm for, mathematical physics. Vec multiplicative inverses have conserved symbolic determinant factors as partial-fraction denominators*. In hoop multiplication and division two vecs produce a third vec and two (often null) remainder* vecs carrying different symmetries in different sub-algebras. This emulates symmetry-conserving particle interactions and eliminates division by zero by projection* (as in h-deformed algebras) into a sub-algebra when a size approaches zero. Multi-phase sinusoidal* polar-duals* (generalising Cartesian-polar duality) appear when quadratic sizes are conserved. Some hoops have stable de-Broglie-like orbits* with Planck-like intrinsic areas*. In Dozal* (the 12-element algebras) 3, 4, 6 & 12-phase orbits have areas that multiply to one (or are zero, as missing symmetries*), possibly explaining particle families, particle stability, ternary quark symmetry, mass, M-theory T-duality, and the “law of large numbers”. Using experimental mathematics, the author has found many analogies between hoop algebras and particle physics. As a 77 yr. old chemical engineer he lacks the skills and time to take the subject much further.
	Hoops complement the use of groups as coordinate transformation matrices. Real, complex, quaternion and octonion algebras are single-symmetry* hoops; their operations are degenerate* cases of more general hoop operations that describe mathematical physics.

1.2. Implementation.

The Hoops concept has been implemented in [3] as a Mathematica package Hoops.m. This provides a databank of over 80 named Hoops (with five counter-example non-hoop Loops), three main procedures hoopTimes, hoopInverse, hoopPower and a few subsidiary procedures. Each hoop is defined by a name (relating it to a group or an algebra), a Cayley index table (a preferred isomorph) and a "shape" (the distinct factors of the inverse symbolic Cayley table). Multiplication, division, and (in many cases) conversion from Cartesian vector form {a,b,...} to polar form {r,θ,...} are provided. The package is written in elementary Mathematica, and includes (* comments *) and "Usage" explaining the details, so it should be readily comprehensible and easy to transcribe into other computer-aided-mathematics systems for independent verification. A larger package (creation and properties of groups, loops, & hoops) is available in [4], which also provides a glossary.
	The figures in this document are self-explanatory output from a Mathematica session (with results of calculations appearing in bold-face) intended to be comprehensible to readers with no knowledge of Mathematica. They are based on the Hoops.m package. Mathematica users can test them with different data, by installing Hoops.nb and running it (to create the Hoops.m package) in the Addons\ExtraPackages directory, and then using the following instructions:-.

Quit[]

<< "Hoops`"

2. Development of Hoop Concepts.

Summary.
	Symmetry-conserving "Hoop Algebras" involve new mathematical concepts, starting with unsigned continuous “Primal Numbers”. Hoop multiplication tables are “folded” (via equivalence relations based on symmetries) from finite groups and Moufang loops; this introduces "generalized signs". Hoops multiply, divide, and add generalized vectors. Their "Frobenius symmetry-conservation" and "Moufang partial-fraction-division" properties make them relevant to physics. A few hoops have stable multi-phase de-Broglie-like "orbits" and Planck-like intrinsic areas; this may explain particle stability. Every vector has a multiplicative inverse, so Hoop multiplication and division use the same "hoopTimes" operation; two vectors produce a third vector and two (often null) "remainder" vectors carrying different symmetries. This is analogous to symmetry-conserving particle interactions. Do not confuse hoops with the traditional use of groups as coordinate transformation matrices for a single vector. Hoop tables are not restricted to complex matrix generators. Many complex operations are degenerate cases of more general hoop operations.

The development of hoops  involves some neglected and some new mathematical concepts. The latter are idistinguished by asterisks:- 

	(1) Primal* unsigned continuous numbers (the half-line, the union of 0 & \(TraditionalForm\`\(ℝ\^+\)\)) can be developed from set theory via natural numbers N, (which have order and a lower bound and allow addition, multiplication, powers, and unique prime factorization) and Landau's [5] unsigned rational numbers \(TraditionalForm\`\(ℚ\^+\)\)(which allow division). A continuity axiom converts \(TraditionalForm\`\(ℚ\^+\)\)to Primal numbers (with unique roots and general factorization). Subtraction and negation are undefined for all these numbers.
	Only the most rigorous mathematicians acknowledge that integers and reals are equivalence relations on pairs of natural or primal numbers; others conflate (unary) negation with (binary) subtraction by working in fields - implicitly assuming that minus one exists and is unique (which is then justified by a circular argument).
	Some physical concepts (time? mass?) should, perhaps, have primal (rather than real) measures.

(2) Algebraic loops are m×m multiplication tables for m-element sets of coefficients (or operators) that define "vecs" (generalized vectors, which are vectors in some algebras) such as A = {(Subscript[a, 1]),(Subscript[a, 2]),...(Subscript[a, m])}, {a,b,...,m}, or (Subscript[a, 1])(Subscript[d, 1])+(Subscript[a, 2])(Subscript[d, 2])+...+(Subscript[a, m])(Subscript[d, m]). Loops can be developed without introducing negation, so the coefficients (Subscript[a, i]) are unsigned. Moufang loops have the Moufang division property zx.yz=(z.xy)z (so every vec has a multiplicative inverse); groups are associative Moufang loops with x(yz)=(xy)z . Frobenius showed [2] that all groups conserve their determinants when they are used as vector multiplication tables; Det[A] Det[B] = ±Det[AB], where Det is the determinant of the inverse multiplication table mapped with the appropriate vector. A few non-associative Moufang loops (including octonions and split-octonions) also have this Frobenius symmetry-conservation* property; general non-associative loops lack it.

(3) Generalized signs* are orthogonal primitive roots of unity* (s^j) (with (s^r) =1) describing r-directional spaces or sub-spaces. Describing roots of unity as cyclotomic numbers distorts their properties by projecting them onto the complex field. Reals, with signs {+.-}, are an r=2 case. Complex numbers are one case of r=4, with signs {+,i,-,-i}. r=3 gives terplex* signs {+, 𝕁, (𝕁^2)} with ternary symmetries relevant to quarks. Many features of complex algebra are specializations of more general features involving generalized signs. Multi-phase sinusoids* are a key example, with (e^(-iθ))=Cos[θ]+i Sin[θ] being collapsed from the r=4 case of E^(sθ)=(∑_(j = 0)^(j = r - 1))(s^j) Cos[θ+2jπ/r].

