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Kaos, Reality and the Kaotic Algebra

Robert G. Wallace

Abstract. Philosophical considerations suggest a dimensional structure of re-ality based on infinite subdivision of singularity via Cayley Dickson doubling.An algebra, labelled the Kaotic algebra, is constructed based on algebrasobtained by Cayley-Dickson doubling. It provides a natural basis for a non-associative geometry with links to the associative geometry of Cl(1, 10). Sub-algebras of the kaotic algebra have a phenomenology corresponding to thatof the fermions of the standard model.

Index

1. Introduction

2. Algebras used in theories of reality

3. The Kaotic Algebra

4. Dimensionality

5. The Kaotic Algebra and the standard model

6. Discussion

7. Acknowledgements

8. Appendix A - Products of unit elements of the kaotic algebra

9. Appendix B: Subalgebras with phenomenology corresponding to fermions

10. Appendix C: The Higgs Mechanism

11. References

2 Robert G. Wallace

1 Introduction

Philosophically, the structure of reality should be an elegant construction based onsimple geometry. However, the observed universe displays a lot of of asymmetry,and current models of its structure are based on odd combinations of dimension-ality, such as four plus six for string theory[1] or four plus seven for M-theory[2].

It would be more elegant for the dimensionality of reality to be a number such as0, 1, or ∞. A singularity at the origin of the universe would have zero dimension-ality. For this to evolve into a dimensionality greater than 1 and for that evolutionto stop at 11 seems arbitrary. It would be more satisfactory for that evolution toproceed ad infinitum. Infinite dimensionality is not apparent in reality. However,if the evolutionary process were to be one of subdivision rather than extension,then it could be thought of as generating a chaotic structure corresponding to aninfinitely fractal quantum foam, which is a feature of quantum theory. The struc-ture of reality as observed might then be found within that foam as a consequenceof properties of substructures with some orderly properties. Several models havebeen proposed featuring emergent spacetime[3].

The Cayley-Dickson construction[4] generates algebras of increasing size which losealgebraic properties as it generates complex numbers from real numbers, quater-nions from complex numbers, octonions from quaternions, and sedenions fromoctonions. After the sedenions there is no further loss of algebraic properties. Infi-nite repetition of the Cayley-Dickson construction could correspond to generationof a chaotic structure corresponding to an infinitely fractal quantum foam. Realitymight then be found within that foam as a consequence of properties of its real,complex, quaternion and octonion subalgebras which have some orderly properties.

Given the T-duality property of string theories, subdivision into an infinitely frac-tal foam may be mathematically equivalent to divergence into infinitely manyworlds. Philosophically, there is a difference, as an infinitely divided singularitywould constitute a more elegant conceptual basis for reality than an infinitely di-verging 11 dimensional manifold.

2 Algebras used in theories of reality

Theories of reality have been based on various algebras.

General relativity[5] is based on four-dimensional space-time, which can be rep-resented by the Clifford algebras Cl(3, 1) or Cl(1, 3). It accounts for gravitationat extensive scales, but does not account for other forces or quantum effects. Itallows the possibility of a scalar field.

Kaos, Reality and the Kaotic Algebra 3

The original Kaluza-Klein model[6] is based on five dimensional space-time, whichcan be represented by the Clifford algebra Cl(1, 4) or Cl(4, 1). The Kaluza-Kleinmodel unifies general relativity with classical electromagnetism, but fails to ac-count for quantum effects.

Yang-Mills theories[7] describe electroweak and strong forces, but not gravity. TheHiggs mechanism[8] features a complex doublet.

String theories extend the Kaluza-Klein model by adding extra dimensions, usinga background of one time-like dimension, three spatial dimensions and six com-pactified dimensions. For M theory an extra dimension is added. Ten and elevendimensions can be represented by the Clifford algebras Cl(p, q), p+ q = 10 or 11.

An octonion based approach using the tensor product C⊗H⊗O has been proposedby G.Dixon[9]. It has been suggested that the non-associative elements of O couldaccount for the probabilistic nature of quantum mechanics. It has been noted thatthere is a correspondence between Feynman vertices[10] which feature two spinorsand a vector component, and the way in which a higher division algebra can beassembled from two copies of a division algebra and a third component[11].

Consider a sequence:

1. From a singularity, generate handings of reals - positive and negative, asso-ciate them with a scalar field.

2. From the reals, generate handings of complex numbers, associate them withthe complex doublet in the Higgs mechanism.

3. From the complex numbers, generate handings of quaternions, whose productis isomorphic to the the Clifford algebra Cl(3, 1), associate it with space-time.

4. From the quaternions generate handings of octonions, associate their productwith the compactified dimensions of string and M-theories.

5. From the octonions generate handings of sedenions, associate them withevents described by vertices in Feynman diagrams.

6. Associate further Cayley Dickson algebras with virtual particles and quantumfoam.

This suggests a model based on a chaotic background represented by an algebraobtained by infinitely repeated Cayley-Dickson constructions, or kaos (naming itafter the primordial void of greek mythology). Observed features of reality - theHiggs field, space-time, particles and events, would then be represented by sub-algebras with more orderly properties such as commutativity, associativity andalternativity.

In this paper an algebra with these qualities, the kaotic algebra, is presented. Ithas been assembled using a modified form of Moufang Loop construction[12].

4 Robert G. Wallace

3 The kaotic algebra

The kaotic algebra is constructed using a modified Moufang loop construction. Ifapplied to a quaternion quartet, it generates a sedenion. When applied to unitmultivector matrices for Cl(1, 4), it generates the Kaotic algebra.

3.1 Moufang Loop construction for octonions

For the Cayley-Dickson construction, octonions are represented by quaternionpairs, where, for p = (a, b), q = (c, d), the product rule is:pq = (ac− db∗, a∗d+ cb) or, equivalently: pq = (ac− d∗b, da+ bc∗)For Moufang loop construction, a dis-association operator, µ, is assigned to thesecond quaternion pair, and the product rule is, for p = (p, µp) and q = (q, µq):p.q = (pq)p.µq = µ(p−1q)µp.q = µ(qp)µp.µq = −(qp−1)

3.2 Modified Moufang Loop construction for sedenions

For sedenions, a similar construction can be assembled, based on quaternion quar-tets. Dis-association operators, µ, ν and λ, are assigned to three of the quaternions,with rules imposed for products of p = (p, µp, νp, λp) and (q, µq, νq, λq) as follows:p.q = (pq)p.µq = µ(p−1q)p.νq = ν(p−1q)p.λq = λ(p−1q)µp.q = µ(qp)νp.q = ν(qp)λp.q = λ(qp)µp.µq = −(qp−1)νp.νq = −(qp−1)λp.λq = −(p−1q)µp.λq = ν(pq)µp.νq = −λ(pq)νp.µq = λ(qp)νp.λq = −µ(qp)λp.νq = µ(pq)λp.µq = −ν(qp)These rules generate anti-commuting products whose sign matrix is a Hadamardmatrix, as is required for Cayley-Dickson sedenions. Note that there are differencesbetween the rules for µ, ν and λ.

A feature of this construction is that, if the signs of products with dissimilar dis-association operators are reversed, the sign matrix is not a Hadamard matrix.


3.3 Construction of the kaotic algebra

To generate the table of products for the Kaotic algebra, the multiplication rulesused in the modified Moufang Loop construction for sedenions are applied to the32 real and imaginary 4× 4 unit matrices for unit multivector components for theClifford algebra Cl(1, 4) factored by 1, µ, ν, or λ, generating an algebra with 128unit elements.

3.4 Notation used to label basis elements for the kaotic algebra

In order to allow compact presentation basis elements, letters have been used tolabel unit 4× 4 matrices as shown in table 1. A prescript is added to denote a dis-association operator and i is added for imaginary matrices, resulting in labels suchas: νiX. When referring to several unit matrices with a common dis-associationoperator, and/or common real/imaginary status, matrices are enclosed in squarebrackets, e.g.: ν [iXiY iZT ] and λi[PQFD].

Table 1. Notation used to label 4× 4 unit matrices

S =

1 0 0 0

0 1 0 00 0 1 0

0 0 0 1

R =

0 0 1 0

0 0 0 1

1 0 0 00 1 0 0

P =

1 0 0 0

0 1 0 0

0 0 −1 00 0 0 −1

M =

0 0 1 0

0 0 0 1

−1 0 0 00 −1 0 0

Y =

0 1 0 01 0 0 0

0 0 0 1

0 0 1 0

E =

1 0 0 00 −1 0 0

0 0 1 0

0 0 0 −1

T =

0 1 0 0−1 0 0 0

0 0 0 1

0 0 −1 0

D =

0 0 0 1

0 0 1 00 1 0 0

1 0 0 0

X =

0 0 1 0

0 0 0 −11 0 0 0

0 −1 0 0

N =

0 0 0 1

0 0 −1 00 1 0 0

−1 0 0 0

F =

0 1 0 0

1 0 0 0

0 0 0 −10 0 −1 0

Z =

1 0 0 0

0 −1 0 0

0 0 −1 00 0 0 1

L =

0 1 0 0

−1 0 0 0

0 0 0 −10 0 1 0

U =

0 0 0 1

0 0 1 00 −1 0 0

−1 0 0 0

V =

0 0 1 00 0 0 −1

−1 0 0 0

0 1 0 0

Q =

0 0 0 10 0 −1 0

0 −1 0 0

1 0 0 0

Note that the positive and negative forms of matrices R,Q,L,X, Y, Z are oppositeto those used in a previous paper by this author, The Pattern of Reality[13], forconsistency with a convention that matrices with a +1 entry in the first row arethe positive form.

6 Robert G. Wallace

4 The Kaotic algebra and Clifford algebras

4.1 Dimensionality

For Clifford algebras, the concept of dimensionality is clear cut, with unit vec-tors assigned to orthogonal directions for each dimension. The unit vectors allanti-commute with each other. For M-theory, if its eleven dimensions are sepa-rated into a Cl(1,3) world manifold and a seven dimensional compactified tangentbundle, the compactified dimensions have to have distinct unit vectors for an asso-ciated Clifford algebra, so Cl(1, 10) is used as the algebra. A Cl(1, 10) multivectorhas 2048 components.

For the kaotic algebra, dimensionality is less clear cut. Sets of four unit elementsthat have no dis-association operator, and which anti-commute with each othercan be found. But, also, sets of seven unit elements that have dis-association op-erators and which anti-commute with each other can be found, such that theyall anti-commute with four elements having no dis-association operator. However,having only 128 unit components, the equivalent of a multivector for the kaoticalgebra should only be equivalent to a Clifford algebra for seven dimensions.

This suggests the hypothesis that, for a model based on the kaotic algebra, theequivalent of the compactified dimensions in M-theory would not be true dimen-sions, but could be thought of as subdivisions of dimensions, or pseudo-dimensions,allowing multivector components equivalent to unit vectors for true dimensions tobe products of three or more unit vector equivalents for pseudo-dimensions. Thiscould provide a basis for a model based on emergent spacetime, a feature of severalattempts to find a fundamental basis for reality[3].

In order to relate physics based on the kaotic algebra to physics based on Cl(1, 10),it is useful to map the true dimensions of a Cl(1, 10) manifold into a combinationof true and pseudo-dimensions for a kaotic algebra manifold. To describe this map,a compact form of notation for 32 × 32 matrices used to represent unit elementsof a Cl(1, 10) multivector is needed.

4.2 Notation for Cl(1, 10) unit matrices

32×32 unit matrices can be assembled as a nesting of a group of 2×2 unit matricesentered into a group of 4 × 4 unit matrices entered into a second group of 4 × 4unit matrices. In this paper 4 × 4 unit matrices are labelled as shown in table 1(in section 3), and 2× 2 matrices are labelled:

s =

[1 00 1

]g =

[1 00 −1

]h =

[0 11 0

]j =

[0 1−1 0

]


This allows a real 32 × 32 matrix to be labelled using a three letter combinationsuch as: gYX, and an imaginary 32× 32 matrix to be labelled using a four lettercombination such as: hDiF . Using this notation, products for two matrices can beobtained by multiplying the nested matrices separately. For instance:gYX ×hD F =g.h Y.D X.FAs for kaotic algebra labels, sets of several matrices with similar nested extensionmatrices can be put in square brackets, such as isS [TXY Z].

4.3 Matrices chosen to represent unit vectors for Cl(1, 10)

sS [iV TXY Z] all anticommute with each other. If sS [TXY Z] are assigned to rep-resent unit vectors for space-time dimensions, [iV ] could be assigned to the fifthdimension used in the original Kaluza-Klein theory. Then, to expand that theoryto eleven dimensions for use in M-theory, it would be logical to select eleven ma-trices to represent unit vectors such as:sS [TXY Z] g[TUV ]V h[LMN ]V ] jSiVThen:sS [TXY Z] can be assigned to space-time dimensions,andg[TUV ]V h[LMN ]V ] jSiV can be assigned to compactified dimensions.

4.4 Mapping between Cl(1, 10) and the kaotic algebra

Space-time dimensions can be regarded as being generated using H⊗H, the prod-uct of a right and a left handed quaternion. This suggests using the product of aright and a left handed octonion to generate compactified dimensions. Considerunit elements from the kaotic algebra such as:µ[TUV ] ν [LMN ] λiSIf sign is ignored, the products for these elements have the same latin square asfor g[TUV ]V h[LMN ]V ] jSiV

This suggests that an equivalent of the Clifford algebra multivector for the com-pactified dimensions of M-theory can be identified for the kaotic algebra elements.

The algebra multivector components for g[LVMV NV ] h[TV UV V V ] ijSV are:Scalar: SSVector: g[TUV ]V , h[LMN ]V , jSiV

Bivector: [TUV LMN ]V , j[XY ZPQRDEF ]V , h[TUV ]iV , g[LMN ]iV

3-vector: j[TUV LMN ]iV , g[SXY ZPQRDEF ]V , h[SXY ZPQRDEF ]V , [XY ZPQRDEF ]iV

4-vector: j[TUV LMN ]V , g[SXY ZPQRDEF ]iV , h[SXY ZPQRDEF ]iV , [XY ZPQRDEF ]V

5-vector: [TUV LMN ]iV , j[XY ZPQRDEF ]iV , h[TUV ]V , g[LMN ]V

6-Vector: g[TUV ]iV , h[LMN ]iV , jSV

Pseudoscalar: SiV

8 Robert G. Wallace

Equivalents of Cl(0, 7) unit multivector components for the kaotic algebra ele-ments can be identified as:Scalar: SVector: µ[TUV ], ν [LMN ], λiS

Bivector: [TUV LMN ], λ[XY ZPQRDEF ], νi[TUV ], µi[LMN ]

3-vector: λi[TUV LMN ], µ[SXY ZPQRDEF ], ν [SXY ZPQRDEF ], i[XY ZPQRDEF ]

4-vector: λ[TUV LMN ], µi[SXY ZPQRDEF ], νi[SXY ZPQRDEF ], [XY ZPQRDEF ]

5-vector: i[TUV LMN ], λi[XY ZPQRDEF ], ν[TUV ], µ[LMN ]

6-Vector: µi[TUV ], µi[LMN ], λS

Pseudoscalar: iS

5 The kaotic algebra and the standard model

For the equivalents of Cl(0, 7) unit multivector components for the kaotic algebraelements as identified above, real elements are the equivalent of Cl(0, 6) productsfor the six elements: µ[TUV ] ν [LMN ], and the remaining elements are their imag-inary counterparts. This allows representation of phase.

For the products of the six elements µ[TUV ] ν [LMN ], the even components are:Scalar: SBivector: [TUV LMN ], λ[XY ZPQRDEF ]

4-vector: λ[TUV LMN ], [XY ZPQRDEF ]

6-vector: λSThese exclude two of the disassociation operators.This suggests consideration of similar arrangements:Scalar: SBivector: [TUV LMN ], µ[XY ZPQRDEF ]

4-vector: µ[TUV LMN ], [XY ZPQRDEF ]

6-vector: µSandScalar: SBivector: [TUV LMN ], ν [XY ZPQRDEF ]

4-vector: ν [TUV LMN ], [XY ZPQRDEF ]

6-vector: νSA combination such as SLMNµ[SLMN ] or SLMNν [SLMN ] is octonionic, but acombination such as SLMNλ[SLMN ] is not. However, subalgebras of SLMNλ[SLMN ]such as SLλ[SL] are quaternionic.

A combination such as SLMN, µ[SLMN ], ν [SLMN ], λ[SLMN ] is sedenionic.It can represent a combination of two octonions with [SLMN ] elements in com-mon - SLMNµ[SLMN ] and SLMNν [SLMN ] together with a non-octonionicsubalgebra - SLMNλ[SLMN ] assembled as a combination of two quaternions -


SLλ[SL]×SMλ[SM ]. This would allow a sedenion to represent a vertex in a Feyn-man diagram, featuring two spinorial components and one component assembledfrom two vectorial components.

Having related elements of the kaotic algebra to multivector components for anassociative compactified manifold, to assemble a model of reality requires the al-gebra to be also be related to a Cl(1, 3) or Cl(3, 1) algebra for space-time. Oneway to do this would be to assemble a tensor product of the kaotic algebra withCl(1, 3) or Cl(3, 1). An alternative scheme would be to redefine the concept ofdimensionality, such that the non-associative manifold is one of sub-divided orpseudo-dimensions, and postulating that products of pseudo-dimensional vectorsgenerate true dimensions for an emergent space-time.

For this scheme, unit elements of the kaotic algebra such as [V TiXiY iZ] or[MNiXiPiD] can be found, all of which anti-commute with each other and anti-commute with all elements of the non-associative manifold. However, these ele-ments have signature (− − − − −), so, to assign an element to time requires athe imaginary counterpart of one element to be used, such as for [iT ].

For [iT iXiY iZ], iX, iY and iZ all anti-commute with all elements of the non-associative manifold, but iT commutes with all elements of the non-associativemanifold. One effect of this would be to create an arrow of time with respect tophenomena associated with the non-associative manifold.

The choice of elements to represent space-time such as [iT iXiY iZ] is arbitrary,[iNiXiP iD] would serve equally well. However, once that choice is made, it breaksa symmetry of the kaotic algebra, as the relationship to [iT iXiY iZ] for the octo-nion (SLMN µ[SLMN ]) is a different to that for the octonion (STUV µ[STUV ]).This symmetry breakage is equivalent to that used to associate permutations ofsubalgebras of Cl(p, q), p + q = 5 with fermions described in a previous paper,The Pattern of Reality[13], a rearranged version of which is shown in Appendix B,table 11. The same permutations of subalgebras can be extended into octonioniccombinations from the kaotic algebra to create a similar phenomenology.

The feature of the kaotic algebra, that, if the signs of products with dissimilardis-association operators are reversed, the sign matrix is not a Hadamard matrixcould account for the matter/anti-matter asymmetry observed in the universe.

10 Robert G. Wallace

6 Discussion

6.1 Introduction

Assembly and analysis of grand unification theories requires an understanding ofcomplex mathematics and physics. However, the goal of such theories is to finda relatively simple algebra at their heart. Physicists have searched exhaustivelyfor that algebra with limited success, suggesting that there may be some elementof conventional physics that misdirects the search. This paper presents the resultof an attempt to find an algebra that achieves the goal by naively focussing onalgebraic patterns, reducing the possibility of any such misdirection. The result,the kaotic algebra, appears to have features that make it a candidate for the algebraof reality. A simplistic analysis of these features is presented in this paper. Furtherwork is required, but if the kaotic algebra is fundamental, it will serve sciencebetter for it to be brought to the attention of the physics community now, ratherthan in a more rigorous way in the distant future.

6.2 Development of the Kaotic algebra

Twenty years after attending university, and twenty years ago, recollection of aremark made by a university friend about the dimensionalities which allow a crossproduct, aroused my curiousity. That led me to an article by A. Eddington[14],which related the metric properties of spacetime to those of a group.

For that group, I assigned letters to matrices representing unit multivector com-ponents for Cl(3, 1) relating them to space-time dimensions, and arranged theirmultiplication table in sets of anti-commuting pentads as shown in table 2. Thistable suggested association of one pentad with gravity, two with electroweak forces,and three with the strong nuclear force.

Table 2. 4× 4 unit matrix products

S V T X Y Z

S S V T X Y Z

V V −S U P Q −RT T −U −S −D E −FX X −P D S N −MY Y −Q −E −N S −LZ Z R F M L S

The description of the electron using even components of the multivector forCl(3, 1) by D. Hestenes[15] suggested searching for a pattern corresponding tothe phenomenology of fermions for sub-algebras of higher Clifford algebras. Thissuggested the scheme of symmetry and symmetry breakage described in a paper,The Pattern of Reality[13].


That paper identified a pattern of subalgebras of Cl(p, q), p+q = 5 as correspond-ing to the phenomenology of the fermions of the standard model together withhypothetical fermions for dark matter. The absence of observation of dark matterfermions made assessment of the validity of the correspondence problematic, asthe known components only constitute a part of the pattern. For further devel-opment, I attempted to find a geometric basis for the Higgs mechanism. In thisendeavour, I have found some properties of matrices that generate an expressionwith the form of the Higgs potential, as set out in Appendix C, but have yet torelate it to the kaotic algebra.

Thinking about bosons led me to consider how to model their interactions withfermions. The nature of protons and neutrons, being combinations of three quarks,suggested investigating combinations of 8-dimensional algebras, such as the octo-nions. The Moufang loop construction enables assembly of octonions using matrixrepresentations of quaternions together with dis-association operators. Genera-tion of a matrix representation of unit multivectors for spacetime from the directproduct of matrix representations of left and right handed quaternions suggestedseeking a similar process for octonions. Octonions are not associative so a groupdirect product is not defined. One possible product is the tensor product of twooctonions, the octo-octonions, but their algebra does not include that of sedenions,which I had been hoping to use to represent combinations of three quarks.

This focussed my efforts on trying to find a way to generate sedenions from oc-tonions using matrix representations of quaternions and dis-association operators,using direct calculation of matrix products for different multiplication rules ona spreadsheet, searching for the right commutation properties for products, andchecking that the sign matrix for products was a Hadamard matrix.

