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CHIRAL ISOMORPHISM OF ABELIAN GROUPS OVER Gf'PnXGfPn

BY

JAMES R. VAN DYKE

jvandyke@slip.net
 

Abstract: Using unitary diagonal matrices as elements (labels), the closed geometrically finite groups of the Tetrahedral and its dual group are combined into the master Abelian rotational group, Gf'PnXGfPn, over a six-dimensional space. The research demonstrates that there exists a geometric unification of left-handed and right-handed systems of coordinates over a spherical projection plane with the matrix algebra representing the regular polytopes on the surface of a hyper-complex sphere. The extension fields take into consideration chirality and invariance over a unified commuting vector field. The orthogonal rotational group Z2,Z2,Z2,Z2,Z2,Z2 is isometric to the symmetric group S6, which is solvable when defined by Gf'P3XGfP3 and labeled with matrices.

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CHIRAL ISOMORPHISM OF ABELIAN GROUPS OVER Gf'PnXGfPn

"There are thus two kinds of Cartesian systems which are designated as 'right-handed' and 'left-handed' systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them."  Albert Einstein, 1922 (1)

1. INTRODUCTION. The purpose of this paper is to demonstrate a geometric unification of left-handed and right-handed systems of coordinates over a spherical surface. The problem consists of finding a system of coordinates, a regular system of points, which commutes globally and models both systems simultaneously. At the heart of this problem is the idea of invariance. An invariant has the property of remaining unaltered by a particular transformation, such as a rotation of the point field. (2,3)

There exists a transitive multiplicative group of degree Pn, which, by means of its properties, defines a discrete, orthogonal, rotational group for the regular polytopes over a spherical surface. These polytopes are contained as subgroups in the Abelian group of order Pn and type (Z2,..,Z2), where Z2 = 1, P = 2 (which is prime), and n = 6. Our goal is geometrically to construct three polytopes: a pair of chiral tetrahedrons and, by combining them, a cube. The handedness of one tetrahedron is distinguished by four colors, and its orthogonal dual is created by its mirror image. We now create a cube, Gf'P3XGfP3, algebraically by the addition of the trace elements of the subgroups of the group Gf64. They are the groups, Gf'8 and its dual Gf8. The first group that is created by the mirror image, models a left-handed tetrahedron and the observer in the mirror. The second group models a right-handed tetrahedron and an observer. Together these chiral groups model handedness, a cube, and a pair of observers. The observer is defined by an operator (a mathematical device), which in this analysis, eliminates the need to specify an absolute value. The mathematical operator we initially use is the identity operator of group theory.
Table 1.1 illustrates the isomorphism between the various fields and groups presented in this paper.

ORDER DECIMAL G8 BINARY Gf'8 Gf8,
1 0 (+++) 000 (- - -+++) (++++++)
2 1 (++-) 001 (- -+++-) (++-++-)
3 2 (+-+) 010 (-+-+-+) (+-++-+)
4 3 (+- -) 011 (-+++- -) (+- -+- -)
5 4 (-++) 100 (+- - -++) (-++-++)
6 5 (-+-) 101 (+-+-+-) (-+- -+-)
7 6 (- -+) 110 (++- - -+) (- -+- -+ )
8 7 (- - -) 111 (+++- - -) (- - - - - -)

Table 1.1 

An isomorphism is to one correspondence between elements in a field or group, such as the decimal numbers, G8, the binary number field modulo (111), Gf8, and Gf'8. Next, we develop the principles of finite fields, then we employ them to demonstrate the existence of a family of multiplicative Abelian groups.

2. FINITE FIELDS. Let u0, u1, u2,...,un-1 be a set of n (n>1) distinct symbols or elements, which may be combined by addition under the following laws:

ui + uj = uj + ui;
ui + (uj + uk) = (ui + uj) + uk.