	(4) "Directors" are sets of primal coefficients with associated "directions", i.e. the (Subscript[d, i])'s in (2). r-fold equivalence relationships on m-element directors "fold" them to "signed vecs", generalized vectors with r-signed coefficients and m/r associated dimensions. Real vecs are the result of 2-folding, (Subscript[r, i])~(Subscript[a, i])- (Subscript[a, i + m/2]); this is the special case "Minus 1 exists and is unique" that allows negation and subtraction to be conflated. (The "directions" concept is due to Sir W.R.Hamilton. He related the eight directions of the quaternion group to the four dimensions of the quaternion algebra, and the sixteen directions of the octonion table to the eight dimensions of the octonion algebra.) Folding is a key concept because it relates generalized signs to loop symmetries. Equivalence relationships on conservative m×m-element loops with r-fold symmetry "fold" them to (m/r)×(m/r) multiplication tables for r-signed vecs. I call such tables "Hoop Algebras". In many cases they are not loops, because some products are "signed elements" that are not members of the defining set. E.g. Quaternions have four elements {e,i,j,ij} with i.j=ij, but j.i=-ij. Each hoop in the database is defined by a name (relating it to a group or an algebra).
	Figure 1 shows “protoloops” (preferred Cayley table isomorphs) for three Moufang loops (the C4C2, quaternion, and C9 groups). The top left corner is repeated in the tables; it also occurs with offsets of 3, 4, or 6, showing the symmetry. These sub-tables are then shown as folded tables C4, Q4, C9J, with the indices replaced by the usual complex or quaternion symbols, or (in the C9J algebra) by signed indices, using the ternary sign J (with (J^3)=1) so that indices 4 & 5 become J1 and J2. Note that the C4C2 table can be folded a second time, to give the complex algebra table with i i = -1. The "Pauli-σ algebra" P4 is also shown; it is 4-folded from a 16-element group (not shown) that can be created by dot-multiplying the Pauli-σ matrices.

Figure 1. Two- Three- & four-fold folding of Moufang Loops, the
resulting signed algebras, and the four-folded Pauli-σ algebra.
  C4C2 Group       Quaternion group     C9 group
1 2 3 4 5 6 7 8    1 2 3 4 5 6 7 8    1 2 3 4 5 6 7 8 9
2 3 4 1 6 7 8 5    2 5 8 3 6 1 4 7    2 3 4 5 6 7 8 9 1
3 4 1 2 7 8 5 6    3 4 5 6 7 8 1 2    3 4 5 6 7 8 9 1 2
4 1 2 3 8 5 6 7    4 7 2 5 8 3 6 1    4 5 6 7 8 9 1 2 3
5 6 7 8 1 2 3 4    5 6 7 8 1 2 3 4    5 6 7 8 9 1 2 3 4
6 7 8 5 2 3 4 1    6 1 4 7 2 5 8 3    6 7 8 9 1 2 3 4 5
7 8 5 6 3 4 1 2    7 8 1 2 3 4 5 6    7 8 9 1 2 3 4 5 6
8 5 6 7 4 1 2 3    8 3 6 1 4 7 2 5    8 9 1 2 3 4 5 6 7
                                      9 1 2 3 4 5 6 7 8

 C4 & Complex    Quaternion      C9J       Pauli-σ
 Algebra C2c     Algebra Qr     Algebra   Algebra P4
 1  i -1 -i     1   i   j -ij   1  2  3   1   2   3   4
 i -1 -i  1     i  -1 -ij   j   2  3 J1   2   1 -i4  i3
-1 -i  1  i     j  ij  -1  -i   3 J1 J2   3  i4   1 -i2
-i  1  i -1    ij  -j   1  -1             4 -i3  i2   1

Octonion algebra         Split-Octonion algebra
1  2  3  4  5  6  7  8     1  2  3  4  5  6  7  8
2 -1  4 -3  6 -5 -8  7     2  1  4  3  6  5 -8 -7
3 -4 -1  2  7  8 -5 -6     3 -4 -1  2  7  8 -5 -6
4  3 -2 -1  8 -7  6 -5     4 -3 -2  1  8  7  6  5
5 -6 -7 -8 -1  2  3  4     5 -6 -7 -8 -1  2  3  4
6  5 -8  7 -2 -1 -4  3     6 -5 -8 -7 -2  1 -4 -3
7  8  5 -6 -3  4 -1 -2     7  8  5 -6 -3  4 -1 -2
8 -7  6  5 -4 -3  2 -1     8  7  6 -5 -4  3  2  1

The two 8-element non-associative algebras are also shown.

(5) Factors of the symbolic inverse table determinant are conserved sizes* (symmetries) for hoop multiplication; this is the defining property for hoop algebras. The results of factorising quadratic or higher factors into complex-conjugate pairs are not conserved properties - complex algebra is only one of many hoop algebras.
	Figure 2 shows the symbolic tables, the factorized determinants and the products hoopTimes[{1,2,...},{a,b,...}] for the algebras in Figure 1. Calculated results are bold.

Figure 2. Symbolic tables, products, factorized determinants.
   C4        C      Quaternion     C9J         Pauli-σ
 a b c d    a  b    a  b  c  d    a  b  c     a   b   c   d
 b c d a    b -a    b -a -d  c    b  c Ja     b   a -id  ic
 c d a b            c  d -a -b    c Ja Jb     c  id   a -ib
 d a b c            d -c  b -a                d -ic  ib   a
Factorized Determinants, Products :-
C4 Factors  (a+b+c+d)(a-b+c-d)((a-c)2+(b-d)2)
hoopTimes[{1,2,3,4},{a,b,c,d}]=
{a+4b+3c+2d,2a+b+4c+3d,3a+2b+c+4d,4a+3b+2c+d}
C  Factor               -a2-b2
hoopTimes[{1,2},{a,b}]={a-2b,2a+b}
Quaternion Factors (a2+b2+c2+d2)2
hoopTimes[{1,2,3,4},{a,b,c,d}]=
{a-2b-3c-4d,2a+b-4c+3d,3a+4b+c-2d,4a-3b+2c+d}
C9J Factor       -c3-b3J+3abcJ-a3J2
hoopTimes[{1,2,3},{a,b,c}]={a+3bJ+2cJ,2a+b+3cJ,3a+2b+c}
Pauli-σ  Factors     (a2-b2-c2-d2)2
hoopTimes[{1,2,3,4},{a,b,c,d}]=
{a+2b+3c+4d,2a+b+4ic-3id,3a-4ib+c+2id,4a+3ib-2ic+d}