After a great deal of trial and error, I found the construction described in section3. Using it, a sedenion is generated from two octonions with a common quater-nionic subalgebra, together with a third 8 dimensional algebra having the samequaternionic subalgebra that is not octonionic. I realised that it made more senseto have a sedenion represent two fermions and a boson combining at a Feynmanvertex than to have it represent a proton or neutron.

Applying the construction to matrices representing unit multivector componentsof Cl(1, 4) resulted in the kaotic algebra. Its manner of construction and numberof elements (128) suggests a relationship with the octonionic projective plane OP2.The differences between the product rules for the three dis-association operatorscreates the possibility of symmetry breakage resulting in the distinction betweenfermions and bosons. The handing of the multiplication rule for elements withdissimilar dis-association operators accounts for the handedness of the universe.


For the kaotic algebra, matrices can be assigned to represent unit vectors for space-time, but the matrix assigned to represent time commutes with matrices assignedto represent compactified dimensions. This would have the effect of creating a dif-ference between positive time and negative time for phenomena associated withcompact dimensions, providing a basis for the arrow of time. The choice of matri-ces to be associated with space-time creates differences between subalgebras whichcorresponds to symmetry breakage distinguishing the electroweak forces from thestrong nuclear force.

In parallel with the search for an algebra incorporating sedenions, I was also lookingat Clifford algebras for 10 and 11 dimensional space-times. The form of notationadopted links the Kaotic algebra with Clifford algebras in a way that suggeststhat the kaotic algebra can be compatible with many of the results obtained usingstring/M-theories.

Being no mathematician or physicist, my analysis of the correspondence betweenthe kaotic algebra and reality is likely to be flawed, but provides an indication ofthe potential of the algebra. It is presented in the hope that others will be able touse it to develop a viable theory of reality.

7 Acknowledgements

Many papers available on arXiv and elsewhere on the internet contributed to thedevelopment of this paper, but most use higher mathematical methods, whichmake them less accessible for a non-mathematician and making it hard to ac-knowledge particular contributions. I thank Wikipedia and its contributors andJohn Baez for information presented on the internet on a more accessible level.


8 Appendix A - Products of unit elements of the kaotic algebra

An overview of the commutation properties of products of the kaotic algebra isshown in condensed form in figure 1, where products that anti-commute have anA in their entry. The multiplication table for the kaotic algebra is shown in tables3 to 10.

Figure 1. Pattern of anticommuting products for the kaotic algebraA A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A

A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A














A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A

A A A A A A A A A A A A A A A A
















































































































Table 3. Kaotic Algebra multiplication table, part A1

S L M N T U V iX iY iZ iP iQ iR iD iE iF iS iL iM iN iT iU iV X Y Z P Q R D E F

S +S +L +M +N +T +U +V +iX +iY +iZ +iP +iQ +iR +iD +iE +iF +iS +iL +iM +iN +iT +iU +iV +X +Y +Z +P +Q +R +D +E +F

L +L −S +N −M −P +X −D −iU +iZ −iY +iT −iR +iQ +iV −iF +iE +iL −iS +iN −iM −iP +iX −iD −U +Z −Y +T −R +Q +V −F +E

M +M −N −S +L +Q −Y −E +iZ +iU −iX −iR −iT +iP +iF +iV −iD +iM −iN −iS +iL +iQ −iY −iE +Z +U −X −R −T +P +F +V −DN +N +M −L −S −R −Z +F −iY +iX +iU −iQ +iP +iT +iE −iD −iV +iN +iM −iL −iS −iR −iZ +iF −Y +X +U −Q +P +T +E −D −VT +T −P +Q −R −S +V −U −iD +iE −iF +iL −iM +iN +iX −iY +iZ +iT −iP +iQ −iR −iS +iV −iU −D +E −F +L −M +N +X −Y +Z

U +U +X −Y −Z −V −S +T −iL +iM +iN −iD +iE +iF +iP −iQ −iR +iU +iX −iY −iZ −iV −iS +iT −L +M +N −D +E +F +P −Q −RV +V −D −E +F +U −T −S +iP +iQ −iR −iX −iY +iZ +iL +iM −iN +iV −iD −iE +iF +iU −iT −iS +P +Q −R −X −Y +Z +L +M −NiX +iX −iU −iZ +iY +iD −iL −iP −S −N +M +V +F −E −T −R +Q −X +U +Z −Y −D +L +P +iS +iN −iM −iV −iF +iE +iT +iR −iQiY +iY −iZ +iU −iX −iE +iM −iQ +N −S +L −F +V −D −R +T −P −Y +Z −U +X +E −M +Q −iN +iS −iL +iF −iV +iD +iR −iT +iP

iZ +iZ +iY +iX +iU +iF +iN +iR −M −L −S −E −D −V −Q −P −T −Z −Y −X −U −F −N −R +iM +iL +iS +iE +iD +iV +iQ +iP +iT

iP +iP +iT +iR +iQ +iL +iD +iX −V −F −E −S −N −M −U −Z −Y −P −T −R −Q −L −D −X +iV +iF +iE +iS +iN +iM +iU +iZ +iY

iQ +iQ +iR −iT −iP −iM −iE +iY +F −V −D +N −S −L −Z +U +X −Q −R +T +P +M +E −Y −iF +iV +iD −iN +iS +iL +iZ −iU −iXiR +iR −iQ −iP +iT +iN −iF −iZ −E −D +V +M +L −S −Y −X +U −R +Q +P −T −N +F +Z +iE +iD −iV −iM −iL +iS +iY +iX −iUiD +iD +iV −iF −iE −iX −iP +iL +T −R −Q +U −Z −Y −S +N +M −D −V +F +E +X +P −L −iT +iR +iQ −iU +iZ +iY +iS −iN −iMiE +iE +iF +iV +iD +iY +iQ +iM −R −T −P −Z −U −X −N −S −L −E −F −V −D −Y −Q −M +iR +iT +iP +iZ +iU +iX +iN +iS +iL

iF +iF −iE +iD −iV −iZ +iR −iN +Q −P +T −Y +X −U −M +L −S −F +E −D +V +Z −R +N −iQ +iP −iT +iY −iX +iU +iM −iL +iS

iS +iS +iL +iM +iN +iT +iU +iV −X −Y −Z −P −Q −R −D −E −F −S −L −M −N −T −U −V +iX +iY +iZ +iP +iQ +iR +iD +iE +iF

iL +iL −iS +iN −iM −iP +iX −iD +U −Z +Y −T +R −Q −V +F −E −L +S −N +M +P −X +D −iU +iZ −iY +iT −iR +iQ +iV −iF +iE

iM +iM −iN −iS +iL +iQ −iY −iE −Z −U +X +R +T −P −F −V +D −M +N +S −L −Q +Y +E +iZ +iU −iX −iR −iT +iP +iF +iV −iDiN +iN +iM −iL −iS −iR −iZ +iF +Y −X −U +Q −P −T −E +D +V −N −M +L +S +R +Z −F −iY +iX +iU −iQ +iP +iT +iE −iD −iViT +iT −iP +iQ −iR −iS +iV −iU +D −E +F −L +M −N −X +Y −Z −T +P −Q +R +S −V +U −iD +iE −iF +iL −iM +iN +iX −iY +iZ

iU +iU +iX −iY −iZ −iV −iS +iT +L −M −N +D −E −F −P +Q +R −U −X +Y +Z +V +S −T −iL +iM +iN −iD +iE +iF +iP −iQ −iRiV +iV −iD −iE +iF +iU −iT −iS −P −Q +R +X +Y −Z −L −M +N −V +D +E −F −U +T +S +iP +iQ −iR −iX −iY +iZ +iL +iM −iNX +X −U −Z +Y +D −L −P +iS +iN −iM −iV −iF +iE +iT +iR −iQ +iX −iU −iZ +iY +iD −iL −iP +S +N −M −V −F +E +T +R −QY +Y −Z +U −X −E +M −Q −iN +iS −iL +iF −iV +iD +iR −iT +iP +iY −iZ +iU −iX −iE +iM −iQ −N +S −L +F −V +D +R −T +P

Z +Z +Y +X +U +F +N +R +iM +iL +iS +iE +iD +iV +iQ +iP +iT +iZ +iY +iX +iU +iF +iN +iR +M +L +S +E +D +V +Q +P +T

P +P +T +R +Q +L +D +X +iV +iF +iE +iS +iN +iM +iU +iZ +iY +iP +iT +iR +iQ +iL +iD +iX +V +F +E +S +N +M +U +Z +Y

Q +Q +R −T −P −M −E +Y −iF +iV +iD −iN +iS +iL +iZ −iU −iX +iQ +iR −iT −iP −iM −iE +iY −F +V +D −N +S +L +Z −U −XR +R −Q −P +T +N −F −Z +iE +iD −iV −iM −iL +iS +iY +iX −iU +iR −iQ −iP +iT +iN −iF −iZ +E +D −V −M −L +S +Y +X −UD +D +V −F −E −X −P +L −iT +iR +iQ −iU +iZ +iY +iS −iN −iM +iD +iV −iF −iE −iX −iP +iL −T +R +Q −U +Z +Y +S −N −ME +E +F +V +D +Y +Q +M +iR +iT +iP +iZ +iU +iX +iN +iS +iL +iE +iF +iV +iD +iY +iQ +iM +R +T +P +Z +U +X +N +S +L

F +F −E +D −V −Z +R −N −iQ +iP −iT +iY −iX +iU +iM −iL +iS +iF −iE +iD −iV −iZ +iR −iN −Q +P −T +Y −X +U +M −L +S

νS +νS +νL +νM +νN +νT +νU +νV +ν iX +ν iY +ν iZ +ν iP +ν iQ +ν iR +ν iD +ν iE +ν iF +ν iS +ν iL +ν iM+ν iN +ν iT +ν iU +ν iV +νX +νY +νZ +νP +νQ +νR +νD +νE +νF

νL +νL −νS −νN +νM −νP +νX −νD −ν iU −ν iZ +ν iY +ν iT +ν iR −ν iQ +ν iV +ν iF −ν iE +ν iL −ν iS −ν iN+ν iM −ν iP +ν iX −ν iD −νU −νZ +νY +νT +νR −νQ +νV +νF −νEνM +νM +νN −νS −νL +νQ −νY −νE −ν iZ +ν iU +ν iX +ν iR −ν iT −ν iP −ν iF +ν iV +ν iD+ν iM+ν iN −ν iS −ν iL +ν iQ −ν iY −ν iE −νZ +νU +νX +νR −νT −νP −νF +νV +νD

νN +νN −νM +νL −νS −νR −νZ +νF +ν iY −ν iX +ν iU +ν iQ −ν iP +ν iT −ν iE +ν iD −ν iV +ν iN−ν iM +ν iL −ν iS −ν iR −ν iZ +ν iF +νY −νX +νU +νQ −νP +νT −νE +νD −νVνT +νT −νP +νQ −νR −νS −νV +νU +ν iD −ν iE +ν iF +ν iL −ν iM+ν iN −ν iX +ν iY −ν iZ +ν iT −ν iP +ν iQ −ν iR −ν iS −ν iV +ν iU +νD −νE +νF +νL −νM +νN −νX +νY −νZνU +νU +νX −νY −νZ +νV −νS −νT −ν iL +ν iM+ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR +ν iU +ν iX −ν iY −ν iZ +ν iV −ν iS −ν iT −νL +νM +νN +νD −νE −νF −νP +νQ +νR

νV +νV −νD −νE +νF −νU +νT −νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ +ν iL +ν iM−ν iN +ν iV −ν iD −ν iE +ν iF −ν iU +ν iT −ν iS −νP −νQ +νR +νX +νY −νZ +νL +νM −νNν iX +ν iX −ν iU +ν iZ −ν iY −ν iD −ν iL +ν iP −νS +νN −νM −νV +νF −νE +νT −νR +νQ −νX +νU −νZ +νY +νD +νL −νP +ν iS −ν iN+ν iM +ν iV −ν iF +ν iE −ν iT +ν iR −ν iQν iY +ν iY +ν iZ +ν iU +ν iX +ν iE +ν iM +ν iQ −νN −νS −νL −νF −νV −νD −νR −νT −νP −νY −νZ −νU −νX −νE −νM −νQ +ν iN +ν iS +ν iL +ν iF +ν iV +ν iD +ν iR +ν iT +ν iP

ν iZ +ν iZ −ν iY −ν iX +ν iU −ν iF +ν iN −ν iR +νM +νL −νS −νE −νD +νV −νQ −νP +νT −νZ +νY +νX −νU +νF −νN +νR −ν iM −ν iL +ν iS +ν iE +ν iD −ν iV +ν iQ +ν iP −ν iTν iP +ν iP +ν iT −ν iR −ν iQ +ν iL −ν iD −ν iX +νV −νF −νE −νS +νN +νM +νU −νZ −νY −νP −νT +νR +νQ −νL +νD +νX −ν iV +ν iF +ν iE +ν iS −ν iN−ν iM −ν iU +ν iZ +ν iY

ν iQ +ν iQ −ν iR −ν iT +ν iP −ν iM +ν iE −ν iY +νF +νV −νD −νN −νS +νL −νZ −νU +νX −νQ +νR +νT −νP +νM −νE +νY −ν iF −ν iV +ν iD +ν iN +ν iS −ν iL +ν iZ +ν iU −ν iXν iR +ν iR +ν iQ +ν iP +ν iT +ν iN +ν iF +ν iZ −νE −νD −νV −νM −νL −νS −νY −νX −νU −νR −νQ −νP −νT −νN −νF −νZ +ν iE +ν iD +ν iV +ν iM +ν iL +ν iS +ν iY +ν iX +ν iU

ν iD +ν iD +ν iV +ν iF +ν iE +ν iX +ν iP +ν iL −νT −νR −νQ −νU −νZ −νY −νS −νN −νM −νD −νV −νF −νE −νX −νP −νL +ν iT +ν iR +ν iQ +ν iU +ν iZ +ν iY +ν iS +ν iN+ν iM

ν iE +ν iE −ν iF +ν iV −ν iD −ν iY −ν iQ+ν iM −νR +νT −νP −νZ +νU −νX +νN −νS +νL −νE +νF −νV +νD +νY +νQ −νM +ν iR −ν iT +ν iP +ν iZ −ν iU +ν iX −ν iN +ν iS −ν iLν iF +ν iF +ν iE −ν iD −ν iV +ν iZ −ν iR −ν iN +νQ −νP −νT −νY +νX +νU +νM −νL −νS −νF −νE +νD +νV −νZ +νR +νN −ν iQ +ν iP +ν iT +ν iY −ν iX −ν iU −ν iM +ν iL +ν iS

ν iS +ν iS +ν iL +ν iM+ν iN +ν iT +ν iU +ν iV −νX −νY −νZ −νP −νQ −νR −νD −νE −νF −νS −νL −νM −νN −νT −νU −νV +ν iX +ν iY +ν iZ +ν iP +ν iQ +ν iR +ν iD +ν iE +ν iF

ν iL +ν iL −ν iS −ν iN+ν iM −ν iP +ν iX −ν iD +νU +νZ −νY −νT −νR +νQ −νV −νF +νE −νL +νS +νN −νM +νP −νX +νD −ν iU −ν iZ +ν iY +ν iT +ν iR −ν iQ +ν iV +ν iF −ν iEν iM +ν iM+ν iN −ν iS −ν iL +ν iQ −ν iY −ν iE +νZ −νU −νX −νR +νT +νP +νF −νV −νD −νM −νN +νS +νL −νQ +νY +νE −ν iZ +ν iU +ν iX +ν iR −ν iT −ν iP −ν iF +ν iV +ν iD

ν iN +ν iN−ν iM +ν iL −ν iS −ν iR −ν iZ +ν iF −νY +νX −νU −νQ +νP −νT +νE −νD +νV −νN +νM −νL +νS +νR +νZ −νF +ν iY −ν iX +ν iU +ν iQ −ν iP +ν iT −ν iE +ν iD −ν iVν iT +ν iT −ν iP +ν iQ −ν iR −ν iS −ν iV +ν iU −νD +νE −νF −νL +νM −νN +νX −νY +νZ −νT +νP −νQ +νR +νS +νV −νU +ν iD −ν iE +ν iF +ν iL −ν iM+ν iN −ν iX +ν iY −ν iZν iU +ν iU +ν iX −ν iY −ν iZ +ν iV −ν iS −ν iT +νL −νM −νN −νD +νE +νF +νP −νQ −νR −νU −νX +νY +νZ −νV +νS +νT −ν iL +ν iM+ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR

ν iV +ν iV −ν iD −ν iE +ν iF −ν iU +ν iT −ν iS +νP +νQ −νR −νX −νY +νZ −νL −νM +νN −νV +νD +νE −νF +νU −νT +νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ +ν iL +ν iM−ν iNνX +νX −νU +νZ −νY −νD −νL +νP +ν iS −ν iN+ν iM +ν iV −ν iF +ν iE −ν iT +ν iR −ν iQ +ν iX −ν iU +ν iZ −ν iY −ν iD −ν iL +ν iP +νS −νN +νM +νV −νF +νE −νT +νR −νQνY +νY +νZ +νU +νX +νE +νM +νQ +ν iN +ν iS +ν iL +ν iF +ν iV +ν iD +ν iR +ν iT +ν iP +ν iY +ν iZ +ν iU +ν iX +ν iE +ν iM +ν iQ +νN +νS +νL +νF +νV +νD +νR +νT +νP

νZ +νZ −νY −νX +νU −νF +νN −νR −ν iM −ν iL +ν iS +ν iE +ν iD −ν iV +ν iQ +ν iP −ν iT +ν iZ −ν iY −ν iX +ν iU −ν iF +ν iN −ν iR −νM −νL +νS +νE +νD −νV +νQ +νP −νTνP +νP +νT −νR −νQ +νL −νD −νX −ν iV +ν iF +ν iE +ν iS −ν iN−ν iM −ν iU +ν iZ +ν iY +ν iP +ν iT −ν iR −ν iQ +ν iL −ν iD −ν iX −νV +νF +νE +νS −νN −νM −νU +νZ +νY

νQ +νQ −νR −νT +νP −νM +νE −νY −ν iF −ν iV +ν iD +ν iN +ν iS −ν iL +ν iZ +ν iU −ν iX +ν iQ −ν iR −ν iT +ν iP −ν iM +ν iE −ν iY −νF −νV +νD +νN +νS −νL +νZ +νU −νXνR +νR +νQ +νP +νT +νN +νF +νZ +ν iE +ν iD +ν iV +ν iM +ν iL +ν iS +ν iY +ν iX +ν iU +ν iR +ν iQ +ν iP +ν iT +ν iN +ν iF +ν iZ +νE +νD +νV +νM +νL +νS +νY +νX +νU

νD +νD +νV +νF +νE +νX +νP +νL +ν iT +ν iR +ν iQ +ν iU +ν iZ +ν iY +ν iS +ν iN+ν iM +ν iD +ν iV +ν iF +ν iE +ν iX +ν iP +ν iL +νT +νR +νQ +νU +νZ +νY +νS +νN +νM

νE +νE −νF +νV −νD −νY −νQ +νM +ν iR −ν iT +ν iP +ν iZ −ν iU +ν iX −ν iN +ν iS −ν iL +ν iE −ν iF +ν iV −ν iD −ν iY −ν iQ+ν iM +νR −νT +νP +νZ −νU +νX −νN +νS −νLνF +νF +νE −νD −νV +νZ −νR −νN −ν iQ +ν iP +ν iT +ν iY −ν iX −ν iU −ν iM +ν iL +ν iS +ν iF +ν iE −ν iD −ν iV +ν iZ −ν iR −ν iN −νQ +νP +νT +νY −νX −νU −νM +νL +νS


Table 4. Kaotic Algebra multiplication table, part A2

S L M N T U V iX iY iZ iP iQ iR iD iE iF iS iL iM iN iT iU iV X Y Z P Q R D E F

µS +µS +µL +µM +µN +µT +µU +µV +µiX +µiY +µiZ +µiP +µiQ +µiR +µiD +µiE +µiF +µiS +µiL +µiM+µiN +µiT +µiU +µiV +µX +µY +µZ +µP +µQ +µR +µD +µE +µF

µL +µL −µS −µN +µM −µP +µX −µD −µiU −µiZ +µiY +µiT +µiR −µiQ +µiV +µiF −µiE +µiL −µiS −µiN+µiM −µiP +µiX −µiD −µU −µZ +µY +µT +µR −µQ +µV +µF −µEµM +µM +µN −µS −µL +µQ −µY −µE −µiZ +µiU +µiX +µiR −µiT −µiP −µiF +µiV +µiD+µiM+µiN −µiS −µiL +µiQ −µiY −µiE −µZ +µU +µX +µR −µT −µP −µF +µV +µD

µN +µN −µM +µL −µS −µR −µZ +µF +µiY −µiX +µiU +µiQ −µiP +µiT −µiE +µiD −µiV +µiN−µiM +µiL −µiS −µiR −µiZ +µiF +µY −µX +µU +µQ −µP +µT −µE +µD −µVµT +µT −µP +µQ −µR −µS −µV +µU +µiD −µiE +µiF +µiL −µiM+µiN −µiX +µiY −µiZ +µiT −µiP +µiQ −µiR −µiS −µiV +µiU +µD −µE +µF +µL −µM +µN −µX +µY −µZµU +µU +µX −µY −µZ +µV −µS −µT −µiL +µiM+µiN +µiD −µiE −µiF −µiP +µiQ +µiR +µiU +µiX −µiY −µiZ +µiV −µiS −µiT −µL +µM +µN +µD −µE −µF −µP +µQ +µR

µV +µV −µD −µE +µF −µU +µT −µS −µiP −µiQ +µiR +µiX +µiY −µiZ +µiL +µiM−µiN +µiV −µiD −µiE +µiF −µiU +µiT −µiS −µP −µQ +µR +µX +µY −µZ +µL +µM −µNµiX +µiX −µiU +µiZ −µiY −µiD −µiL +µiP −µS +µN −µM −µV +µF −µE +µT −µR +µQ −µX +µU −µZ +µY +µD +µL −µP +µiS −µiN+µiM +µiV −µiF +µiE −µiT +µiR −µiQµiY +µiY +µiZ +µiU +µiX +µiE +µiM +µiQ −µN −µS −µL −µF −µV −µD −µR −µT −µP −µY −µZ −µU −µX −µE −µM −µQ +µiN +µiS +µiL +µiF +µiV +µiD +µiR +µiT +µiP

