:

## Symmetries and Properties of Hypercomplex Numbers

Maple

This article is dedicated to a University of Waterloo’s Professor Douglas Wihelm Harder, LEL, M.Math; for without Professor Harder I would not have been able to produce such great Maplesoft Applications.  His advice and guidance gave me the necessary insight to produce great Maplesoft’s Applications.  Thank you Professor, I salute you.

The first world solution in the history of Computer Algebra Software, to the Visualization of Higher Dimensions greater than four, was solved by Maple!

• The Maple Algebra Packages are available on Maplesoft's Application Center:
1. Quaternions by Michael Carter
2. Quaternions, Octonions, and Sedenions by Michael Carter
3. The CayleyDickson Algebra from 4D to 256D by Michael Carter
• The C++ Source Code, for the Visualization, is available at the Computer Science Department of Purdue University.  Feel free to request a copy.  Copyright waiver release delivered to mailbox of Purdue University Professor Aditya Mathur, PhD on Thursday 18 August 2011.

Good mathematicians do not guide the math; they let the math guide them! – Michael Carter

Introduction

Contravariant/Covariant Scientific Theory that unifies Space, Time, and Light

cst := space-time-light in Becquerel units (Bq)

π := transcendental number pi

εr: = relative permittivity

κr: = relative permeability

r := the first 256 Hypercomplex numbers (hyper-scalar r Voudon)

i := the second 256 Hypercomplex numbers (hyper-imaginary i Voudon)

j := the third 256 Hypercomplex numbers (hyper-imaginary j Voudon)

k := the fourth 256 Hypercomplex numbers (hyper-imaginary k Voudon)

The equation we see above is a beautiful simple equation compacted from 1024 base units to four hyper units of 256 base units each.  This equation is the unification of space, time, and light.  Everything is made up of different forms of light; even matter, time, and space are all different forms of light.  This equation is possible because of the symmetries due to certain properties of the Cayley-Dickson Hypercomplex numbers.  This article is about the symmetries and properties of these beautiful numbers.  It is dictated by this equation that the universe is composed of three major region realms: Riemannian, Lobachevskian, and Euclidean.  For sub-atomic particles the region realm is Lobachevskian Region Realm.  For extremely massive matter the space-time is Riemannian Region Realm.  And, the region realm in the middle is the Euclidean Region Realm. In Black Hole Physics the singularity is switch over from Riemannian to Lobachevskian Region Realm.  Dark Energy is evident that the expanding universe is actually spinning, too; spinning from within one region to the next.  The universe is not expanding into a void or spinning within a void, it is spinning within its own cluster regions like three galaxies in clusters.  Dark matter halo is a network of fibers and pure light spinning in a loop rather than on a geodesic.  The Riemannian Region Realm is a source of light (like a waterfall) from the Lobachevskian Region; manifesting itself as loops, the so called dark matter.  The Lobachevskian Region Realm is a sink of light (light a whirlpool) from the Riemannian Region Realm; manifesting itself as rods, the so called black hole.  The Euclidean Region Realm (ERR) is just a light stream flowing in all directions.  Any object made of mass pushing against the stream, when accelerating. Only objects of two dimensions may flow smoothly from the Riemannian Region Realm (RRR) to the Lobachevskian Region Realm (LRR).  There are 11 hyper-dimensions of different sizes: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024.  Gravity is 1D.  Time is 2D.  Space-Time is 4D.  Magnetism is 8D.  Electricity is 16D.  Weak is 32D.  Strong is 64D.  And, Space-Time-Light is 1024D.  Each of the 11 hyper-dimensions is not all the same size!

We needed an algebra + geometry combination that did not require calculus.  We needed this math to allow for singularities (zero-divisors).  We need this math to have a topology of all three Geometries: Lobachevskian, Euclidean, and Riemannian.  This algebra-geometry math is called the Cayley-Dickson Hypercomplex Algebras.  And, it was solved on Maplesoft’s Maple Computer Algebra Software.  Maple solved it!

Properties

We need to discuss seven properties: polychotomy, commutative, associative, alternative, power associative.

Polychotomy Property:  When one discusses the Real numbers we have a property relationship between two Real numbers on the Reals number line:

1. a < b
2. a = b
3. a > b

This is a 3-polychotomy which is called, trichotomy.    When we move to the Ordinary Complex plane we now have the zero-point region (all elements are zero within the tuple coordinates), four axes regions separated by the zero-point region, and four quadrant regions for a total of nine regions.  It had been conventional wisdom that when we move from the Real number line to the complex plane that we loses trichotomy.  However, we actually gain a hyper-polychotomy that is a 9-polychotomy called, enneachotomy (see numbered list below and Table I).