Octonion Factors (a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2)^4
hoopTimes[{1,2,3,4,5,6,7,8},{a,b,c,d,e,f,g,h}]=
{a-2b-3c-4d-5e-6f-7g-8h,2a+b-4c+3d-6e+5f+8g-7h,
 3a+4b+c-2d-7e-8f+5g+6h,4a-3b+2c+d-8e+7f-6g+5h,
 5a+6b+7c+8d+e-2f-3g-4h,6a-5b+8c-7d+2e+f+4g-3h,
 7a-8b-5c+6d+3e-4f+g+2h,8a+7b-6c-5d+4e+3f-2g+h}
Split-Octonion Factors (a^2-b^2+c^2-d^2+e^2-f^2+g^2-h^2)^4
hoopTimes[{1,2,3,4,5,6,7,8},{a,b,c,d,e,f,g,h}]=
{a+2b-3c+4d-5e+6f-7g+8h,2a+b-4c+3d-6e+5f+8g-7h,
 3a-4b+c+2d-7e+8f+5g-6h,4a-3b+2c+d-8e+7f-6g+5h,
 5a-6b+7c-8d+e+2f-3g+4h,6a-5b+8c-7d+2e+f+4g-3h,
 7a+8b-5c-6d+3e+4f+g-2h,8a+7b-6c-5d+4e+3f-2g+h}

(Mathematica users can repeat the calculations with their own input data.) The three C4 sizes fold to the single conserved size (a^2)+(b^2) of Complex algebra. Quaternion, Pauli-σ, octonion and split-octonion algebras conserve their single (but repeated) factors. C9J conserves (a^3)+(b^3)J-3abcJ+(c^3)(J^2), a conjugate of the determinant factor. (Conjugate conservation* appears to be the general rule, because the conserved factors are those of the inverse table determinant. Conjugation has no effect on most sizes.)  In many hoops (including C, H, O, split-octonions, P4, some Clifford algebras), multiplying a vector by its complex conjugate gives a result consisting of the conserved quadratic size, followed by a set of zeroes. This is why complex conjugates are useful. Other hoops have “plex-conjugates” (involutions) that give products related to the conserved sizes. This has allowed the determination of the sizes of some hoops with 16 or more elements.

	(6) The Moufang property ensures that every vec has a multiplicative inverse in each hoop of the same length, merging multiplication and division into one procedure. The inverse Ai solves Ai.A={1,0,..} (the “unit vec” with shape {1,1,...}). The determinant is a divisor for each element of the inverse, so multiple sizes provide denominators for partial-fraction formulations* of the inverse. Although the un-factored determinant becomes zero if one or more size becomes zero, the other factors (and the corresponding partial fractions) remain finite*.
	Figure 3 shows the symbolic C3 table, the C3 (terplex*) product of two symbolic vectors, and the two determinant factors. The inverse is then shown; each term splits into two partial fractions. Then two numeric vectors A & B are multiplied; the sizes are shown as sA, etc. (Mathematica users can repeat the calculations with their own input data.) The sizes of the product AB can be seen to be the products of the sizes of A & B. Ai is a left inverse, so hoopTimes[Ai,AB] recovers B.

Figure 3. C3 Product, Shape, Inverse, Division.

Use["C3"]; 
  {tf[hoopTbl /. {1 -> a, 
      2 -> b, 3 -> c, 4 -> d, 
      5 -> e, 6 -> f}], "\nho\
opTimes[{a,b,c},{d,e,f}=\n", 
   bf[hoopTimes[{a, b, c}, 
     {d, e, f}]], 
   "\nshape =\n", bf[sh], 
   "\nInverse =\n", 
   bf[Apart[hoopInverse[
      {a, b, c, d, e, f}]]]}

tf[{Flatten[{"If  A=", 
     A = {3, 1, -2}, "sA=", 
     sh /. as[A]}], 
   Flatten[{"and B=", 
     B = {1, 0, 2}, "sB=", 
     sh /. as[B]}], 
   Flatten[{"AB=", 
     AB = hoopTimes[A, B], 
     "sAB=", sh /. as[AB]}], 
   Flatten[{"Ai=", 
     Ai = hoopInverse[A], 
     "sAi=", sh /. as[Ai]}], 
   Flatten[{"AiAB=B", 
     AiAB = hoopTimes[Ai, 
       AB], "", "", ""}]}]

It is shown in [3, Example 14] that, in general, loops do not conserve their determinant factors.

(7) Vecs with zero-sizes* exist in sub-algebras* where these sizes are constrained (as a result of conservation) to be zero. Their use in multiplication and division projects* the result into the sub-algebras and "ejects" remainders to maintain size conservation. This eliminates division-by-zero; zeroed sizes are "factored-out" by projection into the constrained sub-algebra. If a size is within an arbitrary hmin of zero hoopTimes sets it to zero and tests whether a remainder is created. Remainders are demonstrated in Figure 4 for the C3 and C4 hoops. The unit vecs in sub-algebras (A1 in Fig.4) are not {1,0,0..} (with shape {1,1,..}) because their shapes must have the appropriate zeroes; A1.A1=A1 correctly. “Ultra-violet divergence” may be eliminated by projection.

Figure 4. If A has a zero size, so do AB and Ai; 
       AB/A only become B after adding a remainder.