Let the set be such that, for every pair of these elements, there exists a single element, uj, such that
ui + uj = uk. Therefore, uj is determined by subtraction and we write uj = uk - ui. The difference of uk and ui is uj. The set then contains every difference ui - ui. These differences have the additive property of the identity, since uj + (ui - ui) = uj. All the differences, ui - ui, are equal since, from the last equation, ui - ui = uj - u j. The notation is chosen such that ui - ui = u0.  The symbol u0 is a unique element having the additive property of zero. Two elements are equal if their difference is u0, otherwise they are distinct. The elements of n form an Abelian group when the law of combination in the group is addition, as here defined with the following consideration. The sum of two elements, ui + uj = uk, is an element in the set followed by a reduction modulo Pn, when uk is greater than Pn. The identity of the group is uo. This group is called the additive group of the field. (4)

Suppose the elements u0, u1, u2,...,un-1 may be combined by multiplication in accordance with the laws

uiuj = ujui; ui(ujuk) = (uiuj)uk;
ui(uj uk) = uiuj uiuk.

Let the product of two elements, uiuj = uk, be an element in the set followed by a reduction modulo Pn, when uk is greater than Pn.  From the relationships

uoui = uiuo = ui(uj - uj) = uiuj - uiuj = u0,

it follows that uo has the multiplicative property of zero. Then, for any two elements, ui and uk, there exists a single unique element, uj, such that uiuj = uk.  Therefore, uj is uniquely determined by division and we may then write uj = uk/ui.   The quotient of uk by ui is uj, if and only if ui   u0.  The set then contains every quotient, ui/ui, and this quotient has the multiplicative property of the identity, I.   From this equation, one sees that ui/ui = uj/uj, if and only if ui  0 or uj  0.  If Iui = ui or  uiI = ui and ui u0, we then have I = ui/ui. Therefore, one element in the set is equal to unity. The notation is chosen so that this element is u1. Then these n - 1 elements, u1, u2, ..., us-1, form an Abelian group in which the law of combination is that of multiplication as here defined. This group is called the multiplicative group of the field. (5)

The set of n distinct elements, u0, u1, u2,...,un-1, satisfying the above conditions, defines a finite field of order n. A finite field is characterized by the property that, when the rational operations of algebra are performed upon the elements in the field, they lead, in every case, to elements in the field followed by a reduction modulo Pn, when ui is greater than Pn. (6) We now provide an example of a finite field. Consider the Abelian group of order eight with the following elements:

u0 = (000), u1 = (001), u2 = (010), u3 = (011),
u4 = (100), u5 = (101), u6 = (110), u7 = (111).

This is a finite set of binary numbers, where addition and multiplication of these elements are the ordinary addition and multiplication of the numbers to which they are equal, followed by a reduction modulo(111) to a number of the set. The property of division is easily verified, when the divisor is different from u0, by observing that the congruence

ax = b mod(111) [a 0 mod(111)].

The congruence always has a unique solution x that is an element in the group, when a and b are given. (7) Under the above rules, the field of binary numbers is closed with respect to the operations of addition and multiplication. The concept of closure is the first rule of group theory. We now review these powerful notions of a group and the rules of the group's operations.

3. THE "ERLANGER PROGRAMM," HYPER-COMPLEX NUMBERS, AND GROUP THEORY. In 1872, Klein first published his essay "Vergleichende Betrachtungen uber neuere geometrische Forschungen" ("Comparing Viewpoints about new geometric Researches"). This pamphlet was published in connection with his appointment to a chair at Erlangen. As a result, the work became known as "The Erlanger Programm." With it, one may classify a geometry by distinguishing its respective transformation group. (8) The transformation group Gf'8XGf8 is distinguished by elements that are left-right invariant over an additive and multiplicative field.   Figure 3.1 is a mirror symmetric image of Gf64, with the tetrahedral groups, Gf'8 and Gf8 highlighted.