1. a = b
2. Both a and b on imaginary axis: a < b
3. Both a and b on imaginary axis: a > b
4. Both a and b on Real axis: a < b
5. Both a and b on Real axis: a > b
6. a and b each on different axis
7. a ≠ b: Both a and b in same quadrant

This make the Ordinary complex plan having a polychotomy called enneachotomy (see Table I).

 Number Dim Regions Axis counting the zero-point region Total Regions Flip quanta Flip quanta start and return back to the initial quantum Reals 1 3 3 = trichotomy 2 360 Complex 2 5 9 = enneachotomy 4 360 Quaternion 4 7 15 = pentakaidecachotomy 8 720 Octonion 8 9 21-polychtomy 16 1440 Sedenion 16 11 27-polychotomy 20 1800 Pathion 32 13 33-polychotomy 28 2520 Chingon 64 15 39-polychotomy 32 2880 Routon 128 17 45-polychtomy 40 3600 Voudon 256 19 51-polychtomy 44 3960 i-Voudon 512 21 57-polychtomy 52 4680 j-Voudon 1024 23 63-polychtomy 56 5040 k-Voudon 2048 25 69-polychtomy 64 5760 Table 1:  Listing polychotomy and flip quanta.

In addition, if we make the zero-point region may act as a pivot.  We may flip from one axis to the next while pivoting on the zero-point region.  On the Real number line we may flip an angle side lying along the positive side of the Real axis to the negative side, which is 180 degrees.  We may flip back to our initial position, which is another 180 degrees.  There are only two flips (360 degrees) to return back to the initial starting location.  Each flip is called a quantum.  When we look at the complex plane, with the zero-point region as the pivot, we have four flip quanta, which is 360 degree.  Each flip quantum is 90 degrees; hence, 4*90 or 360 degrees.  When we move to the quaternion hyperspace we now notice that the pivot must flip 8 times to return back to the starting location, which is 8*90 or 720 degrees (see
Table I).

Commutative Property:  When a*b = b*a, then we say that we have the commutative property for multiplication.  This is the property under the Reals.  The commutative property is not the general product for hypercomplex numbers starting at 4D and higher.  However, this is not a lost to the hypercomplex numbers.  For each imaginary base unit are all anti-commutative when multiplied with another imaginary base unit.  When you transposed the imaginary base units and then apply multiplication one will find the sign will flip; a*b = -b*a.  This is a hyper-commutative property called, anti-commutative.  It is  not a loss of permanence but a gain of permanency to symmetry.

 > x := Qrand(20, 100);   > y := Qrand(20, 100);   > x := Qvector(x);   > y := Qvector(y);   > Qunit(x)^2;   > Qunit(y)^2;   > Qvector(Qunit(x)*Qunit(y));   > Qvector(Qunit(y)*Qunit(x)); Figure 1:  Demonstrating unit vectors behaves like imaginary base units.

When two imaginary units are transposed, then the multiplied operands will produce the imaginary conjugate.  It is the Reals that are self-conjugated.  Within the Reals the sign does not change when one multiplies the transposed of the two operands.  This quality is not just for the imaginary base units.  It also carries over for the vector of two unit hypercomplex numbers being multiplied (see Figure 2).  The imaginary conjugate guarantees that we have symmetry which is a very important property to have in physics, chemistry, biology, and cosmology.  When you add the product of two hypercomplex numbers with its anti-commutative product one will get the sum of zero; hence, the anti-product is the complement of the original product.  This is symmetry.  If we are going to truly study nature and the universe we must move away from just having solutions that are only from the algebraic numbers; we are not studying polynomials.

Associative Property:   The associative, alternative, and power associative Properties all deal with symmetry just like the anti-commutative property does.  According to the Contravariant/Covariant Theory matter is the inverse of space-time.  When one sends a small ship deep into the ocean one will find a great deal of pressure building on that ship.  If the ship goes deep enough it will implode.  On the other hand, if one moves a ship into very deep space (where there is no gravitational warping from any matter), then one will find a great deal of pressure building on that ship.  If the ship goes deep enough it will explode.  This is why large volumes of matter radiate into the shape of a sphere.  Space-time flows in all directions.  When a ship accelerates, in space, it is going against the flow of space-time.  The ship is pushing against the steam of space-time.  It is rising to a higher pressure-stream, which pushes back even harder.  If it was possible for a ship to rise high enough then it will escape space-time completely and not be part of this region of universe as we know it.  A black hole takes it incoming energy and pass it on to a different region of the universe where matter is shaped like involuted.  Matter is no longer shape like spheres but like anti-spheres (spherical shells).  This phase geometry has a different set of physics laws that we will not recognized.  Matter moves faster than the speed of light and it cannot slow down to speeds equal to the speed of light.  Energy that flows within this other universe region (with different physics) halos our current matter.  We detect it as dark matter.