Use["C3"];
tf[{Flatten[{"In C3, if  A=", 
            A0 = {2, -1, -1}, "sA=", sh /. as[A0]}], 
        Flatten[{"and B=", B = {1, 0, 2}, "sB=", sh /. as[B]}], 
        Flatten[{"AB=", AB = hoopTimes[A0, B], "sAB=", 
            sh /. as[AB]}], 
        Flatten[{"Rleft=", Rleft = Rl, "sRleft=", sh /. as[Rleft]}], 
        Flatten[{"Rright=", Rright = Rr, "sRright=", sh /. as[Rr]}], 
        Flatten[{"Ai=", Ai = hoopInverse[A0], "sAi=", sh /. as[Ai]}], 
        Flatten[{"Ai.A (=A1)=", A1 = hoopTimes[Ai, A0], "sA1=", 
            sh /. as[A1]}], 
        Flatten[{"AB/A (=AiAB)=", AiAB = hoopTimes[Ai, AB], "sAiAB=", 
            sh /. as[AiAB]}], 
        Flatten[{"AiAB+Rright=B", R = AiAB + Rright, "s=", 
            sh /. as[R]}]}]
Use["C4"];
tf[{Flatten[{"In C4, if  A=", 
            A = {2, 3, 1, 0}, "sA=", sh /. as[A]}], 
        Flatten[{"and B=", B = {3, -1, -2, 0}, "sB=", 
            sh /. as[B]}], 
        Flatten[{"AB=", AB = hoopTimes[A, B], "sAB=", sh /. as[AB]}], 
        Flatten[{"Rleft=", Rleft = Rl, "sRleft=", sh /. as[Rleft]}], 
        Flatten[{"Rright=", Rright = Rr, "sRright=", sh /. as[Rr]}], 
        Flatten[{"1/A=Ai=", Ai = hoopInverse[A], "sAi=", 
            sh /. as[Ai]}], 
        Flatten[{"Ai.A (=A1)=", A1 = hoopTimes[Ai, A], "sA1=", 
            sh /. as[A1]}], 
        Flatten[{"AB/A (=AiAB)=", AiAB = hoopTimes[Ai, AB], "sAiAB=", 
            sh /. as[AiAB]}], 
        Flatten[{"AiAB+Rright=B", R = AiAB + Rright, "s=", 
            sh /. as[R]}], 
        Flatten[{"1/B=Bi=", Bi = hoopInverse[B], "sBi=", 
            sh /. as[Bi]}], 
        Flatten[{"Bi.B (=B1)=", B1 = hoopTimes[Bi, B], "sB1=", 
            sh /. as[B1]}], 
        Flatten[{"AB/B (=ABBi)=", ABBi = hoopTimes[AB, Bi], "sABBi=", 
            sh /. as[ABBi]}], 
        Flatten[{"ABBi+Rleft=A", R = ABBi + Rleft, "s=", 
            sh /. as[R]}]}]

Note the analogy between integer division, I/J=K+R where J*K+R=I, with Hoop operations, where A.B=C+Rleft+Rright and C/B+Rleft=A, C/A+Rright=B. Rleft carries the sizes that are zero in C but not in A; Rright carries the sizes that are zero in C but not in B. Remainders ensure that conservation is maintained in hoop multiplication (as well as division) when some sizes are zero. There is an obvious analogy with particle interactions, where there may be several products with different symmetries, and to h-deformed and q-deformed algebras.
	Do not confuse loop operations, acting on two vectors to give a new vector and two (possibly null) remainder vectors, with the use of matrices to transform a vector to different coordinates in the “standard model” of particle physics.

(8) Some hoops have quadratic sizes* that provide polar-Cartesian dual* formulations and allow the definition of continuous multi-phase sinusoidal orbits*, as shown in Figures 5 & 7. Quadratic sizes release a degree of freedom that is taken up by an angle, which is a hidden variable* for the Cartesian form. 
	Abelian polar duals have angles that add on vec multiplication*. This leads to powers and roots. Figure 5 shows that the polar form for the C3 vector {a,b,c} consists of expressions for two sizes {α,(ϵ^2)} and an angle σ. The reversion from polar to cartesian form has α as an offset from zero and three equispaced phases. It then demonstrates that vector[polar[A]=A, angles add on C3 multiplication, and taking a square root halves the angle and gives sizes that are square roots of the original sizes. C3 has a prototypical ternary (3-phase) form polar form. Figure 6 shows that C4 has the prototypical 4-phase. This folds to the (asymmetric) {Cos[θ],Sin[θ]} pair. All known polar duals are related to these two prototypes.

Figure 5a. C3 Polar Duals, Reversions, Powers & Roots.

Use["C3"]; "Polar C3"
topol // bf
"Vector C3"
tovec // bf
\({Flatten[{"If  A=", A = {2. , \(-3. \), 5. }, "polarA=", 
            topol /. as[A]}], 
        Flatten[{"Vec(polar(A))=", 
            tovec /. as[
                topol /. as[
                    A]], "", "", "", ""}], 
Flatten[{"and B=", 
            B = {1, \(-5\), 6. }, "polarB=", 
            topol /. as[B]}], 
Flatten[{"AB=", 
            AB = hoopTimes[A, B], "polarAB=", topol /. as[AB]}], 
        Flatten[{\*""(Sqrt[A])="", 
            RootA = hoopPower[A], \*""polar(Sqrt[A])="", 
            topol /. as[RootA]}]} // tf\)

{a + b + c, 
            (1/2) ((((a - b))^2) + (((b - c))^2) + (((-a + c))^2)), ArcTan[2 a - b - c, -(Sqrt[3]) (b - c)]}

{(1/3) (a + 2 (Sqrt[b]) Cos[c]), 
            (1/3) (a + 2 (Sqrt[b]) Cos[c + ((2 π)/3)]), 
            (1/3) (a + 2 (Sqrt[b]) Cos[c - ((2 π)/3)])}

Figure 5b. C4 Polar Duals, Reversions, Powers & Roots.

Use["C4"]; "Polar C4"
topol // bf
"Vector C4"
tovec // bf
\({Flatten[{"If  A=", A = {2. , \(-3. \), 5. , 1. }, "polarA=", 
            topol /. as[A]}], 
        Flatten[{"Vec(polar(A))=", 
            tovec /. as[
                topol /. as[
                    A]], "=A", "", "", "", ""}], 
Flatten[{"and B=", 
            B = {\(-1\), 5, \(-6. \), 3. }, "polarB=", 
            topol /. as[B]}], 
Flatten[{"AB=", 
            AB = hoopTimes[A, B], "polarAB=", topol /. as[AB]}], 
        Flatten[{\*""(Sqrt[A])="", 
            RootA = hoopPower[A], \*""polar(Sqrt[A])="", 
            topol /. as[RootA]}]} // tf\)

Note that the linear sizes are “offsets” that displace the centres of rotation for the sinusoidal terms. They are conserved linear symmetries. Pairs of linear sizes can be combined to give conserved differences of squares, from which hyperbolic duals can be constructed [S, Sect.3.10].
	Non-abelian hoops have repeated roots, often leading to "pseudo-polar" duals, demonstrated in the supplementary material [S,Ex.11]. The “pseudo-angles” do not add on vec multiplication, and the pseudo-powers differ from repeated products. This is discussed in the supplementary material.