µiZ +µiZ −µiY −µiX +µiU −µiF +µiN −µiR +µM +µL −µS −µE −µD +µV −µQ −µP +µT −µZ +µY +µX −µU +µF −µN +µR −µiM −µiL +µiS +µiE +µiD −µiV +µiQ +µiP −µiTµiP +µiP +µiT −µiR −µiQ +µiL −µiD −µiX +µV −µF −µE −µS +µN +µM +µU −µZ −µY −µP −µT +µR +µQ −µL +µD +µX −µiV +µiF +µiE +µiS −µiN−µiM −µiU +µiZ +µiY

µiQ +µiQ −µiR −µiT +µiP −µiM +µiE −µiY +µF +µV −µD −µN −µS +µL −µZ −µU +µX −µQ +µR +µT −µP +µM −µE +µY −µiF −µiV +µiD +µiN +µiS −µiL +µiZ +µiU −µiXµiR +µiR +µiQ +µiP +µiT +µiN +µiF +µiZ −µE −µD −µV −µM −µL −µS −µY −µX −µU −µR −µQ −µP −µT −µN −µF −µZ +µiE +µiD +µiV +µiM +µiL +µiS +µiY +µiX +µiU

µiD +µiD +µiV +µiF +µiE +µiX +µiP +µiL −µT −µR −µQ −µU −µZ −µY −µS −µN −µM −µD −µV −µF −µE −µX −µP −µL +µiT +µiR +µiQ +µiU +µiZ +µiY +µiS +µiN+µiM

µiE +µiE −µiF +µiV −µiD −µiY −µiQ+µiM −µR +µT −µP −µZ +µU −µX +µN −µS +µL −µE +µF −µV +µD +µY +µQ −µM +µiR −µiT +µiP +µiZ −µiU +µiX −µiN +µiS −µiLµiF +µiF +µiE −µiD −µiV +µiZ −µiR −µiN +µQ −µP −µT −µY +µX +µU +µM −µL −µS −µF −µE +µD +µV −µZ +µR +µN −µiQ +µiP +µiT +µiY −µiX −µiU −µiM +µiL +µiS

µiS +µiS +µiL +µiM+µiN +µiT +µiU +µiV −µX −µY −µZ −µP −µQ −µR −µD −µE −µF −µS −µL −µM −µN −µT −µU −µV +µiX +µiY +µiZ +µiP +µiQ +µiR +µiD +µiE +µiF

µiL +µiL −µiS −µiN+µiM −µiP +µiX −µiD +µU +µZ −µY −µT −µR +µQ −µV −µF +µE −µL +µS +µN −µM +µP −µX +µD −µiU −µiZ +µiY +µiT +µiR −µiQ +µiV +µiF −µiEµiM +µiM+µiN −µiS −µiL +µiQ −µiY −µiE +µZ −µU −µX −µR +µT +µP +µF −µV −µD −µM −µN +µS +µL −µQ +µY +µE −µiZ +µiU +µiX +µiR −µiT −µiP −µiF +µiV +µiD

µiN +µiN−µiM +µiL −µiS −µiR −µiZ +µiF −µY +µX −µU −µQ +µP −µT +µE −µD +µV −µN +µM −µL +µS +µR +µZ −µF +µiY −µiX +µiU +µiQ −µiP +µiT −µiE +µiD −µiVµiT +µiT −µiP +µiQ −µiR −µiS −µiV +µiU −µD +µE −µF −µL +µM −µN +µX −µY +µZ −µT +µP −µQ +µR +µS +µV −µU +µiD −µiE +µiF +µiL −µiM+µiN −µiX +µiY −µiZµiU +µiU +µiX −µiY −µiZ +µiV −µiS −µiT +µL −µM −µN −µD +µE +µF +µP −µQ −µR −µU −µX +µY +µZ −µV +µS +µT −µiL +µiM+µiN +µiD −µiE −µiF −µiP +µiQ +µiR

µiV +µiV −µiD −µiE +µiF −µiU +µiT −µiS +µP +µQ −µR −µX −µY +µZ −µL −µM +µN −µV +µD +µE −µF +µU −µT +µS −µiP −µiQ +µiR +µiX +µiY −µiZ +µiL +µiM−µiNµX +µX −µU +µZ −µY −µD −µL +µP +µiS −µiN+µiM +µiV −µiF +µiE −µiT +µiR −µiQ +µiX −µiU +µiZ −µiY −µiD −µiL +µiP +µS −µN +µM +µV −µF +µE −µT +µR −µQµY +µY +µZ +µU +µX +µE +µM +µQ +µiN +µiS +µiL +µiF +µiV +µiD +µiR +µiT +µiP +µiY +µiZ +µiU +µiX +µiE +µiM +µiQ +µN +µS +µL +µF +µV +µD +µR +µT +µP

µZ +µZ −µY −µX +µU −µF +µN −µR −µiM −µiL +µiS +µiE +µiD −µiV +µiQ +µiP −µiT +µiZ −µiY −µiX +µiU −µiF +µiN −µiR −µM −µL +µS +µE +µD −µV +µQ +µP −µTµP +µP +µT −µR −µQ +µL −µD −µX −µiV +µiF +µiE +µiS −µiN−µiM −µiU +µiZ +µiY +µiP +µiT −µiR −µiQ +µiL −µiD −µiX −µV +µF +µE +µS −µN −µM −µU +µZ +µY

µQ +µQ −µR −µT +µP −µM +µE −µY −µiF −µiV +µiD +µiN +µiS −µiL +µiZ +µiU −µiX +µiQ −µiR −µiT +µiP −µiM +µiE −µiY −µF −µV +µD +µN +µS −µL +µZ +µU −µXµR +µR +µQ +µP +µT +µN +µF +µZ +µiE +µiD +µiV +µiM +µiL +µiS +µiY +µiX +µiU +µiR +µiQ +µiP +µiT +µiN +µiF +µiZ +µE +µD +µV +µM +µL +µS +µY +µX +µU

µD +µD +µV +µF +µE +µX +µP +µL +µiT +µiR +µiQ +µiU +µiZ +µiY +µiS +µiN+µiM +µiD +µiV +µiF +µiE +µiX +µiP +µiL +µT +µR +µQ +µU +µZ +µY +µS +µN +µM

µE +µE −µF +µV −µD −µY −µQ +µM +µiR −µiT +µiP +µiZ −µiU +µiX −µiN +µiS −µiL +µiE −µiF +µiV −µiD −µiY −µiQ+µiM +µR −µT +µP +µZ −µU +µX −µN +µS −µLµF +µF +µE −µD −µV +µZ −µR −µN −µiQ +µiP +µiT +µiY −µiX −µiU −µiM +µiL +µiS +µiF +µiE −µiD −µiV +µiZ −µiR −µiN −µQ +µP +µT +µY −µX −µU −µM +µL +µS

λS +λS +λL +λM +λN +λT +λU +λV +λiX +λiY +λiZ +λiP +λiQ +λiR +λiD +λiE +λiF +λiS +λiL +λiM +λiN +λiT +λiU +λiV +λX +λY +λZ +λP +λQ +λR +λD +λE +λF

λL +λL −λS −λN +λM −λP +λX −λD −λiU −λiZ +λiY +λiT +λiR −λiQ +λiV +λiF −λiE +λiL −λiS −λiN +λiM −λiP +λiX −λiD −λU −λZ +λY +λT +λR −λQ +λV +λF −λEλM +λM +λN −λS −λL +λQ −λY −λE −λiZ +λiU +λiX +λiR −λiT −λiP −λiF +λiV +λiD+λiM +λiN −λiS −λiL +λiQ −λiY −λiE −λZ +λU +λX +λR −λT −λP −λF +λV +λD

λN +λN −λM +λL −λS −λR −λZ +λF +λiY −λiX +λiU +λiQ −λiP +λiT −λiE +λiD −λiV +λiN −λiM +λiL −λiS −λiR −λiZ +λiF +λY −λX +λU +λQ −λP +λT −λE +λD −λVλT +λT −λP +λQ −λR −λS −λV +λU +λiD −λiE +λiF +λiL −λiM +λiN −λiX +λiY −λiZ +λiT −λiP +λiQ −λiR −λiS −λiV +λiU +λD −λE +λF +λL −λM +λN −λX +λY −λZλU +λU +λX −λY −λZ +λV −λS −λT −λiL +λiM +λiN +λiD −λiE −λiF −λiP +λiQ +λiR +λiU +λiX −λiY −λiZ +λiV −λiS −λiT −λL +λM +λN +λD −λE −λF −λP +λQ +λR

λV +λV −λD −λE +λF −λU +λT −λS −λiP −λiQ +λiR +λiX +λiY −λiZ +λiL +λiM −λiN +λiV −λiD −λiE +λiF −λiU +λiT −λiS −λP −λQ +λR +λX +λY −λZ +λL +λM −λNλiX +λiX −λiU +λiZ −λiY −λiD −λiL +λiP −λS +λN −λM −λV +λF −λE +λT −λR +λQ −λX +λU −λZ +λY +λD +λL −λP +λiS −λiN +λiM +λiV −λiF +λiE −λiT +λiR −λiQλiY +λiY +λiZ +λiU +λiX +λiE +λiM +λiQ −λN −λS −λL −λF −λV −λD −λR −λT −λP −λY −λZ −λU −λX −λE −λM −λQ +λiN +λiS +λiL +λiF +λiV +λiD +λiR +λiT +λiP

λiZ +λiZ −λiY −λiX +λiU −λiF +λiN −λiR +λM +λL −λS −λE −λD +λV −λQ −λP +λT −λZ +λY +λX −λU +λF −λN +λR −λiM −λiL +λiS +λiE +λiD −λiV +λiQ +λiP −λiTλiP +λiP +λiT −λiR −λiQ +λiL −λiD −λiX +λV −λF −λE −λS +λN +λM +λU −λZ −λY −λP −λT +λR +λQ −λL +λD +λX −λiV +λiF +λiE +λiS −λiN −λiM −λiU +λiZ +λiY

λiQ +λiQ −λiR −λiT +λiP −λiM +λiE −λiY +λF +λV −λD −λN −λS +λL −λZ −λU +λX −λQ +λR +λT −λP +λM −λE +λY −λiF −λiV +λiD +λiN +λiS −λiL +λiZ +λiU −λiXλiR +λiR +λiQ +λiP +λiT +λiN +λiF +λiZ −λE −λD −λV −λM −λL −λS −λY −λX −λU −λR −λQ −λP −λT −λN −λF −λZ +λiE +λiD +λiV +λiM +λiL +λiS +λiY +λiX +λiU

λiD +λiD +λiV +λiF +λiE +λiX +λiP +λiL −λT −λR −λQ −λU −λZ −λY −λS −λN −λM −λD −λV −λF −λE −λX −λP −λL +λiT +λiR +λiQ +λiU +λiZ +λiY +λiS +λiN +λiM

λiE +λiE −λiF +λiV −λiD −λiY −λiQ+λiM −λR +λT −λP −λZ +λU −λX +λN −λS +λL −λE +λF −λV +λD +λY +λQ −λM +λiR −λiT +λiP +λiZ −λiU +λiX −λiN +λiS −λiLλiF +λiF +λiE −λiD −λiV +λiZ −λiR −λiN +λQ −λP −λT −λY +λX +λU +λM −λL −λS −λF −λE +λD +λV −λZ +λR +λN −λiQ +λiP +λiT +λiY −λiX −λiU −λiM +λiL +λiS

λiS +λiS +λiL +λiM +λiN +λiT +λiU +λiV −λX −λY −λZ −λP −λQ −λR −λD −λE −λF −λS −λL −λM −λN −λT −λU −λV +λiX +λiY +λiZ +λiP +λiQ +λiR +λiD +λiE +λiF

λiL +λiL −λiS −λiN +λiM −λiP +λiX −λiD +λU +λZ −λY −λT −λR +λQ −λV −λF +λE −λL +λS +λN −λM +λP −λX +λD −λiU −λiZ +λiY +λiT +λiR −λiQ +λiV +λiF −λiEλiM +λiM +λiN −λiS −λiL +λiQ −λiY −λiE +λZ −λU −λX −λR +λT +λP +λF −λV −λD −λM −λN +λS +λL −λQ +λY +λE −λiZ +λiU +λiX +λiR −λiT −λiP −λiF +λiV +λiD

λiN +λiN −λiM +λiL −λiS −λiR −λiZ +λiF −λY +λX −λU −λQ +λP −λT +λE −λD +λV −λN +λM −λL +λS +λR +λZ −λF +λiY −λiX +λiU +λiQ −λiP +λiT −λiE +λiD −λiVλiT +λiT −λiP +λiQ −λiR −λiS −λiV +λiU −λD +λE −λF −λL +λM −λN +λX −λY +λZ −λT +λP −λQ +λR +λS +λV −λU +λiD −λiE +λiF +λiL −λiM +λiN −λiX +λiY −λiZλiU +λiU +λiX −λiY −λiZ +λiV −λiS −λiT +λL −λM −λN −λD +λE +λF +λP −λQ −λR −λU −λX +λY +λZ −λV +λS +λT −λiL +λiM +λiN +λiD −λiE −λiF −λiP +λiQ +λiR

λiV +λiV −λiD −λiE +λiF −λiU +λiT −λiS +λP +λQ −λR −λX −λY +λZ −λL −λM +λN −λV +λD +λE −λF +λU −λT +λS −λiP −λiQ +λiR +λiX +λiY −λiZ +λiL +λiM −λiNλX +λX −λU +λZ −λY −λD −λL +λP +λiS −λiN +λiM +λiV −λiF +λiE −λiT +λiR −λiQ +λiX −λiU +λiZ −λiY −λiD −λiL +λiP +λS −λN +λM +λV −λF +λE −λT +λR −λQλY +λY +λZ +λU +λX +λE +λM +λQ +λiN +λiS +λiL +λiF +λiV +λiD +λiR +λiT +λiP +λiY +λiZ +λiU +λiX +λiE +λiM +λiQ +λN +λS +λL +λF +λV +λD +λR +λT +λP

λZ +λZ −λY −λX +λU −λF +λN −λR −λiM −λiL +λiS +λiE +λiD −λiV +λiQ +λiP −λiT +λiZ −λiY −λiX +λiU −λiF +λiN −λiR −λM −λL +λS +λE +λD −λV +λQ +λP −λTλP +λP +λT −λR −λQ +λL −λD −λX −λiV +λiF +λiE +λiS −λiN −λiM −λiU +λiZ +λiY +λiP +λiT −λiR −λiQ +λiL −λiD −λiX −λV +λF +λE +λS −λN −λM −λU +λZ +λY

λQ +λQ −λR −λT +λP −λM +λE −λY −λiF −λiV +λiD +λiN +λiS −λiL +λiZ +λiU −λiX +λiQ −λiR −λiT +λiP −λiM +λiE −λiY −λF −λV +λD +λN +λS −λL +λZ +λU −λXλR +λR +λQ +λP +λT +λN +λF +λZ +λiE +λiD +λiV +λiM +λiL +λiS +λiY +λiX +λiU +λiR +λiQ +λiP +λiT +λiN +λiF +λiZ +λE +λD +λV +λM +λL +λS +λY +λX +λU

λD +λD +λV +λF +λE +λX +λP +λL +λiT +λiR +λiQ +λiU +λiZ +λiY +λiS +λiN +λiM +λiD +λiV +λiF +λiE +λiX +λiP +λiL +λT +λR +λQ +λU +λZ +λY +λS +λN +λM

λE +λE −λF +λV −λD −λY −λQ +λM +λiR −λiT +λiP +λiZ −λiU +λiX −λiN +λiS −λiL +λiE −λiF +λiV −λiD −λiY −λiQ+λiM +λR −λT +λP +λZ −λU +λX −λN +λS −λLλF +λF +λE −λD −λV +λZ −λR −λN −λiQ +λiP +λiT +λiY −λiX −λiU −λiM +λiL +λiS +λiF +λiE −λiD −λiV +λiZ −λiR −λiN −λQ +λP +λT +λY −λX −λU −λM +λL +λS


Table 5. Kaotic Algebra multiplication table, part B1

νS νL νM νN νT νU νV ν iX ν iY ν iZ ν iP ν iQ ν iR ν iD ν iE ν iF ν iS ν iL ν iM ν iN ν iT ν iU ν iV νX νY νZ νP νQ νR νD νE νF

S +νS +νL +νM +νN +νT +νU +νV +ν iX +ν iY +ν iZ +ν iP +ν iQ +ν iR +ν iD +ν iE +ν iF +ν iS +ν iL +ν iM+ν iN +ν iT +ν iU +ν iV +νX +νY +νZ +νP +νQ +νR +νD +νE +νF

L −νL +νS −νN +νM +νP −νX +νD +ν iU −ν iZ +ν iY −ν iT +ν iR −ν iQ −ν iV +ν iF −ν iE −ν iL +ν iS −ν iN+ν iM +ν iP −ν iX +ν iD +νU −νZ +νY −νT +νR −νQ −νV +νF −νEM −νM +νN +νS −νL −νQ +νY +νE −ν iZ −ν iU +ν iX +ν iR +ν iT −ν iP −ν iF −ν iV +ν iD−ν iM+ν iN +ν iS −ν iL −ν iQ +ν iY +ν iE −νZ −νU +νX +νR +νT −νP −νF −νV +νD

N −νN −νM +νL +νS +νR +νZ −νF +ν iY −ν iX −ν iU +ν iQ −ν iP −ν iT −ν iE +ν iD +ν iV −ν iN−ν iM +ν iL +ν iS +ν iR +ν iZ −ν iF +νY −νX −νU +νQ −νP −νT −νE +νD +νV

T −νT +νP −νQ +νR +νS −νV +νU +ν iD −ν iE +ν iF −ν iL +ν iM−ν iN −ν iX +ν iY −ν iZ −ν iT +ν iP −ν iQ +ν iR +ν iS −ν iV +ν iU +νD −νE +νF −νL +νM −νN −νX +νY −νZU −νU −νX +νY +νZ +νV +νS −νT +ν iL −ν iM−ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR −ν iU −ν iX +ν iY +ν iZ +ν iV +ν iS −ν iT +νL −νM −νN +νD −νE −νF −νP +νQ +νR

V −νV +νD +νE −νF −νU +νT +νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ −ν iL −ν iM+ν iN −ν iV +ν iD +ν iE −ν iF −ν iU +ν iT +ν iS −νP −νQ +νR +νX +νY −νZ −νL −νM +νN

iX −ν iX +ν iU +ν iZ −ν iY −ν iD +ν iL +ν iP +νS +νN −νM −νV −νF +νE +νT +νR −νQ +νX −νU −νZ +νY +νD −νL −νP −ν iS −ν iN+ν iM +ν iV +ν iF −ν iE −ν iT −ν iR +ν iQ

iY −ν iY +ν iZ −ν iU +ν iX +ν iE −ν iM +ν iQ −νN +νS −νL +νF −νV +νD +νR −νT +νP +νY −νZ +νU −νX −νE +νM −νQ +ν iN −ν iS +ν iL −ν iF +ν iV −ν iD −ν iR +ν iT −ν iPiZ −ν iZ −ν iY −ν iX −ν iU −ν iF −ν iN −ν iR +νM +νL +νS +νE +νD +νV +νQ +νP +νT +νZ +νY +νX +νU +νF +νN +νR −ν iM −ν iL −ν iS −ν iE −ν iD −ν iV −ν iQ −ν iP −ν iTiP −ν iP −ν iT −ν iR −ν iQ −ν iL −ν iD −ν iX +νV +νF +νE +νS +νN +νM +νU +νZ +νY +νP +νT +νR +νQ +νL +νD +νX −ν iV −ν iF −ν iE −ν iS −ν iN−ν iM −ν iU −ν iZ −ν iYiQ −ν iQ −ν iR +ν iT +ν iP +ν iM +ν iE −ν iY −νF +νV +νD −νN +νS +νL +νZ −νU −νX +νQ +νR −νT −νP −νM −νE +νY +ν iF −ν iV −ν iD +ν iN −ν iS −ν iL −ν iZ +ν iU +ν iX

iR −ν iR +ν iQ +ν iP −ν iT −ν iN +ν iF +ν iZ +νE +νD −νV −νM −νL +νS +νY +νX −νU +νR −νQ −νP +νT +νN −νF −νZ −ν iE −ν iD +ν iV +ν iM +ν iL −ν iS −ν iY −ν iX +ν iU

iD −ν iD −ν iV +ν iF +ν iE +ν iX +ν iP −ν iL −νT +νR +νQ −νU +νZ +νY +νS −νN −νM +νD +νV −νF −νE −νX −νP +νL +ν iT −ν iR −ν iQ +ν iU −ν iZ −ν iY −ν iS +ν iN+ν iM

iE −ν iE −ν iF −ν iV −ν iD −ν iY −ν iQ−ν iM +νR +νT +νP +νZ +νU +νX +νN +νS +νL +νE +νF +νV +νD +νY +νQ +νM −ν iR −ν iT −ν iP −ν iZ −ν iU −ν iX −ν iN −ν iS −ν iLiF −ν iF +ν iE −ν iD +ν iV +ν iZ −ν iR +ν iN −νQ +νP −νT +νY −νX +νU +νM −νL +νS +νF −νE +νD −νV −νZ +νR −νN +ν iQ −ν iP +ν iT −ν iY +ν iX −ν iU −ν iM +ν iL −ν iSiS −ν iS −ν iL −ν iM−ν iN −ν iT −ν iU −ν iV +νX +νY +νZ +νP +νQ +νR +νD +νE +νF +νS +νL +νM +νN +νT +νU +νV −ν iX −ν iY −ν iZ −ν iP −ν iQ −ν iR −ν iD −ν iE −ν iFiL +ν iL −ν iS +ν iN−ν iM −ν iP +ν iX −ν iD +νU −νZ +νY −νT +νR −νQ −νV +νF −νE −νL +νS −νN +νM +νP −νX +νD −ν iU +ν iZ −ν iY +ν iT −ν iR +ν iQ +ν iV −ν iF +ν iE

iM +ν iM−ν iN −ν iS +ν iL +ν iQ −ν iY −ν iE −νZ −νU +νX +νR +νT −νP −νF −νV +νD −νM +νN +νS −νL −νQ +νY +νE +ν iZ +ν iU −ν iX −ν iR −ν iT +ν iP +ν iF +ν iV −ν iDiN +ν iN+ν iM −ν iL −ν iS −ν iR −ν iZ +ν iF +νY −νX −νU +νQ −νP −νT −νE +νD +νV −νN −νM +νL +νS +νR +νZ −νF −ν iY +ν iX +ν iU −ν iQ +ν iP +ν iT +ν iE −ν iD −ν iViT +ν iT −ν iP +ν iQ −ν iR −ν iS +ν iV −ν iU +νD −νE +νF −νL +νM −νN −νX +νY −νZ −νT +νP −νQ +νR +νS −νV +νU −ν iD +ν iE −ν iF +ν iL −ν iM+ν iN +ν iX −ν iY +ν iZ

iU +ν iU +ν iX −ν iY −ν iZ −ν iV −ν iS +ν iT +νL −νM −νN +νD −νE −νF −νP +νQ +νR −νU −νX +νY +νZ +νV +νS −νT −ν iL +ν iM+ν iN −ν iD +ν iE +ν iF +ν iP −ν iQ −ν iRiV +ν iV −ν iD −ν iE +ν iF +ν iU −ν iT −ν iS −νP −νQ +νR +νX +νY −νZ −νL −νM +νN −νV +νD +νE −νF −νU +νT +νS +ν iP +ν iQ −ν iR −ν iX −ν iY +ν iZ +ν iL +ν iM−ν iNX +νX −νU −νZ +νY +νD −νL −νP +ν iS +ν iN−ν iM −ν iV −ν iF +ν iE +ν iT +ν iR −ν iQ +ν iX −ν iU −ν iZ +ν iY +ν iD −ν iL −ν iP +νS +νN −νM −νV −νF +νE +νT +νR −νQY +νY −νZ +νU −νX −νE +νM −νQ −ν iN +ν iS −ν iL +ν iF −ν iV +ν iD +ν iR −ν iT +ν iP +ν iY −ν iZ +ν iU −ν iX −ν iE +ν iM −ν iQ −νN +νS −νL +νF −νV +νD +νR −νT +νP