Associative property has two operator and three operands: a*b*c.  We are concern with how these three operands are grouped:  a*(b*c) = (a*b)*c.  Within hypercomplex numbers from 8D and higher there is no associative property as we normally know this property.   It is however a hyper-associative property.   Before we can discuss how hypercomplex numbers of 8D and higher have the hyper-associative property, we must first understand our high school algebra.  When we multiply (a – b)*(b – a) we will received three terms within the product.  We will also received three terms within the product with (a + b)*(b + a) as well.  However, if we take (a – b)*(b + a) we will received only two terms; the middle term is not produced (see figure 2).

 > sort(expand(  (a - b)*(b - a)  ), a);   > sort(expand(  (a + b)*(b + a)  ), a);   > sort(expand(  (a - b)*(b + a)  ), a);   > sort(expand(  (a + b)*(b - a)  ), a); Figure 2:  Multiplying in a way that will remove the middle term in the Real.

 > sort(expand((a - b*I)*(b - a*I)), a);   > sort(expand((a + b*I)*(b + a*I)), a);   > sort(expand((a - b*I)*(b + a*I)), a);   > sort(simplify(expand((a + b*I)*(b - a*I))), a); Figure 3:  When one multiplies in a way to remove the middle term in the Real numbers then this is just the opposite when multiplying that same way with complex numbers.

Using this same technique with complex or Hypercomplex numbers this will do just the opposite (see figure 3).

Now let us look at hypercomplex numbers at 8D or higher.  There are only two ways to group three operands with two operators.   When we used unique imaginary base units in those two grouping manners, then we find that we have an anti-associative property not the associative property.  Note in Figure 4 how we can get a pure Real number.   We are guaranteed to get a pure Real number because we are actually production a product of a number with its conjugate.  An ordinary complex number multiplied by its conjugate gives a pure Real number (see Figures 4a and 4b).

 > setHypercomplex(octonion);   > a := i2*(i6*i7);   > b := (i2*i6)*i7;   > (a - b)*(b - a);   > (b - a)*(a - b); Figure 4a: Multiplying three pure imaginary base units in the two ways of grouping which gives the two products where one is the conjugate of the other one.  We multiply a complex or hypercomplex number with its conjugate which will always produce a Real number.   This is very important for symmetry (see Figure 4b for an Octonions example).

 > setHypercomplex(octonion);   > a := (i2 + i7)*((i3 - i5)*(i2 + i1));   > b := ((i2 + i7)*(i3 - i5))*(i2 + i1);   > (a - b)*(b - a);   > (b - a)*(a - b); Figure 4b: Multiplying three Octonions in the two ways of grouping which gives the two products where one is the conjugate of the other one.  We multiply a complex or hypercomplex number with its conjugate which will always produce a Real number.   This is very important for symmetry (see Figure 4a for an imaginary base units example).

Because we have an anti-associative property with three unique imaginary base units grouped both ways, then we can multiply both products in a way that will give us a pure Real number (see Figure 4c).  This is hard evidence of symmetry.

 > setHypercomplex(octonion);   > x := Orand(-3, 3);   > y := Orand(-3, 3);   > z := Orand(-3, 3);   > s := (x*y)*z;   > t := x*(y*z);   > (s - t)*(t - s);   > (t - s)*(s - t); Figure 4c:  This figure shows the symmetry of the hyper- associative property of hypercomplex numbers from 8D and higher.

Alternative Property:  For hypercomplex numbers of 16D and higher we have another property that helps us deals with symmetry.  These hypercomplex numbers do not have, in all cases, the alternative property.  That particular case when they lose the alternative property is when we have zero divisors.  A zero divisors is when x ≠ 0 and y ≠ 0 but x*y = 0.  We do have a unique division with Hypercomplex numbers 16D and higher, however, because of the zero divisors we cannot classify these algebras as division algebras.  This is perfect for our needs.  We have singularities in black hole physics and at the beginning of the universe.  These zero divisors work just find as representatives of singularities.  The big bang was a Lobachevskian Region Realm (LRR) falling into the Riemannian Region Realm (RRR) .  Our universe is in constant flux.   Black holes are nothing but “Water Falls” of light flowing out of our region realm.  However, within all three Realms all light is conserved.  The universe is not winding down nor is it winding up.  We have flux in Hadrons and leptons in flux.  We geo-systems in flux. We have solar systems in flux.  We have galaxies in flux.  We have clusters of galaxies in flux.  And, we have the three Region Realms in flux.  The entire universe is in flux and everything is still conserved (see figures 5a, 5b, and 5c).  Entropy what?