(9) Some Hoops have stable orbits resembling "multi-phase de Broglie waves"; products of orbits are orbits [S,Ex.35]. They define intrinsic sizes resembling Planck areas and may determine which particles are stable. They have some elements (e.g. ζ in Figure 6) that are neither real nor complex but act as quantum operators [6]. Orbits with 3, 4, & 6 phases (and some multiples) appear to be stable; a 5-phase C5 function proved to have a zero radius, making it unstable.

(10) Four "Dozal" multiplication rules (for vecs with 12 elements [S,Sect.3.3])  may correspond to four types of interaction between particles, with different sizes corresponding to different conserved properties and remainders corresponding to emitted particles. C3K and C3C4 are abelian,  Q12 & D3C2 are non-abelian. As with D3, the non-abelian rules have "pseudo-polar duals" with degrees of freedom that may introduce uncertainty. They are indirect compositions of C2 with (bosonic) abelian groups that appear to be "supersymmetric" to the (fermionic) larger groups. Fermionic hoops have more symmetries (and more particle types) than the bosons. Supersymmetry is not one-to-one. Conjecture - bosons are not subject to the Pauli exclusion principle because each is equivalent to many fermions.
	Figure 6 shows the symbolic vector and polar (orbit) forms for C3K. The orbit combines the two square roots of unity (Dual or Study numbers) of the Klein group K, with the cube root of unity of Terplex numbers. Each line of Figure 6 defines a single element of one form as a function of the elements of the other. Orbits are “unital”; their non-zero sizes multiply to one so that repeated products also multiply to one. Different orbits are obtained by constraining some of the sizes {α,β,γ,δ,ζ,λ,η,κ} to be zero whilst the rest multiply to one. Note that ζ has three phases (repeated four times), whilst {λ,η,κ} each have six; η=κ+π/6 gives twelve phases. The C3C4 orbits ([S,Ex.25]) are closely related to those for C3K, but include a four-phase system in which γ and δ are replaced by (ϵ^2) and an angle σ. The author conjectures that this relates to leptons as 4-phase orbits, with hadrons as 3, 6, and 12-phase quark orbits. Orbits properties include spin, chirality and polarisation.

Figure 6. C3K Polar form and general orbit.

{a + b + c + d + e + f + g + h + i + j + k + l, 
      a + b + c - d - e - f + g + h + i - j - k - l, 
a + 
        b + c + d + e + f - g - h - i - j - k - l, 
      a + b + c - d - e - f - g - h - i + j + k + l, 
      (1/2) ((((a - b + d - e + g - h + j - k))^2) + (((b - c + e - f + h - i + k - l))^2) + (((-a + c - d + f - g + i - j + l))^2)), 
      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)], 
      (1/2) ((((a - b - d + e + g - h - j + k))^2) + (((-a + c + d - f - g + i + j - l))^2) + (((b - c - e + f + h - i - k + l))^2)), 
      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)], 
      (1/2) ((((a - b + d - e - g + h - j + k))^2) + (((-a + c - d + f + g - i + j - l))^2) + (((b - c + e - f - h + i - k + l))^2)), 
      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)], 
      (1/2) ((((a - b - d + e - g + h + j - k))^2) + (((b - c - e + f - h + i + k - l))^2) + (((-a + c + d - f + g - i - j + l))^2)), 
      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)]}

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

The following table (repeated in [S,Ex.30]) defines 39 functions (not all being sizes) that are conserved in one or more of the Dozal operations. The author has attempted to relate them to particle properties, without success.

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

(11) Many Dozal properties disappear when 2-folding converts the vecs to “Hexal”. Figure 7 shows the polar and vector forms for C3C2, which is obtained by to replacing g by -a , etc, in Figure 6. This appears to correspond to supersymmetry, creating bosons as equivalence relations.

Figure 7. C3C2 polar & Vector forms.

Use["C3C2"]; topol
tovec

(12) The real division algebras ℝ, ℂ, ℍ, 𝕆 are degenerate (monosized) hoops without divisors of zero because their sizes (the sums of the squared elements) are non-negative. These hoops operate correctly if their coefficients are complex, but then have divisors of zero. This gives the light-cone in the complex quaternion case.

(13) Hoops subsume, but do not invalidate, complex mathematics. Complex functions are specializations of more general functions. Many standard mathematical operations are specialisations of more general hoop operations ; many "obvious mathematical truths" are special cases restricted to the "real algebras without divisors of zero". E.g. multiphase sinusoids; repeated products and powers (A.A & (A^2)) may differ; vector division is defined; both multiplication and division of vectors may leave remainders.

3. Further Properties.

Note. The Hoops&Physic.doc Word transcript relegates much of the following material to HoopAlgebraSupplement.doc (which is transcribed from HoopSup.nb), references [S,??] correlate the two documents.

Summary. Further properties of Hoop Algebras are demonstrated and discussed. In general, loops do not conserve their determinant factors. C4 folds to Complex algebra. Even powers differ from repeated products if sizes are negative. Dot, cross and wedge products are restrictions on Clifford algebras. Many hoops have “plex-conjugate” involutions that generalize complex conjugation. Products of Abelian orbits are orbits with added angles.

In general, loops do not conserve their determinant factors. Figure 8 [S,Example 14] shows two examples. Their non-conserved factors resemble the sizes of Qr, P4, & D3. It also shows that the sum of elements of a product factorizes into the sums of the multiplicand elements for C6n (a quasi-group) but not for Q4n (a signed table). This distinction appears to be general - the sum of the elements of a product is the product of the multiplicand element sums iff the table is a quasi-group.

Figure 8. Two non-conservative Loops

\(Use["Q4n"]; {"Q4n Table", hoopTbl // tf, "
Shape", 
      sh, 
"
A =   ", 
      A = {4, 2, 3, 5}, 
"
B =   ", 
      B = {1, 6, 2, 2}, 
"
AB=   ", 
      AB = hoopTimes[A, B], "
Shape of A    sa =", 
      sa = sh /. \ as[A], "
Shape of B    sb =", 
      sb = sh /. \ as[B], "
Shape of AB   sab=", \ 
      sh /. \ as[AB], "
This is not sa*sb =", 
      sa\ sb, "
Sum of elements does not factorize", 
      Factor[Plus @@ hoopTimes[{a, b, c, d}, {e, f, g, h}]], 