Z +νZ +νY +νX +νU +νF +νN +νR +ν iM +ν iL +ν iS +ν iE +ν iD +ν iV +ν iQ +ν iP +ν iT +ν iZ +ν iY +ν iX +ν iU +ν iF +ν iN +ν iR +νM +νL +νS +νE +νD +νV +νQ +νP +νT

P +νP +νT +νR +νQ +νL +νD +νX +ν iV +ν iF +ν iE +ν iS +ν iN+ν iM +ν iU +ν iZ +ν iY +ν iP +ν iT +ν iR +ν iQ +ν iL +ν iD +ν iX +νV +νF +νE +νS +νN +νM +νU +νZ +νY

Q +νQ +νR −νT −νP −νM −νE +νY −ν iF +ν iV +ν iD −ν iN +ν iS +ν iL +ν iZ −ν iU −ν iX +ν iQ +ν iR −ν iT −ν iP −ν iM −ν iE +ν iY −νF +νV +νD −νN +νS +νL +νZ −νU −νXR +νR −νQ −νP +νT +νN −νF −νZ +ν iE +ν iD −ν iV −ν iM −ν iL +ν iS +ν iY +ν iX −ν iU +ν iR −ν iQ −ν iP +ν iT +ν iN −ν iF −ν iZ +νE +νD −νV −νM −νL +νS +νY +νX −νUD +νD +νV −νF −νE −νX −νP +νL −ν iT +ν iR +ν iQ −ν iU +ν iZ +ν iY +ν iS −ν iN−ν iM +ν iD +ν iV −ν iF −ν iE −ν iX −ν iP +ν iL −νT +νR +νQ −νU +νZ +νY +νS −νN −νME +νE +νF +νV +νD +νY +νQ +νM +ν iR +ν iT +ν iP +ν iZ +ν iU +ν iX +ν iN +ν iS +ν iL +ν iE +ν iF +ν iV +ν iD +ν iY +ν iQ+ν iM +νR +νT +νP +νZ +νU +νX +νN +νS +νL

F +νF −νE +νD −νV −νZ +νR −νN −ν iQ +ν iP −ν iT +ν iY −ν iX +ν iU +ν iM −ν iL +ν iS +ν iF −ν iE +ν iD −ν iV −ν iZ +ν iR −ν iN −νQ +νP −νT +νY −νX +νU +νM −νL +νS

νS −S −L −M −N −T −U −V −iX −iY −iZ −iP −iQ −iR −iD −iE −iF −iS −iL −iM −iN −iT −iU −iV −X −Y −Z −P −Q −R −D −E −FνL +L −S −N +M −P +X −D −iU −iZ +iY +iT +iR −iQ +iV +iF −iE +iL −iS −iN +iM −iP +iX −iD −U −Z +Y +T +R −Q +V +F −EνM +M +N −S −L +Q −Y −E −iZ +iU +iX +iR −iT −iP −iF +iV +iD +iM +iN −iS −iL +iQ −iY −iE −Z +U +X +R −T −P −F +V +D

νN +N −M +L −S −R −Z +F +iY −iX +iU +iQ −iP +iT −iE +iD −iV +iN −iM +iL −iS −iR −iZ +iF +Y −X +U +Q −P +T −E +D −VνT +T −P +Q −R −S −V +U +iD −iE +iF +iL −iM +iN −iX +iY −iZ +iT −iP +iQ −iR −iS −iV +iU +D −E +F +L −M +N −X +Y −ZνU +U +X −Y −Z +V −S −T −iL +iM +iN +iD −iE −iF −iP +iQ +iR +iU +iX −iY −iZ +iV −iS −iT −L +M +N +D −E −F −P +Q +R

νV +V −D −E +F −U +T −S −iP −iQ +iR +iX +iY −iZ +iL +iM −iN +iV −iD −iE +iF −iU +iT −iS −P −Q +R +X +Y −Z +L +M −Nν iX +iX −iU +iZ −iY −iD −iL +iP −S +N −M −V +F −E +T −R +Q −X +U −Z +Y +D +L −P +iS −iN +iM +iV −iF +iE −iT +iR −iQν iY +iY +iZ +iU +iX +iE +iM +iQ −N −S −L −F −V −D −R −T −P −Y −Z −U −X −E −M −Q +iN +iS +iL +iF +iV +iD +iR +iT +iP

ν iZ +iZ −iY −iX +iU −iF +iN −iR +M +L −S −E −D +V −Q −P +T −Z +Y +X −U +F −N +R −iM −iL +iS +iE +iD −iV +iQ +iP −iTν iP +iP +iT −iR −iQ +iL −iD −iX +V −F −E −S +N +M +U −Z −Y −P −T +R +Q −L +D +X −iV +iF +iE +iS −iN −iM −iU +iZ +iY

ν iQ +iQ −iR −iT +iP −iM +iE −iY +F +V −D −N −S +L −Z −U +X −Q +R +T −P +M −E +Y −iF −iV +iD +iN +iS −iL +iZ +iU −iXν iR +iR +iQ +iP +iT +iN +iF +iZ −E −D −V −M −L −S −Y −X −U −R −Q −P −T −N −F −Z +iE +iD +iV +iM +iL +iS +iY +iX +iU

ν iD +iD +iV +iF +iE +iX +iP +iL −T −R −Q −U −Z −Y −S −N −M −D −V −F −E −X −P −L +iT +iR +iQ +iU +iZ +iY +iS +iN +iM

ν iE +iE −iF +iV −iD −iY −iQ +iM −R +T −P −Z +U −X +N −S +L −E +F −V +D +Y +Q −M +iR −iT +iP +iZ −iU +iX −iN +iS −iLν iF +iF +iE −iD −iV +iZ −iR −iN +Q −P −T −Y +X +U +M −L −S −F −E +D +V −Z +R +N −iQ +iP +iT +iY −iX −iU −iM +iL +iS

ν iS +iS +iL +iM +iN +iT +iU +iV −X −Y −Z −P −Q −R −D −E −F −S −L −M −N −T −U −V +iX +iY +iZ +iP +iQ +iR +iD +iE +iF

ν iL −iL +iS +iN −iM +iP −iX +iD −U −Z +Y +T +R −Q +V +F −E +L −S −N +M −P +X −D +iU +iZ −iY −iT −iR +iQ −iV −iF +iE

ν iM −iM −iN +iS +iL −iQ +iY +iE −Z +U +X +R −T −P −F +V +D +M +N −S −L +Q −Y −E +iZ −iU −iX −iR +iT +iP +iF −iV −iDν iN −iN +iM −iL +iS +iR +iZ −iF +Y −X +U +Q −P +T −E +D −V +N −M +L −S −R −Z +F −iY +iX −iU −iQ +iP −iT +iE −iD +iV

ν iT −iT +iP −iQ +iR +iS +iV −iU +D −E +F +L −M +N −X +Y −Z +T −P +Q −R −S −V +U −iD +iE −iF −iL +iM −iN +iX −iY +iZ

ν iU −iU −iX +iY +iZ −iV +iS +iT −L +M +N +D −E −F −P +Q +R +U +X −Y −Z +V −S −T +iL −iM −iN −iD +iE +iF +iP −iQ −iRν iV −iV +iD +iE −iF +iU −iT +iS −P −Q +R +X +Y −Z +L +M −N +V −D −E +F −U +T −S +iP +iQ −iR −iX −iY +iZ −iL −iM +iN

νX −X +U −Z +Y +D +L −P −iS +iN −iM −iV +iF −iE +iT −iR +iQ −iX +iU −iZ +iY +iD +iL −iP −S +N −M −V +F −E +T −R +Q

νY −Y −Z −U −X −E −M −Q −iN −iS −iL −iF −iV −iD −iR −iT −iP −iY −iZ −iU −iX −iE −iM −iQ −N −S −L −F −V −D −R −T −PνZ −Z +Y +X −U +F −N +R +iM +iL −iS −iE −iD +iV −iQ −iP +iT −iZ +iY +iX −iU +iF −iN +iR +M +L −S −E −D +V −Q −P +T

νP −P −T +R +Q −L +D +X +iV −iF −iE −iS +iN +iM +iU −iZ −iY −iP −iT +iR +iQ −iL +iD +iX +V −F −E −S +N +M +U −Z −YνQ −Q +R +T −P +M −E +Y +iF +iV −iD −iN −iS +iL −iZ −iU +iX −iQ +iR +iT −iP +iM −iE +iY +F +V −D −N −S +L −Z −U +X

νR −R −Q −P −T −N −F −Z −iE −iD −iV −iM −iL −iS −iY −iX −iU −iR −iQ −iP −iT −iN −iF −iZ −E −D −V −M −L −S −Y −X −UνD −D −V −F −E −X −P −L −iT −iR −iQ −iU −iZ −iY −iS −iN −iM −iD −iV −iF −iE −iX −iP −iL −T −R −Q −U −Z −Y −S −N −MνE −E +F −V +D +Y +Q −M −iR +iT −iP −iZ +iU −iX +iN −iS +iL −iE +iF −iV +iD +iY +iQ −iM −R +T −P −Z +U −X +N −S +L

νF −F −E +D +V −Z +R +N +iQ −iP −iT −iY +iX +iU +iM −iL −iS −iF −iE +iD +iV −iZ +iR +iN +Q −P −T −Y +X +U +M −L −S


Table 6. Kaotic Algebra multiplication table, part B2

νS νL νM νN νT νU νV ν iX ν iY ν iZ ν iP ν iQ ν iR ν iD ν iE ν iF ν iS ν iL ν iM ν iN ν iT ν iU ν iV νX νY νZ νP νQ νR νD νE νF

µS −λS −λL −λM −λN −λT −λU −λV −λiX −λiY −λiZ −λiP −λiQ −λiR −λiD −λiE −λiF −λiS −λiL −λiM −λiN −λiT −λiU −λiV −λX −λY −λZ −λP −λQ −λR −λD −λE −λFµL −λL +λS +λN −λM +λP −λX +λD +λiU +λiZ −λiY −λiT −λiR +λiQ −λiV −λiF +λiE −λiL +λiS +λiN −λiM +λiP −λiX +λiD +λU +λZ −λY −λT −λR +λQ −λV −λF +λE

µM −λM −λN +λS +λL −λQ +λY +λE +λiZ −λiU −λiX −λiR +λiT +λiP +λiF −λiV −λiD−λiM −λiN +λiS +λiL −λiQ +λiY +λiE +λZ −λU −λX −λR +λT +λP +λF −λV −λDµN −λN +λM −λL +λS +λR +λZ −λF −λiY +λiX −λiU −λiQ +λiP −λiT +λiE −λiD +λiV −λiN +λiM −λiL +λiS +λiR +λiZ −λiF −λY +λX −λU −λQ +λP −λT +λE −λD +λV

µT −λT +λP −λQ +λR +λS +λV −λU −λiD +λiE −λiF −λiL +λiM −λiN +λiX −λiY +λiZ −λiT +λiP −λiQ +λiR +λiS +λiV −λiU −λD +λE −λF −λL +λM −λN +λX −λY +λZ

µU −λU −λX +λY +λZ −λV +λS +λT +λiL −λiM −λiN −λiD +λiE +λiF +λiP −λiQ −λiR −λiU −λiX +λiY +λiZ −λiV +λiS +λiT +λL −λM −λN −λD +λE +λF +λP −λQ −λRµV −λV +λD +λE −λF +λU −λT +λS +λiP +λiQ −λiR −λiX −λiY +λiZ −λiL −λiM +λiN −λiV +λiD +λiE −λiF +λiU −λiT +λiS +λP +λQ −λR −λX −λY +λZ −λL −λM +λN

µiX −λiX +λiU −λiZ +λiY +λiD +λiL −λiP +λS −λN +λM +λV −λF +λE −λT +λR −λQ +λX −λU +λZ −λY −λD −λL +λP −λiS +λiN −λiM −λiV +λiF −λiE +λiT −λiR +λiQ

µiY −λiY −λiZ −λiU −λiX −λiE −λiM −λiQ +λN +λS +λL +λF +λV +λD +λR +λT +λP +λY +λZ +λU +λX +λE +λM +λQ −λiN −λiS −λiL −λiF −λiV −λiD −λiR −λiT −λiPµiZ −λiZ +λiY +λiX −λiU +λiF −λiN +λiR −λM −λL +λS +λE +λD −λV +λQ +λP −λT +λZ −λY −λX +λU −λF +λN −λR +λiM +λiL −λiS −λiE −λiD +λiV −λiQ −λiP +λiT

µiP −λiP −λiT +λiR +λiQ −λiL +λiD +λiX −λV +λF +λE +λS −λN −λM −λU +λZ +λY +λP +λT −λR −λQ +λL −λD −λX +λiV −λiF −λiE −λiS +λiN +λiM +λiU −λiZ −λiYµiQ −λiQ +λiR +λiT −λiP +λiM −λiE +λiY −λF −λV +λD +λN +λS −λL +λZ +λU −λX +λQ −λR −λT +λP −λM +λE −λY +λiF +λiV −λiD −λiN −λiS +λiL −λiZ −λiU +λiX

µiR −λiR −λiQ −λiP −λiT −λiN −λiF −λiZ +λE +λD +λV +λM +λL +λS +λY +λX +λU +λR +λQ +λP +λT +λN +λF +λZ −λiE −λiD −λiV −λiM −λiL −λiS −λiY −λiX −λiUµiD −λiD −λiV −λiF −λiE −λiX −λiP −λiL +λT +λR +λQ +λU +λZ +λY +λS +λN +λM +λD +λV +λF +λE +λX +λP +λL −λiT −λiR −λiQ −λiU −λiZ −λiY −λiS −λiN −λiMµiE −λiE +λiF −λiV +λiD +λiY +λiQ−λiM +λR −λT +λP +λZ −λU +λX −λN +λS −λL +λE −λF +λV −λD −λY −λQ +λM −λiR +λiT −λiP −λiZ +λiU −λiX +λiN −λiS +λiL

µiF −λiF −λiE +λiD +λiV −λiZ +λiR +λiN −λQ +λP +λT +λY −λX −λU −λM +λL +λS +λF +λE −λD −λV +λZ −λR −λN +λiQ −λiP −λiT −λiY +λiX +λiU +λiM −λiL −λiSµiS −λiS −λiL −λiM −λiN −λiT −λiU −λiV +λX +λY +λZ +λP +λQ +λR +λD +λE +λF +λS +λL +λM +λN +λT +λU +λV −λiX −λiY −λiZ −λiP −λiQ −λiR −λiD −λiE −λiFµiL −λiL +λiS +λiN −λiM +λiP −λiX +λiD −λU −λZ +λY +λT +λR −λQ +λV +λF −λE +λL −λS −λN +λM −λP +λX −λD +λiU +λiZ −λiY −λiT −λiR +λiQ −λiV −λiF +λiE

µiM −λiM −λiN +λiS +λiL −λiQ +λiY +λiE −λZ +λU +λX +λR −λT −λP −λF +λV +λD +λM +λN −λS −λL +λQ −λY −λE +λiZ −λiU −λiX −λiR +λiT +λiP +λiF −λiV −λiDµiN −λiN +λiM −λiL +λiS +λiR +λiZ −λiF +λY −λX +λU +λQ −λP +λT −λE +λD −λV +λN −λM +λL −λS −λR −λZ +λF −λiY +λiX −λiU −λiQ +λiP −λiT +λiE −λiD +λiV

µiT −λiT +λiP −λiQ +λiR +λiS +λiV −λiU +λD −λE +λF +λL −λM +λN −λX +λY −λZ +λT −λP +λQ −λR −λS −λV +λU −λiD +λiE −λiF −λiL +λiM −λiN +λiX −λiY +λiZ

µiU −λiU −λiX +λiY +λiZ −λiV +λiS +λiT −λL +λM +λN +λD −λE −λF −λP +λQ +λR +λU +λX −λY −λZ +λV −λS −λT +λiL −λiM −λiN −λiD +λiE +λiF +λiP −λiQ −λiRµiV −λiV +λiD +λiE −λiF +λiU −λiT +λiS −λP −λQ +λR +λX +λY −λZ +λL +λM −λN +λV −λD −λE +λF −λU +λT −λS +λiP +λiQ −λiR −λiX −λiY +λiZ −λiL −λiM +λiN

µX −λX +λU −λZ +λY +λD +λL −λP −λiS +λiN −λiM −λiV +λiF −λiE +λiT −λiR +λiQ −λiX +λiU −λiZ +λiY +λiD +λiL −λiP −λS +λN −λM −λV +λF −λE +λT −λR +λQ

µY −λY −λZ −λU −λX −λE −λM −λQ −λiN −λiS −λiL −λiF −λiV −λiD −λiR −λiT −λiP −λiY −λiZ −λiU −λiX −λiE −λiM −λiQ −λN −λS −λL −λF −λV −λD −λR −λT −λPµZ −λZ +λY +λX −λU +λF −λN +λR +λiM +λiL −λiS −λiE −λiD +λiV −λiQ −λiP +λiT −λiZ +λiY +λiX −λiU +λiF −λiN +λiR +λM +λL −λS −λE −λD +λV −λQ −λP +λT

µP −λP −λT +λR +λQ −λL +λD +λX +λiV −λiF −λiE −λiS +λiN +λiM +λiU −λiZ −λiY −λiP −λiT +λiR +λiQ −λiL +λiD +λiX +λV −λF −λE −λS +λN +λM +λU −λZ −λYµQ −λQ +λR +λT −λP +λM −λE +λY +λiF +λiV −λiD −λiN −λiS +λiL −λiZ −λiU +λiX −λiQ +λiR +λiT −λiP +λiM −λiE +λiY +λF +λV −λD −λN −λS +λL −λZ −λU +λX

µR −λR −λQ −λP −λT −λN −λF −λZ −λiE −λiD −λiV −λiM −λiL −λiS −λiY −λiX −λiU −λiR −λiQ −λiP −λiT −λiN −λiF −λiZ −λE −λD −λV −λM −λL −λS −λY −λX −λUµD −λD −λV −λF −λE −λX −λP −λL −λiT −λiR −λiQ −λiU −λiZ −λiY −λiS −λiN −λiM −λiD −λiV −λiF −λiE −λiX −λiP −λiL −λT −λR −λQ −λU −λZ −λY −λS −λN −λMµE −λE +λF −λV +λD +λY +λQ −λM −λiR +λiT −λiP −λiZ +λiU −λiX +λiN −λiS +λiL −λiE +λiF −λiV +λiD +λiY +λiQ−λiM −λR +λT −λP −λZ +λU −λX +λN −λS +λL

µF −λF −λE +λD +λV −λZ +λR +λN +λiQ −λiP −λiT −λiY +λiX +λiU +λiM −λiL −λiS −λiF −λiE +λiD +λiV −λiZ +λiR +λiN +λQ −λP −λT −λY +λX +λU +λM −λL −λSλS +µS +µL +µM +µN +µT +µU +µV +µiX +µiY +µiZ +µiP +µiQ +µiR +µiD +µiE +µiF +µiS +µiL +µiM+µiN +µiT +µiU +µiV +µX +µY +µZ +µP +µQ +µR +µD +µE +µF

λL +µL −µS −µN +µM −µP +µX −µD −µiU −µiZ +µiY +µiT +µiR −µiQ +µiV +µiF −µiE +µiL −µiS −µiN+µiM −µiP +µiX −µiD −µU −µZ +µY +µT +µR −µQ +µV +µF −µEλM +µM +µN −µS −µL +µQ −µY −µE −µiZ +µiU +µiX +µiR −µiT −µiP −µiF +µiV +µiD+µiM+µiN −µiS −µiL +µiQ −µiY −µiE −µZ +µU +µX +µR −µT −µP −µF +µV +µD