 > setHypercomplex(sedenion);   > x := (e1 + e11);   > y := (e13 - e15);   > x*y;   > y*y;   > y*x;   > x*(y*y);   > (x*y)*y;   > y*x*y; Figure 5a: Shows the case when hypercomplex numbers 16D and higher do have the alternative property.  It is only when a zero divisor is among the operands is when Sedenions does not have the alternative property.

 > setHypercomplex(sedenion);   > x := (e1 + e10);   > y := (e13 - e6);   > x*y; # zero divisor   > y*y;   > y*x; # zero divisor   > f := x*(y*y);   > g := (x*y)*y;   > h := y*x*y; Figure 5b:  A zero divisor is when x ≠ 0 and y ≠ 0 but x*y = 0.  There is no alternative property in this case.  A zero divisor is an example of a singularity in black hole physics.

 > setHypercomplex(sedenion);   > x := (e1 + e10);   > y := (e13 - e6);   > # Left alternative identity associator > `[x,x,y]` := x*(x*y) - (x*x)*y;     >  # right alternative identity associator > `[y,x,x]` := (y*x)*x - y*(x*x);     > # flexible alternative identity associator > `[x,y,x]` := (x*y)*x - x*(y*x); Figure 5c: There are three associators when you using three operands.  If we have two unique products from the three associators then these operands lack the alternative property.   In this particular case we are lacking the alternative property.

Power Associative:  We are blind and death!  We cannot see the spin of an atom.  We cannot see a black hole.  We cannot see dark matter.  It is a lot we cannot see.  However, we have our seeing eye dog.  Her name is mathematics; and she is a Queen!  If we are going to figure out how to leave this solar system, galaxy, or region realm then we need to get cracking in our mathematics just for the love of it and not for money.  Engineering is booming.  But, mathematics is dying.  We need something like the Power Associative to keep us calm and stable.  Here is where we find our Conservation Law of space-time-light.  Space is not a construct made by man.  Time is not a construct made by man.  And, light is not a construct made by man.  We know how to measure all these concepts.  Yes, they are just concepts.  If we were all born blinded on Earth then we will have a total different way of communication and we will not be aware of light.  If we were all born death then we will have the same situation.  It is our senses that are fooling us.  We need to verify our experiments and we need observations; however, we still must do the math!  Figure 6 is self-explanatory.

It is no fun playing chess anymore.  The computer can beat the best of us with brute force.  Did you catch the hint?  We are still doing our mathematics on chalk boards, pencil, and paper like we are in the 1800s.  There be whales, captain!

Maple!

 > setHypercomplex(pathion);   > x := (p1 + p31);   > > d :=  x*x*x*x;   > > e := (x*x*x)*x;   > > f :=  x*(x*x*x);   > > g :=  x*(x*x)*x;   > > h := (x*x)*(x*x); Figure 6: Power Associative is the weakest of the three different type associative properties: Associative, Alternative, and Power Associative.  All Cayley-Dickson Hypercomplex algebras retain the Power Associative property.  No matter how many powers of the operands exist they will always be power associative.

References

 Carter, Michael.  The Cayley-Dickson Algebras from 4D to 256D.  http://www.maplesoft.com/applications. Maplesoft.  April 23, 2010.

 Carter, Michael.  Visualization of the Cayley-Dickson Hypercomplex Numbers Up to the Chingons (64D).  http://www.mapleprimes.com.  August 19 2011.

 Crane, Keenan.  Ray Tracing Quaternions Julia Set on the GPU.  http://www.cs.caltech.edu/~keenan/project_qjulia.html

 Culbert, Craig.  Cayley-Dickson algebras and loops. Journal of Generalized Lie Theory and Applications. Vo1. 1, No. 1, 1-17,2007.

 de Marrais, Robert P. C. Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions.

 Hanson, Andrew J. Visualizing Quaternions: Series in Interactive 3D Technology. New York, NY: Morgan Kaufmann. 2006.

 Hart, John C. and Sandin, Daniel J. Ray tracing deterministic 3-D fractals. Computer Graphics 23(3), (Proc. SIGGRAPH 89), pp. 289-296. July 1989.

 Kotsireas, Ilias S. and Koukouvinos, Christos. Orthogonal Designs Via Computational Algebra. WileyInterScience. May 4, 2006.

 Mandelbrot, Benoit B. Fractals: Form, Chance and Dimension. W.H.Freeman & Co Ltd. 1977.

 Norton, Alan. Generation and display of geometric fractals in 3-D. Yorktown Heights, NY: IBM Thomas J. Watson Research. 1982. Michael Carter

http://BookOfMichael.com ﻿