      Use["C6n"]; 
"
C6n Table", hoopTbl // tf, "
Shape", 
      sh, "
 A =", A = {4, 2, 3, 5, 1, 6}, "
 B =", 
      B = {0, 6, 2, 4, 3, 2}, "
AB =              ", 
      AB = hoopTimes[A, B], "
shape of A  sa =  ", 
      sa = sh /. \ as[A], "
shape of B  sb =  ", 
      sb = sh /. \ as[B], "
shape of AB sab=  ", 
      sab = sh /. \ as[AB], "
This is not sa*sb ", 
      sa\ sb, "
Sum of elements factorizes", 
Factor[
        Plus @@ hoopTimes[{a, b, c, d, e, f}, {g, h, i, j, k, 
              l}]]}\)

<

Figure 9 [S,Examples.4,5,6] demonstrates that Complex algebra is obtained by folding C4. Multiplication, division, and various powers are shown to match those of the complex numbers created by 2-folding. The shape of the C4 vector includes the squared radius and the polar angle; it has two extra degrees of freedom (conserved linear sizes) that are destroyed by the folding process. Figure 10 also shows that if a size is negative, A.A will differ from (A^2) (though they both fold to the correct complex number; folding is many-to-one). Repeated products will have this size alternately negative and positive, whilst it will be negative for all powers; (A^n) is a continuous function of n.

Figure 9. C4 folds to Complex algebra.

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

"\n\nIf A has a negative size hoopTimes[A,A] differs from hoopPower[A,2] but folds correctly."//bf,"\nIf A=",A = {1. , 1. , -1. , 0. },"with shape ",sh /. as[A],"hoopPower[A,3] gives the cube A3=",A3 = hoopPower[A, 3]; A3 // bf,"\nwhich folds correctly to (2+i)^3=2+11i\nhoopPower[A3,2/3] is",hoopPower[A3, 2/3] // bf,"the 2/3 power of A3, which folds to 3+4i\nand matches hoopPower[A,2]\ni.e. A^2=",A2=hoopPower[A, 2];A2 // bf,"\nwith shape",sh /. as[A2],"\nwhilst the repeated product AA",AA = hoopTimes[A, A]; AA // bf,"\n(which also folds to 3+4i) has the second size positive",sh /. as[AA] // bf,"\npowers are continuous functions:-\n A^1.99=",hoopPower[A, 1.99] // bf,"\n A^2   =",hoopPower[A, 2] // bf,"\n A^2.01=",hoopPower[A, 2.01] // bf}

Figure 10 [S,Ex.33] demonstrates that two and three dimensional dot, cross, and wedge products are obtained by quaternion, octonion and Clifford algebra multiplication of restricted vectors. The restriction is that the scalar and multi-vector components are zero in the multiplicands. The dot product is then the scalar element of the product. The 2D &3D cross products are given by the multi-vectors. The nomenclature matches [4,p 11] for quaternions and CL2, and [4,p 37] for octonions and CL3. More examples are given in the supplementary material. All Clifford algebras are hoops. (The term “univector” is proposed for their basis-vectors, by analogy with bivector and trivector. Basis-vectors are confusingly called vectors or scalars by some authors.) Clifford algebra tables are created in [2] by using the procedure cl or by folding a specific group isomorph. Wedge (exterior) algebras are the bivector part of the Clifford product ([3, p.10]. They multiply univectors using Clifford algebra rules; the scalars and multivectors are restricted to zero in the multiplicands and various products are zero (i.e. these are Grassmann algebras). Finite Lie algebras appear to be Clifford hoops for which the real (scalar) elements have been zeroed. This has yet to be explored.

Figure 10.  Dot,  Cross  & Wedge products are restricted Hoops.

\({Use["Qr"]; "Quaternion" // 
        bf, \*"" hoopTimes[{0,(Subscript[e, 1]),(Subscript[e, 2]),0},{0,(Subscript[u, 1]),(Subscript[u, 2]),0}]\ngives"", 
      hoopTimes[{0, e\_1, e\_2, 0}, {0, u\_1, u\_2, 0}] // 
        bf, "
     Scalar  &   Dot Products", 
      Use["Octr"]; "
Octonion" // 
        bf, \*"" hoopTimes[{0,(Subscript[a, 1]),(Subscript[a, 2]),0,(Subscript[a, 3]),0,0,0},{0,(Subscript[b, 1]),(Subscript[b, 2]),0,(Subscript[b, 3]),0,0,0}]\ngives"", 
      hoopTimes[{0, a\_1, a\_2, 0, a\_3, 0, 0, 0}, {0, b\_1, b\_2, 0, b\_3, 
            0, 0, 0}] // 
        bf, "
Negated Scalar, components of Cross Product", "
Clifford univector products" // bf, " are Wedge Products", 
      Use["CL2"]; "
CL2 ", 
      hoopTimes[{0, u\_1, u\_2, 0}, {0, v\_1, v\_2, 0}] // bf, 
      Use["CL3"]; "
CL3 ", 
      hoopTimes[{0, u\_1, u\_2, 0, u\_3, 0, 0, 0}, {0, v\_1, v\_2, 0, v\_3, 
            0, 0, 0}] // bf, 
Use["CL4"]; "
CL4 ", 
      hoopTimes[{0, e\_1, e\_2, 0, e\_3, 0, 0, e\_4}, {0, u\_1, u\_2, 0, 
            u\_3, 0, 0, u\_4}] // bf}\)

Complex conjugation is an involution that replaces i by - i; multiplying a vector by its conjugate gives a real number (the conserved quadratic size) in complex, quaternion, and Octonion algebras. The other elements in the product are zero. Figure 11 [S,Ex.34] demonstrates that this also applies to the split-octonion hoop and to three Clifford Algebra hoops (though Clifford[3] gives a non-zero trivector element and the size is the sum of the squared terms). "Plex-conjugates" are generalizations of complex conjugation to other hoops. They are involutions that give size-related results on multiplication by the original vec. This is  demonstrated for C3; the second and third elements are interchanged to give the plex in hoops with ternary symmetry, corresponding to exchanging the signs J & (J^2). The sum and the signed sum of the product elements give the sizes (squared in the first case). C3C2 behaves similarly. Some plex-conjugates can be found easily; this has been used to discover the shapes of a few hoops with more than twelve elements.

Figure 11.  Some conjugate products.