λN +µN −µM +µL −µS −µR −µZ +µF +µiY −µiX +µiU +µiQ −µiP +µiT −µiE +µiD −µiV +µiN−µiM +µiL −µiS −µiR −µiZ +µiF +µY −µX +µU +µQ −µP +µT −µE +µD −µVλT +µT −µP +µQ −µR −µS −µV +µU +µiD −µiE +µiF +µiL −µiM+µiN −µiX +µiY −µiZ +µiT −µiP +µiQ −µiR −µiS −µiV +µiU +µD −µE +µF +µL −µM +µN −µX +µY −µZλU +µU +µX −µY −µZ +µV −µS −µT −µiL +µiM+µiN +µiD −µiE −µiF −µiP +µiQ +µiR +µiU +µiX −µiY −µiZ +µiV −µiS −µiT −µL +µM +µN +µD −µE −µF −µP +µQ +µR

λV +µV −µD −µE +µF −µU +µT −µS −µiP −µiQ +µiR +µiX +µiY −µiZ +µiL +µiM−µiN +µiV −µiD −µiE +µiF −µiU +µiT −µiS −µP −µQ +µR +µX +µY −µZ +µL +µM −µNλiX +µiX −µiU +µiZ −µiY −µiD −µiL +µiP −µS +µN −µM −µV +µF −µE +µT −µR +µQ −µX +µU −µZ +µY +µD +µL −µP +µiS −µiN+µiM +µiV −µiF +µiE −µiT +µiR −µiQλiY +µiY +µiZ +µiU +µiX +µiE +µiM +µiQ −µN −µS −µL −µF −µV −µD −µR −µT −µP −µY −µZ −µU −µX −µE −µM −µQ +µiN +µiS +µiL +µiF +µiV +µiD +µiR +µiT +µiP

λiZ +µiZ −µiY −µiX +µiU −µiF +µiN −µiR +µM +µL −µS −µE −µD +µV −µQ −µP +µT −µZ +µY +µX −µU +µF −µN +µR −µiM −µiL +µiS +µiE +µiD −µiV +µiQ +µiP −µiTλiP +µiP +µiT −µiR −µiQ +µiL −µiD −µiX +µV −µF −µE −µS +µN +µM +µU −µZ −µY −µP −µT +µR +µQ −µL +µD +µX −µiV +µiF +µiE +µiS −µiN−µiM −µiU +µiZ +µiY

λiQ +µiQ −µiR −µiT +µiP −µiM +µiE −µiY +µF +µV −µD −µN −µS +µL −µZ −µU +µX −µQ +µR +µT −µP +µM −µE +µY −µiF −µiV +µiD +µiN +µiS −µiL +µiZ +µiU −µiXλiR +µiR +µiQ +µiP +µiT +µiN +µiF +µiZ −µE −µD −µV −µM −µL −µS −µY −µX −µU −µR −µQ −µP −µT −µN −µF −µZ +µiE +µiD +µiV +µiM +µiL +µiS +µiY +µiX +µiU

λiD +µiD +µiV +µiF +µiE +µiX +µiP +µiL −µT −µR −µQ −µU −µZ −µY −µS −µN −µM −µD −µV −µF −µE −µX −µP −µL +µiT +µiR +µiQ +µiU +µiZ +µiY +µiS +µiN+µiM

λiE +µiE −µiF +µiV −µiD −µiY −µiQ+µiM −µR +µT −µP −µZ +µU −µX +µN −µS +µL −µE +µF −µV +µD +µY +µQ −µM +µiR −µiT +µiP +µiZ −µiU +µiX −µiN +µiS −µiLλiF +µiF +µiE −µiD −µiV +µiZ −µiR −µiN +µQ −µP −µT −µY +µX +µU +µM −µL −µS −µF −µE +µD +µV −µZ +µR +µN −µiQ +µiP +µiT +µiY −µiX −µiU −µiM +µiL +µiS

λiS +µiS +µiL +µiM+µiN +µiT +µiU +µiV −µX −µY −µZ −µP −µQ −µR −µD −µE −µF −µS −µL −µM −µN −µT −µU −µV +µiX +µiY +µiZ +µiP +µiQ +µiR +µiD +µiE +µiF

λiL +µiL −µiS −µiN+µiM −µiP +µiX −µiD +µU +µZ −µY −µT −µR +µQ −µV −µF +µE −µL +µS +µN −µM +µP −µX +µD −µiU −µiZ +µiY +µiT +µiR −µiQ +µiV +µiF −µiEλiM +µiM+µiN −µiS −µiL +µiQ −µiY −µiE +µZ −µU −µX −µR +µT +µP +µF −µV −µD −µM −µN +µS +µL −µQ +µY +µE −µiZ +µiU +µiX +µiR −µiT −µiP −µiF +µiV +µiD

λiN +µiN−µiM +µiL −µiS −µiR −µiZ +µiF −µY +µX −µU −µQ +µP −µT +µE −µD +µV −µN +µM −µL +µS +µR +µZ −µF +µiY −µiX +µiU +µiQ −µiP +µiT −µiE +µiD −µiVλiT +µiT −µiP +µiQ −µiR −µiS −µiV +µiU −µD +µE −µF −µL +µM −µN +µX −µY +µZ −µT +µP −µQ +µR +µS +µV −µU +µiD −µiE +µiF +µiL −µiM+µiN −µiX +µiY −µiZλiU +µiU +µiX −µiY −µiZ +µiV −µiS −µiT +µL −µM −µN −µD +µE +µF +µP −µQ −µR −µU −µX +µY +µZ −µV +µS +µT −µiL +µiM+µiN +µiD −µiE −µiF −µiP +µiQ +µiR

λiV +µiV −µiD −µiE +µiF −µiU +µiT −µiS +µP +µQ −µR −µX −µY +µZ −µL −µM +µN −µV +µD +µE −µF +µU −µT +µS −µiP −µiQ +µiR +µiX +µiY −µiZ +µiL +µiM−µiNλX +µX −µU +µZ −µY −µD −µL +µP +µiS −µiN+µiM +µiV −µiF +µiE −µiT +µiR −µiQ +µiX −µiU +µiZ −µiY −µiD −µiL +µiP +µS −µN +µM +µV −µF +µE −µT +µR −µQλY +µY +µZ +µU +µX +µE +µM +µQ +µiN +µiS +µiL +µiF +µiV +µiD +µiR +µiT +µiP +µiY +µiZ +µiU +µiX +µiE +µiM +µiQ +µN +µS +µL +µF +µV +µD +µR +µT +µP

λZ +µZ −µY −µX +µU −µF +µN −µR −µiM −µiL +µiS +µiE +µiD −µiV +µiQ +µiP −µiT +µiZ −µiY −µiX +µiU −µiF +µiN −µiR −µM −µL +µS +µE +µD −µV +µQ +µP −µTλP +µP +µT −µR −µQ +µL −µD −µX −µiV +µiF +µiE +µiS −µiN−µiM −µiU +µiZ +µiY +µiP +µiT −µiR −µiQ +µiL −µiD −µiX −µV +µF +µE +µS −µN −µM −µU +µZ +µY

λQ +µQ −µR −µT +µP −µM +µE −µY −µiF −µiV +µiD +µiN +µiS −µiL +µiZ +µiU −µiX +µiQ −µiR −µiT +µiP −µiM +µiE −µiY −µF −µV +µD +µN +µS −µL +µZ +µU −µXλR +µR +µQ +µP +µT +µN +µF +µZ +µiE +µiD +µiV +µiM +µiL +µiS +µiY +µiX +µiU +µiR +µiQ +µiP +µiT +µiN +µiF +µiZ +µE +µD +µV +µM +µL +µS +µY +µX +µU

λD +µD +µV +µF +µE +µX +µP +µL +µiT +µiR +µiQ +µiU +µiZ +µiY +µiS +µiN+µiM +µiD +µiV +µiF +µiE +µiX +µiP +µiL +µT +µR +µQ +µU +µZ +µY +µS +µN +µM

λE +µE −µF +µV −µD −µY −µQ +µM +µiR −µiT +µiP +µiZ −µiU +µiX −µiN +µiS −µiL +µiE −µiF +µiV −µiD −µiY −µiQ+µiM +µR −µT +µP +µZ −µU +µX −µN +µS −µLλF +µF +µE −µD −µV +µZ −µR −µN −µiQ +µiP +µiT +µiY −µiX −µiU −µiM +µiL +µiS +µiF +µiE −µiD −µiV +µiZ −µiR −µiN −µQ +µP +µT +µY −µX −µU −µM +µL +µS


Table 7. Kaotic Algebra multiplication table, part C1

µS µL µM µN µT µU µV µiX µiY µiZ µiP µiQ µiR µiD µiE µiF µiS µiL µiM µiN µiT µiU µiV µX µY µZ µP µQ µR µD µE µF

S +µS +µL +µM +µN +µT +µU +µV +µiX +µiY +µiZ +µiP +µiQ +µiR +µiD +µiE +µiF +µiS +µiL +µiM+µiN +µiT +µiU +µiV +µX +µY +µZ +µP +µQ +µR +µD +µE +µF

L −µL +µS −µN +µM +µP −µX +µD +µiU −µiZ +µiY −µiT +µiR −µiQ −µiV +µiF −µiE −µiL +µiS −µiN+µiM +µiP −µiX +µiD +µU −µZ +µY −µT +µR −µQ −µV +µF −µEM −µM +µN +µS −µL −µQ +µY +µE −µiZ −µiU +µiX +µiR +µiT −µiP −µiF −µiV +µiD−µiM+µiN +µiS −µiL −µiQ +µiY +µiE −µZ −µU +µX +µR +µT −µP −µF −µV +µD

N −µN −µM +µL +µS +µR +µZ −µF +µiY −µiX −µiU +µiQ −µiP −µiT −µiE +µiD +µiV −µiN−µiM +µiL +µiS +µiR +µiZ −µiF +µY −µX −µU +µQ −µP −µT −µE +µD +µV

T −µT +µP −µQ +µR +µS −µV +µU +µiD −µiE +µiF −µiL +µiM−µiN −µiX +µiY −µiZ −µiT +µiP −µiQ +µiR +µiS −µiV +µiU +µD −µE +µF −µL +µM −µN −µX +µY −µZU −µU −µX +µY +µZ +µV +µS −µT +µiL −µiM−µiN +µiD −µiE −µiF −µiP +µiQ +µiR −µiU −µiX +µiY +µiZ +µiV +µiS −µiT +µL −µM −µN +µD −µE −µF −µP +µQ +µR

V −µV +µD +µE −µF −µU +µT +µS −µiP −µiQ +µiR +µiX +µiY −µiZ −µiL −µiM+µiN −µiV +µiD +µiE −µiF −µiU +µiT +µiS −µP −µQ +µR +µX +µY −µZ −µL −µM +µN

iX −µiX +µiU +µiZ −µiY −µiD +µiL +µiP +µS +µN −µM −µV −µF +µE +µT +µR −µQ +µX −µU −µZ +µY +µD −µL −µP −µiS −µiN+µiM +µiV +µiF −µiE −µiT −µiR +µiQ

iY −µiY +µiZ −µiU +µiX +µiE −µiM +µiQ −µN +µS −µL +µF −µV +µD +µR −µT +µP +µY −µZ +µU −µX −µE +µM −µQ +µiN −µiS +µiL −µiF +µiV −µiD −µiR +µiT −µiPiZ −µiZ −µiY −µiX −µiU −µiF −µiN −µiR +µM +µL +µS +µE +µD +µV +µQ +µP +µT +µZ +µY +µX +µU +µF +µN +µR −µiM −µiL −µiS −µiE −µiD −µiV −µiQ −µiP −µiTiP −µiP −µiT −µiR −µiQ −µiL −µiD −µiX +µV +µF +µE +µS +µN +µM +µU +µZ +µY +µP +µT +µR +µQ +µL +µD +µX −µiV −µiF −µiE −µiS −µiN−µiM −µiU −µiZ −µiYiQ −µiQ −µiR +µiT +µiP +µiM +µiE −µiY −µF +µV +µD −µN +µS +µL +µZ −µU −µX +µQ +µR −µT −µP −µM −µE +µY +µiF −µiV −µiD +µiN −µiS −µiL −µiZ +µiU +µiX

iR −µiR +µiQ +µiP −µiT −µiN +µiF +µiZ +µE +µD −µV −µM −µL +µS +µY +µX −µU +µR −µQ −µP +µT +µN −µF −µZ −µiE −µiD +µiV +µiM +µiL −µiS −µiY −µiX +µiU

iD −µiD −µiV +µiF +µiE +µiX +µiP −µiL −µT +µR +µQ −µU +µZ +µY +µS −µN −µM +µD +µV −µF −µE −µX −µP +µL +µiT −µiR −µiQ +µiU −µiZ −µiY −µiS +µiN+µiM

iE −µiE −µiF −µiV −µiD −µiY −µiQ−µiM +µR +µT +µP +µZ +µU +µX +µN +µS +µL +µE +µF +µV +µD +µY +µQ +µM −µiR −µiT −µiP −µiZ −µiU −µiX −µiN −µiS −µiLiF −µiF +µiE −µiD +µiV +µiZ −µiR +µiN −µQ +µP −µT +µY −µX +µU +µM −µL +µS +µF −µE +µD −µV −µZ +µR −µN +µiQ −µiP +µiT −µiY +µiX −µiU −µiM +µiL −µiSiS −µiS −µiL −µiM−µiN −µiT −µiU −µiV +µX +µY +µZ +µP +µQ +µR +µD +µE +µF +µS +µL +µM +µN +µT +µU +µV −µiX −µiY −µiZ −µiP −µiQ −µiR −µiD −µiE −µiFiL +µiL −µiS +µiN−µiM −µiP +µiX −µiD +µU −µZ +µY −µT +µR −µQ −µV +µF −µE −µL +µS −µN +µM +µP −µX +µD −µiU +µiZ −µiY +µiT −µiR +µiQ +µiV −µiF +µiE

iM +µiM−µiN −µiS +µiL +µiQ −µiY −µiE −µZ −µU +µX +µR +µT −µP −µF −µV +µD −µM +µN +µS −µL −µQ +µY +µE +µiZ +µiU −µiX −µiR −µiT +µiP +µiF +µiV −µiDiN +µiN+µiM −µiL −µiS −µiR −µiZ +µiF +µY −µX −µU +µQ −µP −µT −µE +µD +µV −µN −µM +µL +µS +µR +µZ −µF −µiY +µiX +µiU −µiQ +µiP +µiT +µiE −µiD −µiViT +µiT −µiP +µiQ −µiR −µiS +µiV −µiU +µD −µE +µF −µL +µM −µN −µX +µY −µZ −µT +µP −µQ +µR +µS −µV +µU −µiD +µiE −µiF +µiL −µiM+µiN +µiX −µiY +µiZ

iU +µiU +µiX −µiY −µiZ −µiV −µiS +µiT +µL −µM −µN +µD −µE −µF −µP +µQ +µR −µU −µX +µY +µZ +µV +µS −µT −µiL +µiM+µiN −µiD +µiE +µiF +µiP −µiQ −µiRiV +µiV −µiD −µiE +µiF +µiU −µiT −µiS −µP −µQ +µR +µX +µY −µZ −µL −µM +µN −µV +µD +µE −µF −µU +µT +µS +µiP +µiQ −µiR −µiX −µiY +µiZ +µiL +µiM−µiNX +µX −µU −µZ +µY +µD −µL −µP +µiS +µiN−µiM −µiV −µiF +µiE +µiT +µiR −µiQ +µiX −µiU −µiZ +µiY +µiD −µiL −µiP +µS +µN −µM −µV −µF +µE +µT +µR −µQY +µY −µZ +µU −µX −µE +µM −µQ −µiN +µiS −µiL +µiF −µiV +µiD +µiR −µiT +µiP +µiY −µiZ +µiU −µiX −µiE +µiM −µiQ −µN +µS −µL +µF −µV +µD +µR −µT +µP

Z +µZ +µY +µX +µU +µF +µN +µR +µiM +µiL +µiS +µiE +µiD +µiV +µiQ +µiP +µiT +µiZ +µiY +µiX +µiU +µiF +µiN +µiR +µM +µL +µS +µE +µD +µV +µQ +µP +µT

P +µP +µT +µR +µQ +µL +µD +µX +µiV +µiF +µiE +µiS +µiN+µiM +µiU +µiZ +µiY +µiP +µiT +µiR +µiQ +µiL +µiD +µiX +µV +µF +µE +µS +µN +µM +µU +µZ +µY

Q +µQ +µR −µT −µP −µM −µE +µY −µiF +µiV +µiD −µiN +µiS +µiL +µiZ −µiU −µiX +µiQ +µiR −µiT −µiP −µiM −µiE +µiY −µF +µV +µD −µN +µS +µL +µZ −µU −µXR +µR −µQ −µP +µT +µN −µF −µZ +µiE +µiD −µiV −µiM −µiL +µiS +µiY +µiX −µiU +µiR −µiQ −µiP +µiT +µiN −µiF −µiZ +µE +µD −µV −µM −µL +µS +µY +µX −µUD +µD +µV −µF −µE −µX −µP +µL −µiT +µiR +µiQ −µiU +µiZ +µiY +µiS −µiN−µiM +µiD +µiV −µiF −µiE −µiX −µiP +µiL −µT +µR +µQ −µU +µZ +µY +µS −µN −µME +µE +µF +µV +µD +µY +µQ +µM +µiR +µiT +µiP +µiZ +µiU +µiX +µiN +µiS +µiL +µiE +µiF +µiV +µiD +µiY +µiQ+µiM +µR +µT +µP +µZ +µU +µX +µN +µS +µL

F +µF −µE +µD −µV −µZ +µR −µN −µiQ +µiP −µiT +µiY −µiX +µiU +µiM −µiL +µiS +µiF −µiE +µiD −µiV −µiZ +µiR −µiN −µQ +µP −µT +µY −µX +µU +µM −µL +µS

νS +λS +λL +λM +λN +λT +λU +λV +λiX +λiY +λiZ +λiP +λiQ +λiR +λiD +λiE +λiF +λiS +λiL +λiM +λiN +λiT +λiU +λiV +λX +λY +λZ +λP +λQ +λR +λD +λE +λF

νL +λL −λS +λN −λM −λP +λX −λD −λiU +λiZ −λiY +λiT −λiR +λiQ +λiV −λiF +λiE +λiL −λiS +λiN −λiM −λiP +λiX −λiD −λU +λZ −λY +λT −λR +λQ +λV −λF +λE

νM +λM −λN −λS +λL +λQ −λY −λE +λiZ +λiU −λiX −λiR −λiT +λiP +λiF +λiV −λiD+λiM −λiN −λiS +λiL +λiQ −λiY −λiE +λZ +λU −λX −λR −λT +λP +λF +λV −λDνN +λN +λM −λL −λS −λR −λZ +λF −λiY +λiX +λiU −λiQ +λiP +λiT +λiE −λiD −λiV +λiN +λiM −λiL −λiS −λiR −λiZ +λiF −λY +λX +λU −λQ +λP +λT +λE −λD −λVνT +λT −λP +λQ −λR −λS +λV −λU −λiD +λiE −λiF +λiL −λiM +λiN +λiX −λiY +λiZ +λiT −λiP +λiQ −λiR −λiS +λiV −λiU −λD +λE −λF +λL −λM +λN +λX −λY +λZ

νU +λU +λX −λY −λZ −λV −λS +λT −λiL +λiM +λiN −λiD +λiE +λiF +λiP −λiQ −λiR +λiU +λiX −λiY −λiZ −λiV −λiS +λiT −λL +λM +λN −λD +λE +λF +λP −λQ −λRνV +λV −λD −λE +λF +λU −λT −λS +λiP +λiQ −λiR −λiX −λiY +λiZ +λiL +λiM −λiN +λiV −λiD −λiE +λiF +λiU −λiT −λiS +λP +λQ −λR −λX −λY +λZ +λL +λM −λNν iX +λiX −λiU −λiZ +λiY +λiD −λiL −λiP −λS −λN +λM +λV +λF −λE −λT −λR +λQ −λX +λU +λZ −λY −λD +λL +λP +λiS +λiN −λiM −λiV −λiF +λiE +λiT +λiR −λiQν iY +λiY −λiZ +λiU −λiX −λiE +λiM −λiQ +λN −λS +λL −λF +λV −λD −λR +λT −λP −λY +λZ −λU +λX +λE −λM +λQ −λiN +λiS −λiL +λiF −λiV +λiD +λiR −λiT +λiP

ν iZ +λiZ +λiY +λiX +λiU +λiF +λiN +λiR −λM −λL −λS −λE −λD −λV −λQ −λP −λT −λZ −λY −λX −λU −λF −λN −λR +λiM +λiL +λiS +λiE +λiD +λiV +λiQ +λiP +λiT

ν iP +λiP +λiT +λiR +λiQ +λiL +λiD +λiX −λV −λF −λE −λS −λN −λM −λU −λZ −λY −λP −λT −λR −λQ −λL −λD −λX +λiV +λiF +λiE +λiS +λiN +λiM +λiU +λiZ +λiY

ν iQ +λiQ +λiR −λiT −λiP −λiM −λiE +λiY +λF −λV −λD +λN −λS −λL −λZ +λU +λX −λQ −λR +λT +λP +λM +λE −λY −λiF +λiV +λiD −λiN +λiS +λiL +λiZ −λiU −λiXν iR +λiR −λiQ −λiP +λiT +λiN −λiF −λiZ −λE −λD +λV +λM +λL −λS −λY −λX +λU −λR +λQ +λP −λT −λN +λF +λZ +λiE +λiD −λiV −λiM −λiL +λiS +λiY +λiX −λiUν iD +λiD +λiV −λiF −λiE −λiX −λiP +λiL +λT −λR −λQ +λU −λZ −λY −λS +λN +λM −λD −λV +λF +λE +λX +λP −λL −λiT +λiR +λiQ −λiU +λiZ +λiY +λiS −λiN −λiMν iE +λiE +λiF +λiV +λiD +λiY +λiQ+λiM −λR −λT −λP −λZ −λU −λX −λN −λS −λL −λE −λF −λV −λD −λY −λQ −λM +λiR +λiT +λiP +λiZ +λiU +λiX +λiN +λiS +λiL