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Figure 3.1

The axioms of group theory allow an incident geometry to be defined by physical objects (the regular polytopes) by mapping them to the dual surfaces of a hyper-complex sphere. These axioms also provide the rules and the operations for the group algebra. In the axiomatic geometry that follows, we take points and lines as our fundamental objects. Incidence is the fundamental relation they share, such as M lies on the line L, or L passes through M. (9) What is a hyper-complex sphere? It is a sphere which accommodates simultaneous mapping of interrelated systems of coordinates. The hyper-complex Sphere is defined by Gf8 over the outside surface of a three-sphere. A point on this surface is defined by three parameters. Augmented with its inside surface Gf'8, which is also a three-sphere, a membrane paradigm model defines a six-dimensional hyper-complex sphere. In this analysis, the multiplicative field, which is defined upon these two normalized and unitary surfaces, does not contain a zero point. When these two six-dimensional fields are combined and mapped to the outside surface, a cubic twelve-dimensional hexahedron is created on the surface of a hyper-complex sphere.   Figure 3.2 illustrates this hyper-complex hexahedron. The twelve vectors, from the center of the sphere to the midpoints of the tessellated edges of the cube, represent these twelve dimensions.

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Figure 3.2

Hyper-complex numbers result from an extension of the idea of complex numbers, modeled as points in a plane, to the idea of hyper-complex numbers, modeled as points over a spherical surface. Last century, after the invention in 1843 of the quaternion by Hamilton, this area of mathematics became an active field of research. (10,11) Hyper-complex numbers, as defined by Hamilton, were normalized to a unitary sphere. However, his system only defined the left-handed space. Maxwell and Tait were advocates of Hamilton's quaternion and demonstrated its representation over a right-handed space. (12) This geometry is constructed with the axioms of group theory over a discrete point field that combines these two spaces and uncovers two additional spaces.

A group is a set of elements for which an operation is defined by the following four axioms:

A-1 CLOSURE. For every element a, b, in the group G, the result of ab is also in the group G.

A-2 ASSOCIATIVELY. For all a, b, c, in G, (ab)c = a(bc).

A-3 IDENTITY ELEMENT. For every a in G, there exists some element I, such that
Ia = a, [for P6 I = (+1,+1,+1,+1,+1,+1,+1)].

A-4 INVERSE ELEMENT. For every a in G, there exists an element a-1, such that
a-1a = I. (13)

The problem of determining the possible arrangements of the regular polytopes projected on the surface of a sphere is purely geometrical. These objects are considered as arrangements of points and such an arrangement as a regular system of points. A regular system of points is defined by three properties:

DEFINITION 3.1. A regular system of points in space is to contain infinitely many points, and the number of points of the system contained inside a sphere is to go to infinity as the cube of the radius.

DEFINITION 3.2. Any finite region of a regular system of points is to contain only a finite number of points.

DEFINITION 3.3. There exists a symmetry operation for each point of a regular system of points, such that any point may be moved to coincide with any other point, leaving the point field invariant. (14)

The first two defining properties are clear without any further explanation. The third may be elaborated upon to insure the proper understanding. An observer situated at some particular point of the system cannot determine, by performing some measurements, at which point of the system he is positioned. The reason for this phenomenon is the position of every point, relative to any other point, is the same. To bring any point of the system into coincidence with any other point of the system, there exists a motion through space, such that every position occupied by a point of the system before the motion is also occupied by a point of the system after the motion. This type of motion leaves the point system unchanged, or what is known as invariant. The movement is called a symmetry transformation and all such movements form a transformation group. (15) Using a theorem of extension, we construct a point field that unites the conventional left-handed and right-handed orthogonal fields over a single hyper-complex field. 

4. THEOREMS FOR EXTENSION FIELDS. The following theorems provide the building blocks for our finite point fields, as defined by the above definitions and later for defining geodesic lines.

THEOREM 4.1. The Hyper-complex sphere Gf'PnXGfPn , on whose surfaces an incident geometry is defined, is determined over the sphere's finite point field by the following condition:

Gf'PnXGfPn = [x e Gf'PnXGfPn ] | x2 = I.

With the surfaces of the hyper-complex sphere being the locus of all points, the distance I of 60 degrees, from a given central point. (16)

COROLLARY 4.2. A great circle of 360 degrees is the locus of all points, the distance I of 60 degrees, from a given central point, and coplanar with a hyper-plane incident to the given central point.

The result of any element multiplied by itself, being equal to the identity element, I, is a remarkable fact of this geometry. The algebraic formula defines a sphere encompassing space. A great circle is defined by any hyper-plane that is incident with the sphere's center.