{Use["Qr"]; "Quaternion conjugate is" // bf,plex // bf, "product with {a,b,c,d}= ",hoopTimes[{a, b, c, d}, plex]//bf,Use["Octr"]; "\nOctonion conjugate is" // bf,plex // bf, "product with {a,b,c,d,e,f,g,h}= ",hoopTimes[{a, b, c, d, e, f, g, h}, plex]//bf,Use["Octi"]; "\nSplit-octonion conjugate is" // bf,plex // bf, "product with {a,b,c,d,e,f,g,h}= ",hoopTimes[{a, b, c, d, e, f, g, h}, plex]//bf,
Use["CL2"]; "\nClifford[2] conjugate \
is" // bf,plex // bf, "product with {a,b,c,d}= ",hoopTimes[{a, b, c, d}, plex] // bf,
Use["CL3"]; "\nClifford[3] conjugate \
is" // bf,plex // bf, "product with {a,b,c,d,e,f,g,h}= ",hoopTimes[{a, b, c, d, e, f, g, h}, plex]//bf,Use["C3"]; "\nC3 conjugate is" // bf,plex // bf,"product with {a,b,c}=",res = hoopTimes[{a, b, c}, plex]; res // bf,"\nThis sums to the square of the first size",Factor [Plus @@ res] // bf,"\nThe (+,-,-) signed sum gives the quadratic size",Factor [res[[1]] - res[[2]] - res[[3]]] // bf,Use["C3C2"]; "\nC3C2 conjugate is" // bf,plex // bf, "product with {a,b,c,d,e,f}= ",Pr=hoopTimes[{a, b, c, d, e, f}, plex];Pr // bf,"\nThis sums to the square of the first size",Factor [Plus @@ Pr] // bf,"\nThe third and fourth sizes are Pr[[1]]-Pr[[2]]±Pr[[4]]∓Pr[[5]]",Expand[sh[[3]]] == 
            Expand[Pr[[1]] - Pr[[2]] + Pr[[4]] - Pr[[5]]] // 
          bf,Expand[sh[[4]]] == 
            Expand[Pr[[1]] - Pr[[2]] - Pr[[4]] + Pr[[5]]] // 
          bf}

Orbits are sub-algebras with the product of non-zero sizes multiplying to one; they have angle parameters that add on abelian multiplication, and products of orbits are orbits. This is demonstrated for two C3 orbits in Figure 12. [S,Fig.35]

Figure 12.  Products of Orbits are Orbits.

{Use["C3"]; "OC301 polar form is ", OC301 = {0, 1, τ}; 
      OC301 // bf, "with vector form", 
      v01 = tovec /. as[{0, 1, τ}, {a, b, c}]; 
      v01 // bf, "\nOC311 polar form is ", OC311 = {1, 1, ϕ}; 
      OC311 // bf, "with vector form", 
      v11 = tovec /. as[{1, 1, ϕ}, {a, b, c}]; 
      v11 // bf, "\nThe product vector form is ", 
      vv = Simplify[hoopTimes[v01, v11]]; vv // bf, "\nwith polar form ", 
      Simplify[topol /. as[vv]] // 
        bf, "\nMathematica fails to simplify to ", {1, 
          1, τ + ϕ} // bf}

4. Controversial concepts discussed in the supplementary material.

Summary.
	The supplementary material discusses further properties of Hoop Algebras that are not present in conventional algebras. Subtraction generalizes to “additive elimination”. Exponents of generalized signs define multi-phase sinusoids, and arise from sets of differential equations that allow resonances. Half-spin quantum operators are neither real nor complex. Dozal orbits have three radii that multiply to 1 and could define three size regimes - point particle size, Planck size, cosmic size. The unit velocity wave equation may create mass via velocity being split between spatial dimensions and Kaluza-Klein field directions. The 12-directional Schrodinger equation might describe particles as non-dispersing wave packets.

Negation "-a" and subtraction "b-a" are conflated and treated as a single fundamental process by most mathematicians, but [4] requires over a hundred pages of "Post-Modern Algebra" before subtraction can be tackled. In hoops, ordinary negation is the result of r=2 folding. This equivalences {(Subscript[a, i]),(Subscript[a, i + m/2])} to a real number (Subscript[a, i]) with {(Subscript[a, i]),(Subscript[a, i])} ~ -(Subscript[a, i]) and {(Subscript[a, i]),(Subscript[a, i])} ~ 0. The first size ∑(Subscript[a, i]) is destroyed, as it equivalences to zero. Subtraction can now be seen as "additive elimination", adding a set of coefficients that make all the terms of an un-folded vector identical, so that they equivalence to zero: {a,b}+{b,a} ={a+b,a+b} ~ 0. (The commutative property of addition is inherited from the natural numbers.)
	When r is 3, the equivalence relationship is (Subscript[t, j]) ~ {(Subscript[a, j]),(Subscript[a, j + m/3]),(Subscript[a, j + 2  m/3])} ~ {(Subscript[b, j]),(Subscript[b, j + m/3]),(Subscript[b, j + 2  m/3])} iff ((Subscript[a, j])+(Subscript[b, j + m/3]))=((Subscript[a, j + m/3])+(Subscript[b, j + 2  m/3]))=((Subscript[a, j + 2  m/3])+(Subscript[b, j])) . The "terplex number" {a,b,c} has a left negation (or rotation), {b,c,a} and a right negation {c,a,b}, with {a,b,c} +{b,c,a} +{c,a,b} ={a+b+c, a+b+c, a+b+c} ~0. A number is eliminated by adding the two rotations of its primal form. This extends to other r values; a number is eliminated by adding the r-1 rotations of a primal form. Simultaneous equations can be solved in this way in hoop algebras; this is demonstrated in Mathsource/4894, but suffers from multiple solutions.

Orbits are related to the interpretation of hoops as banded sets of differential equations with multi-phase cyclic solutions; resonances at multiples of the fundamental frequency may account for the resonances of particle physics. The function (Subscript[x, j])=c[t]+(∑_p)(Subscript[c, p]) Sin[2a Sin[2k (Subscript[f, p]) π/m] t+(Subscript[ϕ, p])+2  j k π/m]/(Subscript[f, p]) is a solution to 
D[(Subscript[x, j]), t] = c'[t] + 
            a ((Subscript[x, Mod[j + k, m]]) - (Subscript[x, Mod[j - k, m]])), 
        j = 0.  ..  m - 1, k < m/2.  Here  j is one of the m phases; k is a phase step; c[t] gives freedom from the origin; the integer (Subscript[f, p]) is a resonance number, being 1 for the fundamental wave; (Subscript[c, p]) and (Subscript[ϕ, p]) are phase amplitude and phase offset constants. The subscript p is needed because independent sets can arise. Sin[2k π/m] is a constant that determines the frequency & period.  When m=4, k must be 1; folding then removes the c[t] term and reproduces the ordinary sinusoid equation-pair with double the amplitude. Other values of m and k give a frequency reduction. The period is 2π/Sin[2k π/m]/(Subscript[f, p]), and the function is non-sinusoidal (constant) if 2/m is integral; m=12 gives half-spin with a period of 4π if (Subscript[f, p])=1. Spin-2 is (Sin^2) with period π; this suggests that the spin-2 sinusoids (gravity waves) are essentially non-negative. The relationships between orbit resonances and the resonances of particle physics has not been explored (Neither has that between branes and multi-radius orbits.)