ν iF +λiF −λiE +λiD −λiV −λiZ +λiR −λiN +λQ −λP +λT −λY +λX −λU −λM +λL −λS −λF +λE −λD +λV +λZ −λR +λN −λiQ +λiP −λiT +λiY −λiX +λiU +λiM −λiL +λiS

ν iS +λiS +λiL +λiM +λiN +λiT +λiU +λiV −λX −λY −λZ −λP −λQ −λR −λD −λE −λF −λS −λL −λM −λN −λT −λU −λV +λiX +λiY +λiZ +λiP +λiQ +λiR +λiD +λiE +λiF

ν iL +λiL −λiS +λiN −λiM −λiP +λiX −λiD +λU −λZ +λY −λT +λR −λQ −λV +λF −λE −λL +λS −λN +λM +λP −λX +λD −λiU +λiZ −λiY +λiT −λiR +λiQ +λiV −λiF +λiE

ν iM +λiM −λiN −λiS +λiL +λiQ −λiY −λiE −λZ −λU +λX +λR +λT −λP −λF −λV +λD −λM +λN +λS −λL −λQ +λY +λE +λiZ +λiU −λiX −λiR −λiT +λiP +λiF +λiV −λiDν iN +λiN +λiM −λiL −λiS −λiR −λiZ +λiF +λY −λX −λU +λQ −λP −λT −λE +λD +λV −λN −λM +λL +λS +λR +λZ −λF −λiY +λiX +λiU −λiQ +λiP +λiT +λiE −λiD −λiVν iT +λiT −λiP +λiQ −λiR −λiS +λiV −λiU +λD −λE +λF −λL +λM −λN −λX +λY −λZ −λT +λP −λQ +λR +λS −λV +λU −λiD +λiE −λiF +λiL −λiM +λiN +λiX −λiY +λiZ

ν iU +λiU +λiX −λiY −λiZ −λiV −λiS +λiT +λL −λM −λN +λD −λE −λF −λP +λQ +λR −λU −λX +λY +λZ +λV +λS −λT −λiL +λiM +λiN −λiD +λiE +λiF +λiP −λiQ −λiRν iV +λiV −λiD −λiE +λiF +λiU −λiT −λiS −λP −λQ +λR +λX +λY −λZ −λL −λM +λN −λV +λD +λE −λF −λU +λT +λS +λiP +λiQ −λiR −λiX −λiY +λiZ +λiL +λiM −λiNνX +λX −λU −λZ +λY +λD −λL −λP +λiS +λiN −λiM −λiV −λiF +λiE +λiT +λiR −λiQ +λiX −λiU −λiZ +λiY +λiD −λiL −λiP +λS +λN −λM −λV −λF +λE +λT +λR −λQνY +λY −λZ +λU −λX −λE +λM −λQ −λiN +λiS −λiL +λiF −λiV +λiD +λiR −λiT +λiP +λiY −λiZ +λiU −λiX −λiE +λiM −λiQ −λN +λS −λL +λF −λV +λD +λR −λT +λP

νZ +λZ +λY +λX +λU +λF +λN +λR +λiM +λiL +λiS +λiE +λiD +λiV +λiQ +λiP +λiT +λiZ +λiY +λiX +λiU +λiF +λiN +λiR +λM +λL +λS +λE +λD +λV +λQ +λP +λT

νP +λP +λT +λR +λQ +λL +λD +λX +λiV +λiF +λiE +λiS +λiN +λiM +λiU +λiZ +λiY +λiP +λiT +λiR +λiQ +λiL +λiD +λiX +λV +λF +λE +λS +λN +λM +λU +λZ +λY

νQ +λQ +λR −λT −λP −λM −λE +λY −λiF +λiV +λiD −λiN +λiS +λiL +λiZ −λiU −λiX +λiQ +λiR −λiT −λiP −λiM −λiE +λiY −λF +λV +λD −λN +λS +λL +λZ −λU −λXνR +λR −λQ −λP +λT +λN −λF −λZ +λiE +λiD −λiV −λiM −λiL +λiS +λiY +λiX −λiU +λiR −λiQ −λiP +λiT +λiN −λiF −λiZ +λE +λD −λV −λM −λL +λS +λY +λX −λUνD +λD +λV −λF −λE −λX −λP +λL −λiT +λiR +λiQ −λiU +λiZ +λiY +λiS −λiN −λiM +λiD +λiV −λiF −λiE −λiX −λiP +λiL −λT +λR +λQ −λU +λZ +λY +λS −λN −λMνE +λE +λF +λV +λD +λY +λQ +λM +λiR +λiT +λiP +λiZ +λiU +λiX +λiN +λiS +λiL +λiE +λiF +λiV +λiD +λiY +λiQ+λiM +λR +λT +λP +λZ +λU +λX +λN +λS +λL

νF +λF −λE +λD −λV −λZ +λR −λN −λiQ +λiP −λiT +λiY −λiX +λiU +λiM −λiL +λiS +λiF −λiE +λiD −λiV −λiZ +λiR −λiN −λQ +λP −λT +λY −λX +λU +λM −λL +λS


Table 8. Kaotic Algebra multiplication table, part C2

µS µL µM µN µT µU µV µiX µiY µiZ µiP µiQ µiR µiD µiE µiF µiS µiL µiM µiN µiT µiU µiV µX µY µZ µP µQ µR µD µE µF

µS −S −L −M −N −T −U −V −iX −iY −iZ −iP −iQ −iR −iD −iE −iF −iS −iL −iM −iN −iT −iU −iV −X −Y −Z −P −Q −R −D −E −FµL +L −S −N +M −P +X −D −iU −iZ +iY +iT +iR −iQ +iV +iF −iE +iL −iS −iN +iM −iP +iX −iD −U −Z +Y +T +R −Q +V +F −EµM +M +N −S −L +Q −Y −E −iZ +iU +iX +iR −iT −iP −iF +iV +iD +iM +iN −iS −iL +iQ −iY −iE −Z +U +X +R −T −P −F +V +D

µN +N −M +L −S −R −Z +F +iY −iX +iU +iQ −iP +iT −iE +iD −iV +iN −iM +iL −iS −iR −iZ +iF +Y −X +U +Q −P +T −E +D −VµT +T −P +Q −R −S −V +U +iD −iE +iF +iL −iM +iN −iX +iY −iZ +iT −iP +iQ −iR −iS −iV +iU +D −E +F +L −M +N −X +Y −ZµU +U +X −Y −Z +V −S −T −iL +iM +iN +iD −iE −iF −iP +iQ +iR +iU +iX −iY −iZ +iV −iS −iT −L +M +N +D −E −F −P +Q +R

µV +V −D −E +F −U +T −S −iP −iQ +iR +iX +iY −iZ +iL +iM −iN +iV −iD −iE +iF −iU +iT −iS −P −Q +R +X +Y −Z +L +M −NµiX +iX −iU +iZ −iY −iD −iL +iP −S +N −M −V +F −E +T −R +Q −X +U −Z +Y +D +L −P +iS −iN +iM +iV −iF +iE −iT +iR −iQµiY +iY +iZ +iU +iX +iE +iM +iQ −N −S −L −F −V −D −R −T −P −Y −Z −U −X −E −M −Q +iN +iS +iL +iF +iV +iD +iR +iT +iP

µiZ +iZ −iY −iX +iU −iF +iN −iR +M +L −S −E −D +V −Q −P +T −Z +Y +X −U +F −N +R −iM −iL +iS +iE +iD −iV +iQ +iP −iTµiP +iP +iT −iR −iQ +iL −iD −iX +V −F −E −S +N +M +U −Z −Y −P −T +R +Q −L +D +X −iV +iF +iE +iS −iN −iM −iU +iZ +iY

µiQ +iQ −iR −iT +iP −iM +iE −iY +F +V −D −N −S +L −Z −U +X −Q +R +T −P +M −E +Y −iF −iV +iD +iN +iS −iL +iZ +iU −iXµiR +iR +iQ +iP +iT +iN +iF +iZ −E −D −V −M −L −S −Y −X −U −R −Q −P −T −N −F −Z +iE +iD +iV +iM +iL +iS +iY +iX +iU

µiD +iD +iV +iF +iE +iX +iP +iL −T −R −Q −U −Z −Y −S −N −M −D −V −F −E −X −P −L +iT +iR +iQ +iU +iZ +iY +iS +iN +iM

µiE +iE −iF +iV −iD −iY −iQ +iM −R +T −P −Z +U −X +N −S +L −E +F −V +D +Y +Q −M +iR −iT +iP +iZ −iU +iX −iN +iS −iLµiF +iF +iE −iD −iV +iZ −iR −iN +Q −P −T −Y +X +U +M −L −S −F −E +D +V −Z +R +N −iQ +iP +iT +iY −iX −iU −iM +iL +iS

µiS +iS +iL +iM +iN +iT +iU +iV −X −Y −Z −P −Q −R −D −E −F −S −L −M −N −T −U −V +iX +iY +iZ +iP +iQ +iR +iD +iE +iF

µiL −iL +iS +iN −iM +iP −iX +iD −U −Z +Y +T +R −Q +V +F −E +L −S −N +M −P +X −D +iU +iZ −iY −iT −iR +iQ −iV −iF +iE

µiM −iM −iN +iS +iL −iQ +iY +iE −Z +U +X +R −T −P −F +V +D +M +N −S −L +Q −Y −E +iZ −iU −iX −iR +iT +iP +iF −iV −iDµiN −iN +iM −iL +iS +iR +iZ −iF +Y −X +U +Q −P +T −E +D −V +N −M +L −S −R −Z +F −iY +iX −iU −iQ +iP −iT +iE −iD +iV

µiT −iT +iP −iQ +iR +iS +iV −iU +D −E +F +L −M +N −X +Y −Z +T −P +Q −R −S −V +U −iD +iE −iF −iL +iM −iN +iX −iY +iZ

µiU −iU −iX +iY +iZ −iV +iS +iT −L +M +N +D −E −F −P +Q +R +U +X −Y −Z +V −S −T +iL −iM −iN −iD +iE +iF +iP −iQ −iRµiV −iV +iD +iE −iF +iU −iT +iS −P −Q +R +X +Y −Z +L +M −N +V −D −E +F −U +T −S +iP +iQ −iR −iX −iY +iZ −iL −iM +iN

µX −X +U −Z +Y +D +L −P −iS +iN −iM −iV +iF −iE +iT −iR +iQ −iX +iU −iZ +iY +iD +iL −iP −S +N −M −V +F −E +T −R +Q

µY −Y −Z −U −X −E −M −Q −iN −iS −iL −iF −iV −iD −iR −iT −iP −iY −iZ −iU −iX −iE −iM −iQ −N −S −L −F −V −D −R −T −PµZ −Z +Y +X −U +F −N +R +iM +iL −iS −iE −iD +iV −iQ −iP +iT −iZ +iY +iX −iU +iF −iN +iR +M +L −S −E −D +V −Q −P +T

µP −P −T +R +Q −L +D +X +iV −iF −iE −iS +iN +iM +iU −iZ −iY −iP −iT +iR +iQ −iL +iD +iX +V −F −E −S +N +M +U −Z −YµQ −Q +R +T −P +M −E +Y +iF +iV −iD −iN −iS +iL −iZ −iU +iX −iQ +iR +iT −iP +iM −iE +iY +F +V −D −N −S +L −Z −U +X

µR −R −Q −P −T −N −F −Z −iE −iD −iV −iM −iL −iS −iY −iX −iU −iR −iQ −iP −iT −iN −iF −iZ −E −D −V −M −L −S −Y −X −UµD −D −V −F −E −X −P −L −iT −iR −iQ −iU −iZ −iY −iS −iN −iM −iD −iV −iF −iE −iX −iP −iL −T −R −Q −U −Z −Y −S −N −MµE −E +F −V +D +Y +Q −M −iR +iT −iP −iZ +iU −iX +iN −iS +iL −iE +iF −iV +iD +iY +iQ −iM −R +T −P −Z +U −X +N −S +L

µF −F −E +D +V −Z +R +N +iQ −iP −iT −iY +iX +iU +iM −iL −iS −iF −iE +iD +iV −iZ +iR +iN +Q −P −T −Y +X +U +M −L −SλS −νS −νL −νM −νN −νT −νU −νV −ν iX −ν iY −ν iZ −ν iP −ν iQ −ν iR −ν iD −ν iE −ν iF −ν iS −ν iL −ν iM−ν iN −ν iT −ν iU −ν iV −νX −νY −νZ −νP −νQ −νR −νD −νE −νFλL −νL +νS −νN +νM +νP −νX +νD +ν iU −ν iZ +ν iY −ν iT +ν iR −ν iQ −ν iV +ν iF −ν iE −ν iL +ν iS −ν iN+ν iM +ν iP −ν iX +ν iD +νU −νZ +νY −νT +νR −νQ −νV +νF −νEλM −νM +νN +νS −νL −νQ +νY +νE −ν iZ −ν iU +ν iX +ν iR +ν iT −ν iP −ν iF −ν iV +ν iD−ν iM+ν iN +ν iS −ν iL −ν iQ +ν iY +ν iE −νZ −νU +νX +νR +νT −νP −νF −νV +νD

λN −νN −νM +νL +νS +νR +νZ −νF +ν iY −ν iX −ν iU +ν iQ −ν iP −ν iT −ν iE +ν iD +ν iV −ν iN−ν iM +ν iL +ν iS +ν iR +ν iZ −ν iF +νY −νX −νU +νQ −νP −νT −νE +νD +νV

λT −νT +νP −νQ +νR +νS −νV +νU +ν iD −ν iE +ν iF −ν iL +ν iM−ν iN −ν iX +ν iY −ν iZ −ν iT +ν iP −ν iQ +ν iR +ν iS −ν iV +ν iU +νD −νE +νF −νL +νM −νN −νX +νY −νZλU −νU −νX +νY +νZ +νV +νS −νT +ν iL −ν iM−ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR −ν iU −ν iX +ν iY +ν iZ +ν iV +ν iS −ν iT +νL −νM −νN +νD −νE −νF −νP +νQ +νR

λV −νV +νD +νE −νF −νU +νT +νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ −ν iL −ν iM+ν iN −ν iV +ν iD +ν iE −ν iF −ν iU +ν iT +ν iS −νP −νQ +νR +νX +νY −νZ −νL −νM +νN

λiX −ν iX +ν iU +ν iZ −ν iY −ν iD +ν iL +ν iP +νS +νN −νM −νV −νF +νE +νT +νR −νQ +νX −νU −νZ +νY +νD −νL −νP −ν iS −ν iN+ν iM +ν iV +ν iF −ν iE −ν iT −ν iR +ν iQ

λiY −ν iY +ν iZ −ν iU +ν iX +ν iE −ν iM +ν iQ −νN +νS −νL +νF −νV +νD +νR −νT +νP +νY −νZ +νU −νX −νE +νM −νQ +ν iN −ν iS +ν iL −ν iF +ν iV −ν iD −ν iR +ν iT −ν iPλiZ −ν iZ −ν iY −ν iX −ν iU −ν iF −ν iN −ν iR +νM +νL +νS +νE +νD +νV +νQ +νP +νT +νZ +νY +νX +νU +νF +νN +νR −ν iM −ν iL −ν iS −ν iE −ν iD −ν iV −ν iQ −ν iP −ν iTλiP −ν iP −ν iT −ν iR −ν iQ −ν iL −ν iD −ν iX +νV +νF +νE +νS +νN +νM +νU +νZ +νY +νP +νT +νR +νQ +νL +νD +νX −ν iV −ν iF −ν iE −ν iS −ν iN−ν iM −ν iU −ν iZ −ν iYλiQ −ν iQ −ν iR +ν iT +ν iP +ν iM +ν iE −ν iY −νF +νV +νD −νN +νS +νL +νZ −νU −νX +νQ +νR −νT −νP −νM −νE +νY +ν iF −ν iV −ν iD +ν iN −ν iS −ν iL −ν iZ +ν iU +ν iX

λiR −ν iR +ν iQ +ν iP −ν iT −ν iN +ν iF +ν iZ +νE +νD −νV −νM −νL +νS +νY +νX −νU +νR −νQ −νP +νT +νN −νF −νZ −ν iE −ν iD +ν iV +ν iM +ν iL −ν iS −ν iY −ν iX +ν iU

λiD −ν iD −ν iV +ν iF +ν iE +ν iX +ν iP −ν iL −νT +νR +νQ −νU +νZ +νY +νS −νN −νM +νD +νV −νF −νE −νX −νP +νL +ν iT −ν iR −ν iQ +ν iU −ν iZ −ν iY −ν iS +ν iN+ν iM

λiE −ν iE −ν iF −ν iV −ν iD −ν iY −ν iQ−ν iM +νR +νT +νP +νZ +νU +νX +νN +νS +νL +νE +νF +νV +νD +νY +νQ +νM −ν iR −ν iT −ν iP −ν iZ −ν iU −ν iX −ν iN −ν iS −ν iLλiF −ν iF +ν iE −ν iD +ν iV +ν iZ −ν iR +ν iN −νQ +νP −νT +νY −νX +νU +νM −νL +νS +νF −νE +νD −νV −νZ +νR −νN +ν iQ −ν iP +ν iT −ν iY +ν iX −ν iU −ν iM +ν iL −ν iSλiS −ν iS −ν iL −ν iM−ν iN −ν iT −ν iU −ν iV +νX +νY +νZ +νP +νQ +νR +νD +νE +νF +νS +νL +νM +νN +νT +νU +νV −ν iX −ν iY −ν iZ −ν iP −ν iQ −ν iR −ν iD −ν iE −ν iFλiL −ν iL +ν iS −ν iN+ν iM +ν iP −ν iX +ν iD −νU +νZ −νY +νT −νR +νQ +νV −νF +νE +νL −νS +νN −νM −νP +νX −νD +ν iU −ν iZ +ν iY −ν iT +ν iR −ν iQ −ν iV +ν iF −ν iEλiM −ν iM+ν iN +ν iS −ν iL −ν iQ +ν iY +ν iE +νZ +νU −νX −νR −νT +νP +νF +νV −νD +νM −νN −νS +νL +νQ −νY −νE −ν iZ −ν iU +ν iX +ν iR +ν iT −ν iP −ν iF −ν iV +ν iD

λiN −ν iN−ν iM +ν iL +ν iS +ν iR +ν iZ −ν iF −νY +νX +νU −νQ +νP +νT +νE −νD −νV +νN +νM −νL −νS −νR −νZ +νF +ν iY −ν iX −ν iU +ν iQ −ν iP −ν iT −ν iE +ν iD +ν iV

λiT −ν iT +ν iP −ν iQ +ν iR +ν iS −ν iV +ν iU −νD +νE −νF +νL −νM +νN +νX −νY +νZ +νT −νP +νQ −νR −νS +νV −νU +ν iD −ν iE +ν iF −ν iL +ν iM−ν iN −ν iX +ν iY −ν iZλiU −ν iU −ν iX +ν iY +ν iZ +ν iV +ν iS −ν iT −νL +νM +νN −νD +νE +νF +νP −νQ −νR +νU +νX −νY −νZ −νV −νS +νT +ν iL −ν iM−ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR

λiV −ν iV +ν iD +ν iE −ν iF −ν iU +ν iT +ν iS +νP +νQ −νR −νX −νY +νZ +νL +νM −νN +νV −νD −νE +νF +νU −νT −νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ −ν iL −ν iM+ν iN

λX −νX +νU +νZ −νY −νD +νL +νP −ν iS −ν iN+ν iM +ν iV +ν iF −ν iE −ν iT −ν iR +ν iQ −ν iX +ν iU +ν iZ −ν iY −ν iD +ν iL +ν iP −νS −νN +νM +νV +νF −νE −νT −νR +νQ

λY −νY +νZ −νU +νX +νE −νM +νQ +ν iN −ν iS +ν iL −ν iF +ν iV −ν iD −ν iR +ν iT −ν iP −ν iY +ν iZ −ν iU +ν iX +ν iE −ν iM +ν iQ +νN −νS +νL −νF +νV −νD −νR +νT −νPλZ −νZ −νY −νX −νU −νF −νN −νR −ν iM −ν iL −ν iS −ν iE −ν iD −ν iV −ν iQ −ν iP −ν iT −ν iZ −ν iY −ν iX −ν iU −ν iF −ν iN −ν iR −νM −νL −νS −νE −νD −νV −νQ −νP −νTλP −νP −νT −νR −νQ −νL −νD −νX −ν iV −ν iF −ν iE −ν iS −ν iN−ν iM −ν iU −ν iZ −ν iY −ν iP −ν iT −ν iR −ν iQ −ν iL −ν iD −ν iX −νV −νF −νE −νS −νN −νM −νU −νZ −νYλQ −νQ −νR +νT +νP +νM +νE −νY +ν iF −ν iV −ν iD +ν iN −ν iS −ν iL −ν iZ +ν iU +ν iX −ν iQ −ν iR +ν iT +ν iP +ν iM +ν iE −ν iY +νF −νV −νD +νN −νS −νL −νZ +νU +νX

λR −νR +νQ +νP −νT −νN +νF +νZ −ν iE −ν iD +ν iV +ν iM +ν iL −ν iS −ν iY −ν iX +ν iU −ν iR +ν iQ +ν iP −ν iT −ν iN +ν iF +ν iZ −νE −νD +νV +νM +νL −νS −νY −νX +νU

λD −νD −νV +νF +νE +νX +νP −νL +ν iT −ν iR −ν iQ +ν iU −ν iZ −ν iY −ν iS +ν iN+ν iM −ν iD −ν iV +ν iF +ν iE +ν iX +ν iP −ν iL +νT −νR −νQ +νU −νZ −νY −νS +νN +νM

λE −νE −νF −νV −νD −νY −νQ −νM −ν iR −ν iT −ν iP −ν iZ −ν iU −ν iX −ν iN −ν iS −ν iL −ν iE −ν iF −ν iV −ν iD −ν iY −ν iQ−ν iM −νR −νT −νP −νZ −νU −νX −νN −νS −νLλF −νF +νE −νD +νV +νZ −νR +νN +ν iQ −ν iP +ν iT −ν iY +ν iX −ν iU −ν iM +ν iL −ν iS −ν iF +ν iE −ν iD +ν iV +ν iZ −ν iR +ν iN +νQ −νP +νT −νY +νX −νU −νM +νL −νS