COROLLARY 4.3. Two points that are separated by an arc of 90 degrees are perpendicular points. Two unit vectors from the given central point to these points are perpendicular vectors.

This theorem and its related corollaries establish the ground work for the idea of extension, the basis for a geodesic line, and a representation for its metric. We build the first point field from the midpoints of a tetrahedron's edges and its complex of six unitary, orthogonal vectors that are defined by the group Z2,Z2,Z2.

THEOREM 4.4. Let u0, u1, u2,...,un-1 or X0, X1, X2,...,Xn-1be a set of unitary, orthogonal Tetrahedral-vectors, represented by a diagonal matrix, of the Abelian group Z2,Z2,Z2, or Gf8. These T-vectors define the six midpoints of the tetrahedron's edges. The matrix represents an orientation with respect to I, an observer. The observer is introduced and represented by the identity element, I, and its conjugate, -I, which defines an axis of orientation for the observer. The identity element also defines an absolute direction for up, upon which all observers must agree. Furthermore, let each matrix represent a permutation of this set of six orthogonal T-vectors, which may be thought of as rays emanating from the center of the tetrahedron, passing through the midpoints of the edges, and tracing the midpoints' projection upon the surface of a hyper-complex sphere. Therefore, this group of elements, Z2,Z2,Z2, has the capability of being extended to Gf'PnXGfPn , and having the form Z2,...,Zn, where Z2 = 1 and p = 2, by the action of multiplication and/or addition.   Figure 4.1 illustrates these concepts of a spherical tetrahedron presented by theorem 4.4.

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Figure 4.1

X0 = (+1,+1,+1); X1 = (+1,+1,-1); X2 = (+1,-1,+1); X3 = (+1,-1,-1),
X4 = (-1,+1,+1); X5 = (-1,+1,-1); X6 = (-1,-1,+1); X7 = (-1,-1,-1).

Next, we introduce an axiom related to the identity that we introduced in the above theorem.

A-5 THE CONJUGATE IDENTITY ELEMENT. Conjugate points are defined as two diametrically opposite points on a sphere, separated by 180 degrees, and when they are multiplied together they determine a unique conjugate identity element, -I, [for P6, -I = (-1,-1,-1,-1,-1,-1)].  This element has the ability to generate one of three 32-element homomorphs, HA u HC, of Gf64. The products, which results from multiplying -I with the elements of the alternate group, HA, form a homomorphic group, HC, which geometrically represents the side of an object the observer is unable to see, (refer to Figure 5.2 below).

DEFINITION 4.5. When H and G are groups, a homomorphism of H into G is a function, such that
f:H G preserves the group axioms; i.e., for all h1,h2 e H and g1,g2 e G, where

     (h1,h2 )f h1f,h2f g1f,g2f (g1,g2 )f.      (4.6)

The kernel of f is the set of all h e H, such that (h)f is the identity, I, in G. (17)

DEFINITION 4.7. Let f be a homomorphism of H into G. If f is one to one, then f is called an isomorphism of H into G. If, in addition, (H)f = G, then f is called an isomorphism of H onto G or an isomorphism between H and G, and, in this case, only H and G are called isomorphic groups, such that

     (h1,h2 )f h1f,h2f g1f,g2f  (g1,g2 )f.      (4.8)

Isomorphic groups define a transformation in both directions, or what is known as a bijection. (18)

DEFINITION 4.9. Let HA be the alternate group with the identity element, I, and elements that are distinguished by having positive bits in bit positions n, such that n = 2,5,8,... of its array; for example P6 is (0,1,2,3,4,5). If (HA)f = HA, then f is called an identical isomorphism, or an automorphism, which maps each element to itself.(19)

We take the positive z axis as the position of the observer's viewpoint. The array (+i,+j,+k,+x,+y,+z) represents each axis and the positive k axis represents the observer's viewpoint in the mirror. The proofs to the above theorems and axioms will follow after we introduce our chiral matrix groups and a theorem that includes the observer in the algebra.