The Dozal sizes with ternary symmetry could have three sizes that multiply to 1, one corresponding to a Planck area (P^2) and the others in a reciprocal relationship, two squared radii (r^2) and (R^2) corresponding to the dimensions of "pseudo-point" particles and the universe. p.r.R=1 gives three size regimes. More work is needed.

The basic wave equation in many dimensions gives a unit velocity, which is projected onto the different dimensions as velocity components. Massless particle waves would be restricted to space-like dimensions and would travel a the speed of light. Others would have velocities in Kaluza-Klein dimensions and their spatial velocities would be less than that of light.

Generalizing the Schrodinger equation to 12 directions (instead of the 4 implied by using i) might lead to a more general description of "information about particles", and a non-dispersing form (including non-linear terms) could describe stable particles.

5. Outstanding problems.

The author, as an amateur experimental mathematician who interprets Goëdel’s theorem as “Truth transcends Proof”, has only concerned himself with the demonstrability of hoop algebras and their properties. Rigorous analysis, based on primal numbers rather than real numbers, is required.
	The only known general “proof” that a table is conservative is by demonstration using random vectors; proof by factorization is only possible for small tables. Frobenius’s proof has not been extended to folded tables; it does not apply to general Moufang loops. Proof is needed for the observation that the Moufang property ensures that vecs are invertible.
	Non abelian algebras are not well understood. Some non-abelian hoops have "pseudo-roots" that may introduce uncertainty because their angles do not add on hoop multiplication. They have repeated quadratic roots that release extra degrees of freedom. D3 is the smallest non-commutative group. The supplementary material demonstrates that AB differs from BA in the D3 algebra; B & A are recovered by pre-multiplication by the appropriate inverse. The "pseudo-power" A\(TraditionalForm\`\^2\) differs from the repeated product, but the "pseudo-root" recovers A.
	Conservation and orbits are “emergent properties”; others can be expected when techniques are developed to find the shapes of larger hoops. Higher-order sizes may lead to new emergent properties. They may be difficult to identify - C3 & C4 have duals with only a single angle each, but several years of effort were needed before “L4”, a quartic size of C3C4, could be factorized and shown to define two orbits. A4 has one linear size, one quadratic size (giving an orbit) and a repeated cubic size that appears to discriminate between other types of Dozal orbit. This is a conserved symmetry; is there a related force or particle? Quartic sizes are common; if (like “L4”) they have a minimum of zero, they may be factorizable. A4 has one linear size, one quadratic size (giving an orbit) and a repeated cubic size that appears to discriminate between other types of Dozal orbit. This is a conserved symmetry; is there a related force or particle?
	The (complex) Schrödinger equation describes “information about particles” and undergoes dispersion. Replacing i by the 12th root of unity could lead to a Dozal description of non-dispersing (solitonic) particles with 4-phase leptons and {3,6,12}-phase hadrons.
	Orbit instability is a vague concept. It may be related to the occurrence of determinant factors that are larger than quadratic.
Dozal may eventually lead to descriptions of particles as stable multi-phase wave-packets in bosonic and leptonic fields. Their space-time description may involve much larger groups, possibly composed from the 12-element Dozal groups and the groups (derived from Pauli-sigma matrices) that conserve the Minkowski metric. There are no known techniques to find the non-linear conserved properties of such large groups.

6. Review.

Hoops provide a unified approach to many of the algebras that are relevant to physics. They introduce new concepts that transcend the limitations of real and complex fields. They are obtained by folding those Cayley tables that define multiplication for sets of unsigned (primal) coefficients, provided that they have the Frobenius conservation property for their symbolic table inverse determinant factors or “sizes”. Folding introduces generalized signs as symmetries. As these tables all have the Moufang division property, hoops are vector division algebras that project results into constrained sub-algebras whenever any sizes are zero. Remainders then have to be ejected to maintain the conservative property. Multi-phase polar-duals have been found for some hoops; these provide stable unital orbits in a few cases - possibly related to the limited number of stable particles. 12-element “dozal” algebras have some elements that act as half-spin quantum operators; the author conjectures that their 3, 4, 6, and 12-phase orbits are related to lepton and hadron wave-packets, and that three ternary orbits are related to M-theory dimensions and T-duality.
	Many problems remain to be solved before Hoops can provide the improved description of physics that is hinted at by their known properties.

7. Validation.

The self-explanatory examples should need no knowledge of Mathematica. Mathematica users can access the programmes and data, and test the examples with different data, by installing Hoops.nb in the Addons\ExtraPackages directory, and running it to create the Hoops.m package. This is written in elementary Mathematica, and includes “Usage” and (*comments*) explaining the details, so it should be readily comprehensible and easy to transcribe into other computer-aided-mathematics systems for independent validation.
	A larger package is available in [4]. It details the creation and properties of groups, loops, & hoops and provides further examples and a glossary.

8. References

[1] G. Frobenius, Uber die Primfactoren der Gruppendeterminante, Sitzungsber.Preuss. Akad. Wiss. Berlin Phys. Math,KL. 1896,(985-1021)
[2] J.D.H. Smith, A.B. Romanowska, Post-Modern Algebra, Wiley Interscience 1999.
[3]  http://library.wolfram.com/infocenter/MathSource/6198/  R.H.Beresford 2003-7.
[4]  http://library.wolfram.com/infocenter/MathSource/4894/  R.H.Beresford 2003-6.
[5]  H.-D Ebbinghaus et al, Numbers, Springer-Verlag N.Y. 1991.
[6] J.J.Hamilton, Hypercomplex numbers and the description of spin states, J.Math.Phys. 38(10) Oct. 1997. pp4914-4928.
[7] P. Lounesto, Clifford Algebras & Spinors 2nd ed., Cambridge University Press 2001.
[S] Supplementary material hoopSup.nb in [3].