Table 9. Kaotic Algebra multiplication table, part D1

λS λL λM λN λT λU λV λiX λiY λiZ λiP λiQ λiR λiD λiE λiF λiS λiL λiM λiN λiT λiU λiV λX λY λZ λP λQ λR λD λE λF

S +λS +λL +λM +λN +λT +λU +λV +λiX +λiY +λiZ +λiP +λiQ +λiR +λiD +λiE +λiF +λiS +λiL +λiM +λiN +λiT +λiU +λiV +λX +λY +λZ +λP +λQ +λR +λD +λE +λF

L −λL +λS −λN +λM +λP −λX +λD +λiU −λiZ +λiY −λiT +λiR −λiQ −λiV +λiF −λiE −λiL +λiS −λiN +λiM +λiP −λiX +λiD +λU −λZ +λY −λT +λR −λQ −λV +λF −λEM −λM +λN +λS −λL −λQ +λY +λE −λiZ −λiU +λiX +λiR +λiT −λiP −λiF −λiV +λiD−λiM +λiN +λiS −λiL −λiQ +λiY +λiE −λZ −λU +λX +λR +λT −λP −λF −λV +λD

N −λN −λM +λL +λS +λR +λZ −λF +λiY −λiX −λiU +λiQ −λiP −λiT −λiE +λiD +λiV −λiN −λiM +λiL +λiS +λiR +λiZ −λiF +λY −λX −λU +λQ −λP −λT −λE +λD +λV

T −λT +λP −λQ +λR +λS −λV +λU +λiD −λiE +λiF −λiL +λiM −λiN −λiX +λiY −λiZ −λiT +λiP −λiQ +λiR +λiS −λiV +λiU +λD −λE +λF −λL +λM −λN −λX +λY −λZU −λU −λX +λY +λZ +λV +λS −λT +λiL −λiM −λiN +λiD −λiE −λiF −λiP +λiQ +λiR −λiU −λiX +λiY +λiZ +λiV +λiS −λiT +λL −λM −λN +λD −λE −λF −λP +λQ +λR

V −λV +λD +λE −λF −λU +λT +λS −λiP −λiQ +λiR +λiX +λiY −λiZ −λiL −λiM +λiN −λiV +λiD +λiE −λiF −λiU +λiT +λiS −λP −λQ +λR +λX +λY −λZ −λL −λM +λN

iX −λiX +λiU +λiZ −λiY −λiD +λiL +λiP +λS +λN −λM −λV −λF +λE +λT +λR −λQ +λX −λU −λZ +λY +λD −λL −λP −λiS −λiN +λiM +λiV +λiF −λiE −λiT −λiR +λiQ

iY −λiY +λiZ −λiU +λiX +λiE −λiM +λiQ −λN +λS −λL +λF −λV +λD +λR −λT +λP +λY −λZ +λU −λX −λE +λM −λQ +λiN −λiS +λiL −λiF +λiV −λiD −λiR +λiT −λiPiZ −λiZ −λiY −λiX −λiU −λiF −λiN −λiR +λM +λL +λS +λE +λD +λV +λQ +λP +λT +λZ +λY +λX +λU +λF +λN +λR −λiM −λiL −λiS −λiE −λiD −λiV −λiQ −λiP −λiTiP −λiP −λiT −λiR −λiQ −λiL −λiD −λiX +λV +λF +λE +λS +λN +λM +λU +λZ +λY +λP +λT +λR +λQ +λL +λD +λX −λiV −λiF −λiE −λiS −λiN −λiM −λiU −λiZ −λiYiQ −λiQ −λiR +λiT +λiP +λiM +λiE −λiY −λF +λV +λD −λN +λS +λL +λZ −λU −λX +λQ +λR −λT −λP −λM −λE +λY +λiF −λiV −λiD +λiN −λiS −λiL −λiZ +λiU +λiX

iR −λiR +λiQ +λiP −λiT −λiN +λiF +λiZ +λE +λD −λV −λM −λL +λS +λY +λX −λU +λR −λQ −λP +λT +λN −λF −λZ −λiE −λiD +λiV +λiM +λiL −λiS −λiY −λiX +λiU

iD −λiD −λiV +λiF +λiE +λiX +λiP −λiL −λT +λR +λQ −λU +λZ +λY +λS −λN −λM +λD +λV −λF −λE −λX −λP +λL +λiT −λiR −λiQ +λiU −λiZ −λiY −λiS +λiN +λiM

iE −λiE −λiF −λiV −λiD −λiY −λiQ−λiM +λR +λT +λP +λZ +λU +λX +λN +λS +λL +λE +λF +λV +λD +λY +λQ +λM −λiR −λiT −λiP −λiZ −λiU −λiX −λiN −λiS −λiLiF −λiF +λiE −λiD +λiV +λiZ −λiR +λiN −λQ +λP −λT +λY −λX +λU +λM −λL +λS +λF −λE +λD −λV −λZ +λR −λN +λiQ −λiP +λiT −λiY +λiX −λiU −λiM +λiL −λiSiS −λiS −λiL −λiM −λiN −λiT −λiU −λiV +λX +λY +λZ +λP +λQ +λR +λD +λE +λF +λS +λL +λM +λN +λT +λU +λV −λiX −λiY −λiZ −λiP −λiQ −λiR −λiD −λiE −λiFiL +λiL −λiS +λiN −λiM −λiP +λiX −λiD +λU −λZ +λY −λT +λR −λQ −λV +λF −λE −λL +λS −λN +λM +λP −λX +λD −λiU +λiZ −λiY +λiT −λiR +λiQ +λiV −λiF +λiE

iM +λiM −λiN −λiS +λiL +λiQ −λiY −λiE −λZ −λU +λX +λR +λT −λP −λF −λV +λD −λM +λN +λS −λL −λQ +λY +λE +λiZ +λiU −λiX −λiR −λiT +λiP +λiF +λiV −λiDiN +λiN +λiM −λiL −λiS −λiR −λiZ +λiF +λY −λX −λU +λQ −λP −λT −λE +λD +λV −λN −λM +λL +λS +λR +λZ −λF −λiY +λiX +λiU −λiQ +λiP +λiT +λiE −λiD −λiViT +λiT −λiP +λiQ −λiR −λiS +λiV −λiU +λD −λE +λF −λL +λM −λN −λX +λY −λZ −λT +λP −λQ +λR +λS −λV +λU −λiD +λiE −λiF +λiL −λiM +λiN +λiX −λiY +λiZ

iU +λiU +λiX −λiY −λiZ −λiV −λiS +λiT +λL −λM −λN +λD −λE −λF −λP +λQ +λR −λU −λX +λY +λZ +λV +λS −λT −λiL +λiM +λiN −λiD +λiE +λiF +λiP −λiQ −λiRiV +λiV −λiD −λiE +λiF +λiU −λiT −λiS −λP −λQ +λR +λX +λY −λZ −λL −λM +λN −λV +λD +λE −λF −λU +λT +λS +λiP +λiQ −λiR −λiX −λiY +λiZ +λiL +λiM −λiNX +λX −λU −λZ +λY +λD −λL −λP +λiS +λiN −λiM −λiV −λiF +λiE +λiT +λiR −λiQ +λiX −λiU −λiZ +λiY +λiD −λiL −λiP +λS +λN −λM −λV −λF +λE +λT +λR −λQY +λY −λZ +λU −λX −λE +λM −λQ −λiN +λiS −λiL +λiF −λiV +λiD +λiR −λiT +λiP +λiY −λiZ +λiU −λiX −λiE +λiM −λiQ −λN +λS −λL +λF −λV +λD +λR −λT +λP

Z +λZ +λY +λX +λU +λF +λN +λR +λiM +λiL +λiS +λiE +λiD +λiV +λiQ +λiP +λiT +λiZ +λiY +λiX +λiU +λiF +λiN +λiR +λM +λL +λS +λE +λD +λV +λQ +λP +λT

P +λP +λT +λR +λQ +λL +λD +λX +λiV +λiF +λiE +λiS +λiN +λiM +λiU +λiZ +λiY +λiP +λiT +λiR +λiQ +λiL +λiD +λiX +λV +λF +λE +λS +λN +λM +λU +λZ +λY

Q +λQ +λR −λT −λP −λM −λE +λY −λiF +λiV +λiD −λiN +λiS +λiL +λiZ −λiU −λiX +λiQ +λiR −λiT −λiP −λiM −λiE +λiY −λF +λV +λD −λN +λS +λL +λZ −λU −λXR +λR −λQ −λP +λT +λN −λF −λZ +λiE +λiD −λiV −λiM −λiL +λiS +λiY +λiX −λiU +λiR −λiQ −λiP +λiT +λiN −λiF −λiZ +λE +λD −λV −λM −λL +λS +λY +λX −λUD +λD +λV −λF −λE −λX −λP +λL −λiT +λiR +λiQ −λiU +λiZ +λiY +λiS −λiN −λiM +λiD +λiV −λiF −λiE −λiX −λiP +λiL −λT +λR +λQ −λU +λZ +λY +λS −λN −λME +λE +λF +λV +λD +λY +λQ +λM +λiR +λiT +λiP +λiZ +λiU +λiX +λiN +λiS +λiL +λiE +λiF +λiV +λiD +λiY +λiQ+λiM +λR +λT +λP +λZ +λU +λX +λN +λS +λL

F +λF −λE +λD −λV −λZ +λR −λN −λiQ +λiP −λiT +λiY −λiX +λiU +λiM −λiL +λiS +λiF −λiE +λiD −λiV −λiZ +λiR −λiN −λQ +λP −λT +λY −λX +λU +λM −λL +λS

νS −µS −µL −µM −µN −µT −µU −µV −µiX −µiY −µiZ −µiP −µiQ −µiR −µiD −µiE −µiF −µiS −µiL −µiM−µiN −µiT −µiU −µiV −µX −µY −µZ −µP −µQ −µR −µD −µE −µFνL −µL +µS −µN +µM +µP −µX +µD +µiU −µiZ +µiY −µiT +µiR −µiQ −µiV +µiF −µiE −µiL +µiS −µiN+µiM +µiP −µiX +µiD +µU −µZ +µY −µT +µR −µQ −µV +µF −µEνM −µM +µN +µS −µL −µQ +µY +µE −µiZ −µiU +µiX +µiR +µiT −µiP −µiF −µiV +µiD−µiM+µiN +µiS −µiL −µiQ +µiY +µiE −µZ −µU +µX +µR +µT −µP −µF −µV +µD

νN −µN −µM +µL +µS +µR +µZ −µF +µiY −µiX −µiU +µiQ −µiP −µiT −µiE +µiD +µiV −µiN−µiM +µiL +µiS +µiR +µiZ −µiF +µY −µX −µU +µQ −µP −µT −µE +µD +µV

νT −µT +µP −µQ +µR +µS −µV +µU +µiD −µiE +µiF −µiL +µiM−µiN −µiX +µiY −µiZ −µiT +µiP −µiQ +µiR +µiS −µiV +µiU +µD −µE +µF −µL +µM −µN −µX +µY −µZνU −µU −µX +µY +µZ +µV +µS −µT +µiL −µiM−µiN +µiD −µiE −µiF −µiP +µiQ +µiR −µiU −µiX +µiY +µiZ +µiV +µiS −µiT +µL −µM −µN +µD −µE −µF −µP +µQ +µR

νV −µV +µD +µE −µF −µU +µT +µS −µiP −µiQ +µiR +µiX +µiY −µiZ −µiL −µiM+µiN −µiV +µiD +µiE −µiF −µiU +µiT +µiS −µP −µQ +µR +µX +µY −µZ −µL −µM +µN

ν iX −µiX +µiU +µiZ −µiY −µiD +µiL +µiP +µS +µN −µM −µV −µF +µE +µT +µR −µQ +µX −µU −µZ +µY +µD −µL −µP −µiS −µiN+µiM +µiV +µiF −µiE −µiT −µiR +µiQ

ν iY −µiY +µiZ −µiU +µiX +µiE −µiM +µiQ −µN +µS −µL +µF −µV +µD +µR −µT +µP +µY −µZ +µU −µX −µE +µM −µQ +µiN −µiS +µiL −µiF +µiV −µiD −µiR +µiT −µiPν iZ −µiZ −µiY −µiX −µiU −µiF −µiN −µiR +µM +µL +µS +µE +µD +µV +µQ +µP +µT +µZ +µY +µX +µU +µF +µN +µR −µiM −µiL −µiS −µiE −µiD −µiV −µiQ −µiP −µiTν iP −µiP −µiT −µiR −µiQ −µiL −µiD −µiX +µV +µF +µE +µS +µN +µM +µU +µZ +µY +µP +µT +µR +µQ +µL +µD +µX −µiV −µiF −µiE −µiS −µiN−µiM −µiU −µiZ −µiYν iQ −µiQ −µiR +µiT +µiP +µiM +µiE −µiY −µF +µV +µD −µN +µS +µL +µZ −µU −µX +µQ +µR −µT −µP −µM −µE +µY +µiF −µiV −µiD +µiN −µiS −µiL −µiZ +µiU +µiX

ν iR −µiR +µiQ +µiP −µiT −µiN +µiF +µiZ +µE +µD −µV −µM −µL +µS +µY +µX −µU +µR −µQ −µP +µT +µN −µF −µZ −µiE −µiD +µiV +µiM +µiL −µiS −µiY −µiX +µiU

ν iD −µiD −µiV +µiF +µiE +µiX +µiP −µiL −µT +µR +µQ −µU +µZ +µY +µS −µN −µM +µD +µV −µF −µE −µX −µP +µL +µiT −µiR −µiQ +µiU −µiZ −µiY −µiS +µiN+µiM

ν iE −µiE −µiF −µiV −µiD −µiY −µiQ−µiM +µR +µT +µP +µZ +µU +µX +µN +µS +µL +µE +µF +µV +µD +µY +µQ +µM −µiR −µiT −µiP −µiZ −µiU −µiX −µiN −µiS −µiLν iF −µiF +µiE −µiD +µiV +µiZ −µiR +µiN −µQ +µP −µT +µY −µX +µU +µM −µL +µS +µF −µE +µD −µV −µZ +µR −µN +µiQ −µiP +µiT −µiY +µiX −µiU −µiM +µiL −µiSν iS −µiS −µiL −µiM−µiN −µiT −µiU −µiV +µX +µY +µZ +µP +µQ +µR +µD +µE +µF +µS +µL +µM +µN +µT +µU +µV −µiX −µiY −µiZ −µiP −µiQ −µiR −µiD −µiE −µiFν iL −µiL +µiS −µiN+µiM +µiP −µiX +µiD −µU +µZ −µY +µT −µR +µQ +µV −µF +µE +µL −µS +µN −µM −µP +µX −µD +µiU −µiZ +µiY −µiT +µiR −µiQ −µiV +µiF −µiEν iM −µiM+µiN +µiS −µiL −µiQ +µiY +µiE +µZ +µU −µX −µR −µT +µP +µF +µV −µD +µM −µN −µS +µL +µQ −µY −µE −µiZ −µiU +µiX +µiR +µiT −µiP −µiF −µiV +µiD

ν iN −µiN−µiM +µiL +µiS +µiR +µiZ −µiF −µY +µX +µU −µQ +µP +µT +µE −µD −µV +µN +µM −µL −µS −µR −µZ +µF +µiY −µiX −µiU +µiQ −µiP −µiT −µiE +µiD +µiV

ν iT −µiT +µiP −µiQ +µiR +µiS −µiV +µiU −µD +µE −µF +µL −µM +µN +µX −µY +µZ +µT −µP +µQ −µR −µS +µV −µU +µiD −µiE +µiF −µiL +µiM−µiN −µiX +µiY −µiZν iU −µiU −µiX +µiY +µiZ +µiV +µiS −µiT −µL +µM +µN −µD +µE +µF +µP −µQ −µR +µU +µX −µY −µZ −µV −µS +µT +µiL −µiM−µiN +µiD −µiE −µiF −µiP +µiQ +µiR

ν iV −µiV +µiD +µiE −µiF −µiU +µiT +µiS +µP +µQ −µR −µX −µY +µZ +µL +µM −µN +µV −µD −µE +µF +µU −µT −µS −µiP −µiQ +µiR +µiX +µiY −µiZ −µiL −µiM+µiN

νX −µX +µU +µZ −µY −µD +µL +µP −µiS −µiN+µiM +µiV +µiF −µiE −µiT −µiR +µiQ −µiX +µiU +µiZ −µiY −µiD +µiL +µiP −µS −µN +µM +µV +µF −µE −µT −µR +µQ

νY −µY +µZ −µU +µX +µE −µM +µQ +µiN −µiS +µiL −µiF +µiV −µiD −µiR +µiT −µiP −µiY +µiZ −µiU +µiX +µiE −µiM +µiQ +µN −µS +µL −µF +µV −µD −µR +µT −µPνZ −µZ −µY −µX −µU −µF −µN −µR −µiM −µiL −µiS −µiE −µiD −µiV −µiQ −µiP −µiT −µiZ −µiY −µiX −µiU −µiF −µiN −µiR −µM −µL −µS −µE −µD −µV −µQ −µP −µTνP −µP −µT −µR −µQ −µL −µD −µX −µiV −µiF −µiE −µiS −µiN−µiM −µiU −µiZ −µiY −µiP −µiT −µiR −µiQ −µiL −µiD −µiX −µV −µF −µE −µS −µN −µM −µU −µZ −µYνQ −µQ −µR +µT +µP +µM +µE −µY +µiF −µiV −µiD +µiN −µiS −µiL −µiZ +µiU +µiX −µiQ −µiR +µiT +µiP +µiM +µiE −µiY +µF −µV −µD +µN −µS −µL −µZ +µU +µX

νR −µR +µQ +µP −µT −µN +µF +µZ −µiE −µiD +µiV +µiM +µiL −µiS −µiY −µiX +µiU −µiR +µiQ +µiP −µiT −µiN +µiF +µiZ −µE −µD +µV +µM +µL −µS −µY −µX +µU

νD −µD −µV +µF +µE +µX +µP −µL +µiT −µiR −µiQ +µiU −µiZ −µiY −µiS +µiN+µiM −µiD −µiV +µiF +µiE +µiX +µiP −µiL +µT −µR −µQ +µU −µZ −µY −µS +µN +µM

νE −µE −µF −µV −µD −µY −µQ −µM −µiR −µiT −µiP −µiZ −µiU −µiX −µiN −µiS −µiL −µiE −µiF −µiV −µiD −µiY −µiQ−µiM −µR −µT −µP −µZ −µU −µX −µN −µS −µLνF −µF +µE −µD +µV +µZ −µR +µN +µiQ −µiP +µiT −µiY +µiX −µiU −µiM +µiL −µiS −µiF +µiE −µiD +µiV +µiZ −µiR +µiN +µQ −µP +µT −µY +µX −µU −µM +µL −µS


Table 10. Kaotic Algebra multiplication table, part D2

λS λL λM λN λT λU λV λiX λiY λiZ λiP λiQ λiR λiD λiE λiF λiS λiL λiM λiN λiT λiU λiV λX λY λZ λP λQ λR λD λE λF

µS +νS +νL +νM +νN +νT +νU +νV +ν iX +ν iY +ν iZ +ν iP +ν iQ +ν iR +ν iD +ν iE +ν iF +ν iS +ν iL +ν iM+ν iN +ν iT +ν iU +ν iV +νX +νY +νZ +νP +νQ +νR +νD +νE +νF

µL +νL −νS −νN +νM −νP +νX −νD −ν iU −ν iZ +ν iY +ν iT +ν iR −ν iQ +ν iV +ν iF −ν iE +ν iL −ν iS −ν iN+ν iM −ν iP +ν iX −ν iD −νU −νZ +νY +νT +νR −νQ +νV +νF −νEµM +νM +νN −νS −νL +νQ −νY −νE −ν iZ +ν iU +ν iX +ν iR −ν iT −ν iP −ν iF +ν iV +ν iD+ν iM+ν iN −ν iS −ν iL +ν iQ −ν iY −ν iE −νZ +νU +νX +νR −νT −νP −νF +νV +νD

µN +νN −νM +νL −νS −νR −νZ +νF +ν iY −ν iX +ν iU +ν iQ −ν iP +ν iT −ν iE +ν iD −ν iV +ν iN−ν iM +ν iL −ν iS −ν iR −ν iZ +ν iF +νY −νX +νU +νQ −νP +νT −νE +νD −νVµT +νT −νP +νQ −νR −νS −νV +νU +ν iD −ν iE +ν iF +ν iL −ν iM+ν iN −ν iX +ν iY −ν iZ +ν iT −ν iP +ν iQ −ν iR −ν iS −ν iV +ν iU +νD −νE +νF +νL −νM +νN −νX +νY −νZµU +νU +νX −νY −νZ +νV −νS −νT −ν iL +ν iM+ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR +ν iU +ν iX −ν iY −ν iZ +ν iV −ν iS −ν iT −νL +νM +νN +νD −νE −νF −νP +νQ +νR

µV +νV −νD −νE +νF −νU +νT −νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ +ν iL +ν iM−ν iN +ν iV −ν iD −ν iE +ν iF −ν iU +ν iT −ν iS −νP −νQ +νR +νX +νY −νZ +νL +νM −νNµiX +ν iX −ν iU +ν iZ −ν iY −ν iD −ν iL +ν iP −νS +νN −νM −νV +νF −νE +νT −νR +νQ −νX +νU −νZ +νY +νD +νL −νP +ν iS −ν iN+ν iM +ν iV −ν iF +ν iE −ν iT +ν iR −ν iQµiY +ν iY +ν iZ +ν iU +ν iX +ν iE +ν iM +ν iQ −νN −νS −νL −νF −νV −νD −νR −νT −νP −νY −νZ −νU −νX −νE −νM −νQ +ν iN +ν iS +ν iL +ν iF +ν iV +ν iD +ν iR +ν iT +ν iP