5. CHIRAL ISOMORPHISM AND DUAL TETRAHEDRAL GROUPS. In nature there exist objects which, in all respects are identical, except for their spatial orientations. This chiral property is known as handedness, and the difference between a person's two hands best represents this concept. An isomorphism is a one to one correspondence between elements in two different groups. We create a new group by making the substitution of +1 for zero and -1 for one in the group example (given in section 2), which is isomorphic to the former group. This results in the group Gf8 with the following elements: (20,21)

u0 = (+1,+1,+1); u1 = (+1,+1,-1); u2 = (+1,-1,+1); u3 = (+1,-1,-1),
u4 = (-1,+1,+1); u5 = (-1,+1,-1); u6 = (-1,-1,+1); u1 = (-1,-1,-1).

In this new group, the properties of the elements have changed. We define the arrays as the main diagonal of a 3X3 matrix, representing the trace, and having the following law of multiplication with the asterisk indicating the multiplication:

(a1,a2,a3)*(b1,b2,b3) = (a1b1,a2b2.a3b3) = (c1,c2,c3).    (5.1)

From the relationships

uoui = uiuo = ui; uiui = uo; ui/ui = uo; ui/u0 = ui;

it follows that, in this group, uo has the multiplicative property of the identity, I. With this choice of notation, zero is excluded from the multiplicative field in a natural way. The most interesting properties are that division by any element is the same as multiplication by that element and the product of any element squared is the identity, I. This group has the additional advantage of not requiring a reduction modulo Pn. The unique properties are the result of the normalization to a unitary hyper-complex sphere. The group is Abelian that has an order Pn, and type (Z2,..,Z2), where Z2 = 1, P = 2, and n = 3.

When the elements are assigned to the midpoints of the edges of a spherical tetrahedron, they provide a discrete, orthogonal, rotational group. In addition to the six edges, the identity, I, and its conjugate, -I, define an axis for the observer. The orientation of the tetrahedron is therefore related to the observer's axis. With this tetrahedron defined over a right-handed system of coordinates and then using a mirror as a tool of inversion, a dual space is created over a left-handed system of coordinates. This is illustrated in Figure 5.1, with both a traditional tetrahedron, marked with numbers, and a spherical one, marked with matrices.

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Figure 5.1

The dual space of the left-handed tetrahedron is isomorphic to the right-handed space. In the space of the left-handed tetrahedron, the matrices are those of the secondary diagonals, which may be converted into a main diagonal by exchanging the first and last columns. In all cases, this action produces negative matrices and, when they are converted back to binary numbers, we obtain the negative number field module Pn. When the two groups are multiplied and/or added together, the two groups become the combined group Gf'8XGf8, which has the form (-Z2,-Z2,-Z2 + Z2,Z2,Z2). The group is Abelian, of order Pn, and has the form (Z2,..,Z2), where Z2 = 1, P = 2, and n = 6. The first example of extension given by theorem 4.4 is demonstrated by the group Gf'8XGf8 or the new group Gf64. The elements of the mirror image do not form a group. However, they do form a semi-group. When the complex conjugate identity operator (-1,-1,-1,+1,+1,+1) is included in the algebraic law of multiplication, the group properties return to the mirrored elements. We therefore introduce the following axioms and related theorem:

A-6 THE COMPLEX CONJUGATE IDENTITY ELEMENT. A unique complex conjugate identity element, CI, [for P6 CI = (-1,-1,-1,+1,+1,+1)], defines, for any given point, a mirrored point, which is perpendicular to the given point. The complex conjugate semi-group's unique identity operator has the property of creating one of three 32-element homomorphs, HA u HD of Gf64. The products, which result from multiplying CI with the elements of the alternate group, HA, form a homomorphic group, HD, which geometrically represents the mirror image of the object, and the side of the object the observer in the mirror is able to see.