µiZ +ν iZ −ν iY −ν iX +ν iU −ν iF +ν iN −ν iR +νM +νL −νS −νE −νD +νV −νQ −νP +νT −νZ +νY +νX −νU +νF −νN +νR −ν iM −ν iL +ν iS +ν iE +ν iD −ν iV +ν iQ +ν iP −ν iTµiP +ν iP +ν iT −ν iR −ν iQ +ν iL −ν iD −ν iX +νV −νF −νE −νS +νN +νM +νU −νZ −νY −νP −νT +νR +νQ −νL +νD +νX −ν iV +ν iF +ν iE +ν iS −ν iN−ν iM −ν iU +ν iZ +ν iY

µiQ +ν iQ −ν iR −ν iT +ν iP −ν iM +ν iE −ν iY +νF +νV −νD −νN −νS +νL −νZ −νU +νX −νQ +νR +νT −νP +νM −νE +νY −ν iF −ν iV +ν iD +ν iN +ν iS −ν iL +ν iZ +ν iU −ν iXµiR +ν iR +ν iQ +ν iP +ν iT +ν iN +ν iF +ν iZ −νE −νD −νV −νM −νL −νS −νY −νX −νU −νR −νQ −νP −νT −νN −νF −νZ +ν iE +ν iD +ν iV +ν iM +ν iL +ν iS +ν iY +ν iX +ν iU

µiD +ν iD +ν iV +ν iF +ν iE +ν iX +ν iP +ν iL −νT −νR −νQ −νU −νZ −νY −νS −νN −νM −νD −νV −νF −νE −νX −νP −νL +ν iT +ν iR +ν iQ +ν iU +ν iZ +ν iY +ν iS +ν iN+ν iM

µiE +ν iE −ν iF +ν iV −ν iD −ν iY −ν iQ+ν iM −νR +νT −νP −νZ +νU −νX +νN −νS +νL −νE +νF −νV +νD +νY +νQ −νM +ν iR −ν iT +ν iP +ν iZ −ν iU +ν iX −ν iN +ν iS −ν iLµiF +ν iF +ν iE −ν iD −ν iV +ν iZ −ν iR −ν iN +νQ −νP −νT −νY +νX +νU +νM −νL −νS −νF −νE +νD +νV −νZ +νR +νN −ν iQ +ν iP +ν iT +ν iY −ν iX −ν iU −ν iM +ν iL +ν iS

µiS +ν iS +ν iL +ν iM+ν iN +ν iT +ν iU +ν iV −νX −νY −νZ −νP −νQ −νR −νD −νE −νF −νS −νL −νM −νN −νT −νU −νV +ν iX +ν iY +ν iZ +ν iP +ν iQ +ν iR +ν iD +ν iE +ν iF

µiL +ν iL −ν iS −ν iN+ν iM −ν iP +ν iX −ν iD +νU +νZ −νY −νT −νR +νQ −νV −νF +νE −νL +νS +νN −νM +νP −νX +νD −ν iU −ν iZ +ν iY +ν iT +ν iR −ν iQ +ν iV +ν iF −ν iEµiM +ν iM+ν iN −ν iS −ν iL +ν iQ −ν iY −ν iE +νZ −νU −νX −νR +νT +νP +νF −νV −νD −νM −νN +νS +νL −νQ +νY +νE −ν iZ +ν iU +ν iX +ν iR −ν iT −ν iP −ν iF +ν iV +ν iD

µiN +ν iN−ν iM +ν iL −ν iS −ν iR −ν iZ +ν iF −νY +νX −νU −νQ +νP −νT +νE −νD +νV −νN +νM −νL +νS +νR +νZ −νF +ν iY −ν iX +ν iU +ν iQ −ν iP +ν iT −ν iE +ν iD −ν iVµiT +ν iT −ν iP +ν iQ −ν iR −ν iS −ν iV +ν iU −νD +νE −νF −νL +νM −νN +νX −νY +νZ −νT +νP −νQ +νR +νS +νV −νU +ν iD −ν iE +ν iF +ν iL −ν iM+ν iN −ν iX +ν iY −ν iZµiU +ν iU +ν iX −ν iY −ν iZ +ν iV −ν iS −ν iT +νL −νM −νN −νD +νE +νF +νP −νQ −νR −νU −νX +νY +νZ −νV +νS +νT −ν iL +ν iM+ν iN +ν iD −ν iE −ν iF −ν iP +ν iQ +ν iR

µiV +ν iV −ν iD −ν iE +ν iF −ν iU +ν iT −ν iS +νP +νQ −νR −νX −νY +νZ −νL −νM +νN −νV +νD +νE −νF +νU −νT +νS −ν iP −ν iQ +ν iR +ν iX +ν iY −ν iZ +ν iL +ν iM−ν iNµX +νX −νU +νZ −νY −νD −νL +νP +ν iS −ν iN+ν iM +ν iV −ν iF +ν iE −ν iT +ν iR −ν iQ +ν iX −ν iU +ν iZ −ν iY −ν iD −ν iL +ν iP +νS −νN +νM +νV −νF +νE −νT +νR −νQµY +νY +νZ +νU +νX +νE +νM +νQ +ν iN +ν iS +ν iL +ν iF +ν iV +ν iD +ν iR +ν iT +ν iP +ν iY +ν iZ +ν iU +ν iX +ν iE +ν iM +ν iQ +νN +νS +νL +νF +νV +νD +νR +νT +νP

µZ +νZ −νY −νX +νU −νF +νN −νR −ν iM −ν iL +ν iS +ν iE +ν iD −ν iV +ν iQ +ν iP −ν iT +ν iZ −ν iY −ν iX +ν iU −ν iF +ν iN −ν iR −νM −νL +νS +νE +νD −νV +νQ +νP −νTµP +νP +νT −νR −νQ +νL −νD −νX −ν iV +ν iF +ν iE +ν iS −ν iN−ν iM −ν iU +ν iZ +ν iY +ν iP +ν iT −ν iR −ν iQ +ν iL −ν iD −ν iX −νV +νF +νE +νS −νN −νM −νU +νZ +νY

µQ +νQ −νR −νT +νP −νM +νE −νY −ν iF −ν iV +ν iD +ν iN +ν iS −ν iL +ν iZ +ν iU −ν iX +ν iQ −ν iR −ν iT +ν iP −ν iM +ν iE −ν iY −νF −νV +νD +νN +νS −νL +νZ +νU −νXµR +νR +νQ +νP +νT +νN +νF +νZ +ν iE +ν iD +ν iV +ν iM +ν iL +ν iS +ν iY +ν iX +ν iU +ν iR +ν iQ +ν iP +ν iT +ν iN +ν iF +ν iZ +νE +νD +νV +νM +νL +νS +νY +νX +νU

µD +νD +νV +νF +νE +νX +νP +νL +ν iT +ν iR +ν iQ +ν iU +ν iZ +ν iY +ν iS +ν iN+ν iM +ν iD +ν iV +ν iF +ν iE +ν iX +ν iP +ν iL +νT +νR +νQ +νU +νZ +νY +νS +νN +νM

µE +νE −νF +νV −νD −νY −νQ +νM +ν iR −ν iT +ν iP +ν iZ −ν iU +ν iX −ν iN +ν iS −ν iL +ν iE −ν iF +ν iV −ν iD −ν iY −ν iQ+ν iM +νR −νT +νP +νZ −νU +νX −νN +νS −νLµF +νF +νE −νD −νV +νZ −νR −νN −ν iQ +ν iP +ν iT +ν iY −ν iX −ν iU −ν iM +ν iL +ν iS +ν iF +ν iE −ν iD −ν iV +ν iZ −ν iR −ν iN −νQ +νP +νT +νY −νX −νU −νM +νL +νS

λS −S −L −M −N −T −U −V −iX −iY −iZ −iP −iQ −iR −iD −iE −iF −iS −iL −iM −iN −iT −iU −iV −X −Y −Z −P −Q −R −D −E −FλL +L −S +N −M −P +X −D −iU +iZ −iY +iT −iR +iQ +iV −iF +iE +iL −iS +iN −iM −iP +iX −iD −U +Z −Y +T −R +Q +V −F +E

λM +M −N −S +L +Q −Y −E +iZ +iU −iX −iR −iT +iP +iF +iV −iD +iM −iN −iS +iL +iQ −iY −iE +Z +U −X −R −T +P +F +V −DλN +N +M −L −S −R −Z +F −iY +iX +iU −iQ +iP +iT +iE −iD −iV +iN +iM −iL −iS −iR −iZ +iF −Y +X +U −Q +P +T +E −D −VλT +T −P +Q −R −S +V −U −iD +iE −iF +iL −iM +iN +iX −iY +iZ +iT −iP +iQ −iR −iS +iV −iU −D +E −F +L −M +N +X −Y +Z

λU +U +X −Y −Z −V −S +T −iL +iM +iN −iD +iE +iF +iP −iQ −iR +iU +iX −iY −iZ −iV −iS +iT −L +M +N −D +E +F +P −Q −RλV +V −D −E +F +U −T −S +iP +iQ −iR −iX −iY +iZ +iL +iM −iN +iV −iD −iE +iF +iU −iT −iS +P +Q −R −X −Y +Z +L +M −NλiX +iX −iU −iZ +iY +iD −iL −iP −S −N +M +V +F −E −T −R +Q −X +U +Z −Y −D +L +P +iS +iN −iM −iV −iF +iE +iT +iR −iQλiY +iY −iZ +iU −iX −iE +iM −iQ +N −S +L −F +V −D −R +T −P −Y +Z −U +X +E −M +Q −iN +iS −iL +iF −iV +iD +iR −iT +iP

λiZ +iZ +iY +iX +iU +iF +iN +iR −M −L −S −E −D −V −Q −P −T −Z −Y −X −U −F −N −R +iM +iL +iS +iE +iD +iV +iQ +iP +iT

λiP +iP +iT +iR +iQ +iL +iD +iX −V −F −E −S −N −M −U −Z −Y −P −T −R −Q −L −D −X +iV +iF +iE +iS +iN +iM +iU +iZ +iY

λiQ +iQ +iR −iT −iP −iM −iE +iY +F −V −D +N −S −L −Z +U +X −Q −R +T +P +M +E −Y −iF +iV +iD −iN +iS +iL +iZ −iU −iXλiR +iR −iQ −iP +iT +iN −iF −iZ −E −D +V +M +L −S −Y −X +U −R +Q +P −T −N +F +Z +iE +iD −iV −iM −iL +iS +iY +iX −iUλiD +iD +iV −iF −iE −iX −iP +iL +T −R −Q +U −Z −Y −S +N +M −D −V +F +E +X +P −L −iT +iR +iQ −iU +iZ +iY +iS −iN −iMλiE +iE +iF +iV +iD +iY +iQ +iM −R −T −P −Z −U −X −N −S −L −E −F −V −D −Y −Q −M +iR +iT +iP +iZ +iU +iX +iN +iS +iL

λiF +iF −iE +iD −iV −iZ +iR −iN +Q −P +T −Y +X −U −M +L −S −F +E −D +V +Z −R +N −iQ +iP −iT +iY −iX +iU +iM −iL +iS

λiS +iS +iL +iM +iN +iT +iU +iV −X −Y −Z −P −Q −R −D −E −F −S −L −M −N −T −U −V +iX +iY +iZ +iP +iQ +iR +iD +iE +iF

λiL −iL +iS −iN +iM +iP −iX +iD −U +Z −Y +T −R +Q +V −F +E +L −S +N −M −P +X −D +iU −iZ +iY −iT +iR −iQ −iV +iF −iEλiM −iM +iN +iS −iL −iQ +iY +iE +Z +U −X −R −T +P +F +V −D +M −N −S +L +Q −Y −E −iZ −iU +iX +iR +iT −iP −iF −iV +iD

λiN −iN −iM +iL +iS +iR +iZ −iF −Y +X +U −Q +P +T +E −D −V +N +M −L −S −R −Z +F +iY −iX −iU +iQ −iP −iT −iE +iD +iV

λiT −iT +iP −iQ +iR +iS −iV +iU −D +E −F +L −M +N +X −Y +Z +T −P +Q −R −S +V −U +iD −iE +iF −iL +iM −iN −iX +iY −iZλiU −iU −iX +iY +iZ +iV +iS −iT −L +M +N −D +E +F +P −Q −R +U +X −Y −Z −V −S +T +iL −iM −iN +iD −iE −iF −iP +iQ +iR

λiV −iV +iD +iE −iF −iU +iT +iS +P +Q −R −X −Y +Z +L +M −N +V −D −E +F +U −T −S −iP −iQ +iR +iX +iY −iZ −iL −iM +iN

λX −X +U +Z −Y −D +L +P −iS −iN +iM +iV +iF −iE −iT −iR +iQ −iX +iU +iZ −iY −iD +iL +iP −S −N +M +V +F −E −T −R +Q

λY −Y +Z −U +X +E −M +Q +iN −iS +iL −iF +iV −iD −iR +iT −iP −iY +iZ −iU +iX +iE −iM +iQ +N −S +L −F +V −D −R +T −PλZ −Z −Y −X −U −F −N −R −iM −iL −iS −iE −iD −iV −iQ −iP −iT −iZ −iY −iX −iU −iF −iN −iR −M −L −S −E −D −V −Q −P −TλP −P −T −R −Q −L −D −X −iV −iF −iE −iS −iN −iM −iU −iZ −iY −iP −iT −iR −iQ −iL −iD −iX −V −F −E −S −N −M −U −Z −YλQ −Q −R +T +P +M +E −Y +iF −iV −iD +iN −iS −iL −iZ +iU +iX −iQ −iR +iT +iP +iM +iE −iY +F −V −D +N −S −L −Z +U +X

λR −R +Q +P −T −N +F +Z −iE −iD +iV +iM +iL −iS −iY −iX +iU −iR +iQ +iP −iT −iN +iF +iZ −E −D +V +M +L −S −Y −X +U

λD −D −V +F +E +X +P −L +iT −iR −iQ +iU −iZ −iY −iS +iN +iM −iD −iV +iF +iE +iX +iP −iL +T −R −Q +U −Z −Y −S +N +M

λE −E −F −V −D −Y −Q −M −iR −iT −iP −iZ −iU −iX −iN −iS −iL −iE −iF −iV −iD −iY −iQ −iM −R −T −P −Z −U −X −N −S −LλF −F +E −D +V +Z −R +N +iQ −iP +iT −iY +iX −iU −iM +iL −iS −iF +iE −iD +iV +iZ −iR +iN +Q −P +T −Y +X −U −M +L −S


9 Appendix B - Subalgebras with phenomenology corresponding to fermions

In a previous paper, The Pattern of Reality[13], it was found that the phenomenol-ogy of the fermions of the standard model could be associated with that of someSU(2)×U(1) subalgebras of SL(4, C). It was postulated that other SU(2)×U(1)subalgebras of SL(4, C) could describe the phenomenology of fundamental par-ticles for dark matter, labelled varks. These subalgebras were represented as theproduct of matrices isomorphic to the quaternions with matrices isomorphic to thecomplex numbers, as shown in table 2. The same pattern can be assembled with thequaternions extended into octonions, generating right and left handed versions. Forexample, the entry SLMN×SV can be extended to ([SLMN ], µ[SLMN ])× [SV ]or to ([SLMN ], ν [SLMN ])× [SV ].

Table 11. Fermions

Family Texture Flavor Color

Electron e SLMN × SV

µ SLMN × ST

τ SLMN × SU

Neutrino νe SV (TU)× SL SV (TU)× SM SV (TU)× SN

νµ ST (UV )× SL ST (UV )× SM ST (UV )× SN

ντ SU(V T )× SL SU(V T )× SM SU(V T )× SN

red blue green

Up Quark up SiEiFL× SV SiDiFM × SV SiDiEN × SV

charm SiQiRL× ST SiP iRM × ST SiP iQN × ST

bottom SiY iZL× SU SiXiZM × SU SiXiY N × SU

Down Quark down SiXiPV × SL SiY iQV × SM SiZiRV × SN

strange SiDiXT × SL SiEiY T × SM SiF iZT × SN

top SiP iDU × SL SiQiEU × SM SiRiFU × SN

magenta yellow cyan

Sharp vark Rough hard STiEiY × SiP ST iF iZ × SiQ STiDiX × SiR

bright SUiQiE × SiX SUiRiF × SiY SUiP iD × SiZ

cool SV iY iQ× SiD SV iZiR× SiE SV iXiP × SiF

Smooth hard STiF iZ × SiP ST iDiX × SiQ STiEiY × SiR

bright SUiRiF × SiX SUiP iD × SiY SUiQiE × SiZ

cool SV iRiZ × SiD SV iXiP × SiE SV iY iQ× SiF

Round vark Rough soft SiELiF × SiP SiFMiD × SiQ SiDNiE × SiR

dull SiQLiR× SiX SiRMiP × SiY SiPNiQ× SiZ

warm SiY LiZ × SiD SiZMiX × SiE SiXNiY × SiF

Smooth soft SiZLiY × SiP SiXMiZ × SiQ SiY NiX × SiR

dull SiFLiE × SiX SiDMiF × SiY SiENiD × SiZ

warm SiRLiQ× SiD SiPMiR× SiE SiQNiP × SiF


10 Appendix C - The Higgs mechanism

It is not obvious how to generate the phenomenology of the Higgs mechanismfrom simple mathematical structures. Considerations such as the requirement forbosons to have symmetric wave fuctions and the application of Heisenberg groupsto their physics suggest investigation of subgroups of Heisenberg groups.

A simple symmetric matrix with four components is the 4 × 4 diagonal matrix.It’s entries could be set to eiθn , generating the matrix:

eiα 0 0 00 eiβ 0 00 0 eiγ 00 0 0 eiδ

If −δ is set equal to α+ β + γ, the matrix has unit determinant.For multiplication, a matrix with similar properties can be found as a subgroupof the Heisenberg group H3:

1 a b d+ ab0 1 0 b0 0 1 a0 0 0 1

Substituting c = (d+ab) this matrix can be written in terms of unit 4×4 matricesas:S + a(T + Y )/2 + b(M +R)/2 + c(D + U +N +Q)/4

For matrices rows and columns can be manipulated whilst maintaining overallproperties. In terms of the unit 4× 4 matrices, an equivalent manipulation is onewhich preserves commutation relationships, interchanging rows and columns foran array such as:

S Z R M L F

Z S V X Y T

R V S P Q U

M X P S N D

L Y Q N S E

F T U D E S

For this array, [TY ] anticommute with [ZE], [MR] anticommute with [ZP ], [DUNQ]anticommute with [PE]. If rows and columns of the array are interchanged to:


S V T X Y Z

V S U P Q R

T U S D E F

X P D S N M

Y Q E N S L

Z R F M L S

For this array, [QR] anticommute with [V L], [TX] anticommute with [V D]. [EFMN ]anticommute with [LD], and matrices with similar properties to:S + a(T + Y )/2 + b(M +R)/2 + c(D + U +N +Q)/4can be assembled such as:S + a(Q+R)/2 + b(T +X)/2 + c(E + F +M +N)/4Another possibility is:S + a(U + P )/2 + b(Y + Z)/2 + c(E + F +M +N)/4These properties include having unit determinant and forming an abelian groupisomorphic to that formed by:

1 a b d+ ab0 1 0 b0 0 1 a0 0 0 1

However, for: [S] + [QR] + [TX] + [EFMN ] and [S] + [UP ] + [Y Z] + [EFMN ],a matrix with unit determinant can also be represented by:√|(1± 2a2)|[S] + a[Q+R]/2 + b[V + L]/2 + c[E + F +M +N ]/4

This is also the case for [S]+[UP ]+[Y Z]+[EFMN and for any similar combinationfor which the commuting elements diagonally adjacent to the leading diagonal ofthe array have signature (−+−), including complex combinations such as [iZP iE].

For:√|(1− a2/2)|[S] + a[Q+R]/2 + b[V + L]/2 + c[E + F +M +N ]/4,

the term in S is:√

(1− a2/2) 0 0 0

0√

(1− a2/2) 0 0

0 0√

(1− a2/2) 0

0 0 0√

(1− a2/2)

The determinant for this matrix = 1− 2a2 + a4

which has the form of the Higgs potential.

A plot of√|(1− a2/2)| against a in the range 0 to 2 is shown below:


a²

|1-2a²|

0 10

1

1/2

This has a similar form to a plot of sin2θW against momentum transfer as shownbelow.

At this stage I have not investigated the consequences of using octonionic elementsinstead of matrices.


References

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[3] seiberg, N; Emergent Spacetime; arXiv:hep-th/0601234v1 31 Jan 2006

[4] Albert, A.; (1942), Quadratic forms permitting composition, Annals of Mathematics,Second Series 43 (1), 1942, 161177,

[5] Einstein, A.; Feldgleichungen der Gravitation; Preussische Akademie der Wis-senschaften, Sitzungsberichte, 1915 (part 2), 844847

[6] Klein, O.; Quantentheorie und fnfdimensionale Relativittstheorie; Zeitschrift fr PhysikA, 37 (12), 1926, 895906

[7] Yang, C.N. and Mills, H.; In Phys Rev., 1954, 96, 192.

[8] Higgs, P.; In Phys Lett., 1964, 12, 132 and Phys Lett., 1964, 13, 508.

[9] Dixon, G.M.; Division Algebras: Octonions, Quaternions, Complex Numbers and theAlgebraic Design of Physics; Kluwer, Dordrecht, 1994.

[10] Feynman, R.; Space-time approach to quantum electrodynamics, em Physical Review1949b 76:769789.

[11] Baez.J ; arXiv:math/0105155v4 [math.RA] 23 Apr 2002

[12] Moufang, R; Alternativkorper und der Satz vom vollstandigen Vierseit; Abh. Math.Sem. Hamburg 9 (1933), 207222.

[13] Wallace, R.; The Pattern of Reality. In Advances in Applied Clifford Algebras, 2008,18-1, 115-133.

[14] Eddington, A.; The Theory of Groups; In The World of Mathematics, Volume 3;Editor, James R Newman, Simon and Schuster, 1956

[15] Hestenes, D.; Geometry of the Dirac Theory; In A Symposium on the Mathematicsof Physical Space-Time; Facultad de Quimica, Universidad Nacional Autonoma deMexico, Mexico City, 1981, 67-96.

Robert G. Wallace

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