A-7 THE COMPLEX IDENTITY ELEMENT. A unique complex identity element, -CI,
[for  P6  -CI = (+1,+1,+1,-1,-1,-1)], defines, for any given point, a mirrored point, which is perpendicular to the given point. The complex semi-group's unique identity operator, -CI, has the property of creating the last of three 32-element homomorphs, HA u HB, of Gf64. The products, which result from multiplying -CI with the elements of the alternate group, HA, form a homomorphic group, HB, which geometrically represents the side of an object the observer in the mirror is unable to see.

We now state the following theorem, which includes the observer in the algebra. The identity operators are given in the abbreviated form (++++++), where the unitary state (+1,+1,+1,+1,+1,+1) is understood.

THEOREM 5.2. (a) When the alternate group, HA(++++++), is in composition with its complex semi-group, HB(+++- - -), its conjugate semi-group, HC(- - - - - -), and its complex conjugate semi-group, HD(- - -+++), (with the subgroups identity operators given in the parenthesis), they form the triply transitive homomorphic 32-element subgroups of Gf64,

 HA u HB, HA u HC, and HA u HD, (refer to Figure 5.2 below).

(b) The elements of the group HA and these semi-groups obey axiom A-1, and the rest of the group properties, when their identity operators are included in the law of multiplication.

(c) Each transformation of these semi-groups into a group forms a homomorphism and a bijective isomorphism with the alternate group, HA.

(d) The kernel identity subgroup of Gf64, is HA = (++++++), HB = (+++- - -), HC = (- - - - - -), and
HD = (- - -+++), the four identity operators, and when they are in composition with the alternate group, HA, they regenerate Gf64.

(e) And there exists three additional transitive homomorphic 32-element subgroups of Gf64,
HC u HB, HD u HB, and HC u HD, which geometrically represent the mirror images of the first triply transitive homomorphic groups. (22)

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Figure 5.2 (RETURN)

We prove  theorem 5.2 in section 7, where we demonstrate the relationships of the identity operators with the two observers. First, we prove theorem 4.4, which is similar to the theorems of Lagrange and Galois.(23) The theorem provides the method for the algebraic extension of the field. The six additional vectors, determined by the mirror image of the first tetrahedron, are considered to originate from their common point of incidence, (the center) and to terminate at the projections of the edges' midpoints on the inside of the surface. The complex conjugate identity operator's axis (in the mirror) is perpendicular to the identity operator's axis. This completes the definition of the appended field of hyper-complex numbers, which is based on the fields of dual tetrahedrons (the dual in the mirror and the original). The group +Z2,+Z2,+Z2 is equal to the set of trace elements of the main diagonal. The trace elements of the mirror image, which is the secondary diagonal, is transformed using matrix algebra into a main diagonal, -Z2,-Z2,-Z2, and appended with the original trace elements by addition, (-Z2,-Z2,-Z2)+(+Z2,+Z2,+Z2). This follows from Weyl's treatment of the transformation of the principal axis.

Proof of theorem 4.4: Following Weyl's treatment of the transformation of the principal axis, our proof uses the method of mathematical induction over the familiar vector field. We seek a normal coordinate system ei, such that in addition to

r = x1e1 + x2e2 + ... + xnen

     r2 = x21e21 + x22e22 + ... + x2ne2n       (5.2)

we also have

     A(r) = a1 x21e21 + a2x22e22 + ... + anx2ne2n.       (5.3)

That is, A will be brought into normal form 5.3 by means of a multiplicative unitary transformation. An invariant correspondence of the field upon itself is also referred to as a rotation or transformation of the principal axes. The real numbers a1, a2,... , an are called the characteristic numbers of the form A, and e1, e2,...,en are the corresponding characteristic vectors. (24)

We consider the correspondence r - r'= Ar and seek those vectors r  0, which are transformed into multiples r' = l r of themselves by A. We thus obtain the well known "secular equation" (25)

     f( l ) = det( l 1 - A) = 0,      (5.4)

for the multipliers  l. According to the fundamental theorem of algebra, this equation certainly has a root     l = a1, and there exists a non vanishing vector r = e1, which satisfies the equation Ae1 = a1 e1. On multiplying this vector by an appropriate numerical factor so chosen, such that its modulus is unity. Then, e1 may be supplemented by n - 1 further vectors, e2 ,e3 ,..., en, in such a manner that these n vectors then constitute a normal coordinate system. In these coordinates, the formula

        ei' = Aei = S k akiek      (5.5)

for the correspondence A requires, in accordance with the definition on e1, that the following coefficients a21,a31,...,an1 must vanish and a11 = a1, and because of the symmetry conditions aki = aik, the coefficients a12,a13,...,a1n must also vanish. Hence, in the new coordinates, the matrix A takes the form

chiral1.gif (11516 bytes)

and the modified hermitic form becomes

     A(r) = a1x21 + A'(r),      (5.6)

where A' is the modified form containing only the n - 1 variables x2,x3,...,xn. Repeating this process, we establish the validity of theorem 4.4. The characteristic polynomial of equation 4.3 is

     det(  l 1 - A) = ( l - a1)( l - a2)...( l - an).       (5.7)

Thus it follows that the characteristic numbers, a1,a 2,...,an, are uniquely determined, and their sum is the trace of A. (26)

Since these fields are discrete, we may again verify theorem 4.4 with a computer program that uses an algorithm to examine each possible permutation of Pn, which has the form Z2,...,Zn, where Z2 = 1, P = 2, and n = 3, 6, 12. Using this notation, we are able to demonstrate that the symmetric group of six variables is solvable, when they are transformed, first into triplets, and then sextuplets. (27) The following axiom now includes the commutative law of multiplication for these Abelian groups.

A-8 COMMUTATIVE. For every element a, b,... in the group G, the products of any two elements,
ab = ba, have the commutative property of multiplication. (28)

The algorithm verifies that all the axioms and the theorems are valid within these groups and various subgroups. First, each element is multiplied by itself to verify theorem 4.1. Next, each element pair a,b of the groups, Gf8, Gf64, and Gf4096, are multiplied together to check that the result ab and ba are equal, and are also in these groups. This proves axioms A-1 and A-8. Continuing in this manner, the axioms, A-2, A-3, A-4, A-5, A-6, and A-7, are also verified for these groups and semi-groups, thus proving these axioms. Again, using the result ab as a pole point, P, the algorithm finds the subset of all the orthogonal point pairs over the hyper-complex field, which are perpendicular to each other and form the group Gf'64XGf64.

The property of points being perpendicular is very important because we define the property of handedness with three perpendicular base vectors. Two of these perpendicular vectors are identified, and their product defines a third perpendicular vector, the pole point, P. This orthogonal vector triplet forms the complex that models handedness. With this vector triplet, a geodesic line is also defined. By evaluating specific bits in the pole's array, (0,1,2,3,4,5), the direction of the geodesic line is determined by the sense of its pole. When the product is negative, a counterclockwise rotation is determined, which models left-handedness. When the product of bits two and five are positive, a clockwise rotation is determined, which models right-handedness. From the theorems of extension, we develop the ideas to define the points, the geodesic lines, and an incident geometry over the inside and the outside surfaces of a hyper-complex sphere. We now define a geodesic line, which is constructed from the subgroup of perpendicular point pairs, over the hyper-complex number field Gf'64XGf64.

6. THE GEODESIC LINE. A single point, P, on a spherical surface breaks the perfect symmetry of the sphere and determines a unique great circle. All of the points on this great circle are 90 degrees from the point P. A unit vector from the center of the sphere to this point is perpendicular to the plane of the great circle. The point is known as a pole point of the great circle, and it also determines another point diametrically opposite, its conjugate, -P. Together the poles determine an axis of rotation, and a unique, equatorial plane through the center of the hyper-complex sphere. This plane determines a unique great circle, as illustrated in Figure 6.1. (29)

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Figure 6.1

Each point on the hyper-complex sphere determines a unique plane. In space, the point and the plane are dual elements.(30) The following theorem defines geodesic lines with respect to their pole (the point) and to their great circle (the plane). The point and plane duality here provide the basis for the creation of a geodesic line on a hyper-complex sphere.

THEOREM 6.1. Let a geodesic line L exist, such that L is equal to one half of a great circle, and is represented uniquely by its pole, P, its midpoint point, M, and its terminal point, T. When M, T, P are orthogonal unitary base vectors, which define perpendicular points on a surface of the hyper-complex sphere and the asterisk indicates multiplication, then

L = [ M,T,P e Gf'PnXGfPn] | (M*T) = P

is called the geodesic line L with pole P; L is the pole geodesic line of P, such that

P*(M*T) = P*(P) = P2 = I. (30)

We consider the metric, Y = (Pi/2) M*T, of the real variable M*T, as a function defining a geodesic line of 180 degrees, or Pi, incident to a great circle of 360 degrees. These concepts of an incident geometry are illustrated by the geodesic lines in Figure 6.2 that shows their handedness by defining a unique direction.

figure6a2.GIF (40962 bytes)

Figure 6.2

We have met this construction before in vector analysis. It is similar to the cross product. Additional geodesic lines exist, which have special relationships of incidence with the first geodesic line defined, such as the geodesic line in figure 6.2 that is in the mirror. However, their definitions must wait for a future paper that reconstructs all the regular polytopes and proves theorem 6.1. (31) Now, the role of the observer is elaborated upon in order to help clarify the proof of theorem 5.2, which is presented next.

7. THE OBSERVER. Previously we demonstrated that the mirror image forms a semi-group, requiring its identity operator to be included in the algebra. The observer was introduced and defined with the identity operator, (++++++), again shown in its abbreviated form. The observer may change orientation with respect to an object. For example, one may move around behind the object, which is a change of 180 degrees. Therefore, the observer's operator changes to its conjugate, (- - - - - -). When the observer's operator, defining the observer's orientation, is included in the law of multiplication, the result is a description of the object as seen by the observer. With this idea in mind, the algebra is a natural consequence of the observers' viewpoint.

The mirror image is defined on the inside of the hyper-complex sphere. An observer defined on the outside cannot see this tetrahedron. Therefore, the complex conjugate identity operator, (- - -+++), defines the observer on the inside of the hyper-complex sphere. In this manner, the left-handed field is specified and all rotations of the left-handed tetrahedron may be determined without the multiplication changing its field. Three negative numbers multiplied together obviously results in a negative number. In other analysis, the concept of absolute value is introduced to restrict the results to the positive field, but unlike other analysis, the choice of which field the observer is in determines the correct value in this analysis.

Theorem 4.4 and Theorem 5.2 physically place an observer into the mathematics by defining an orientation of an object from the observer's perspective. In the proof of Theorem 5.2, we start with provisions b, c, d, and then the final provision e is proved together with a.

Proof of Theorem 5.2, provision (b). The elements of the group HA and these semi-groups obey axiom A1, and the rest of the group properties, when their identity operators are included in the law of multiplication. We prove this provision by demonstrating the existence of f, as defined by definition 4.5, for each of the four cases where f is respectfully one of the four identity operators,

HA = (++++++), HB = (+++- - -), HC = (- - - - - -), and HD = (- - -+++).

These equations are valid as expressed by equation 4.6,

(h1,h2 )f h1f,h2f g1f,g2f (g1,g2 )f.

when, we make the substitutions, in turn, for h1,h2 e H, using the elements of the alternate group,
a1,a2 e HA and then the semi-groups, b1,b2 e HB,  c1,c2 e HC, and d1,d2 e HD, and multiply by the corresponding identity element f.

Proof of provision (c). Each transformation of these semi-groups into a group forms a homomorphism and a bijective isomorphism with the alternate group HA. The proof follows directly from definition 4.7 and equation 4.8,

(h1,h2 )f h1f,h2f g1f,g2f (g1,g2 )f,

after we make the substitution, a1,a2 e HA for G and then make the next substitutions, in turn, for H, a1,a2 e HA, b1,b2 e HB,  c1,c2 e HC, and d1,d2 e HD. The resulting four equations are valid, when the multiplication by f is replaced, in turn, by the corresponding identity element,
(++++++),   (+++- - -), (- - - - - -), and (- - -+++).

Proof of provision (d). The kernel identity subgroup of Gf64 is

HA = (++++++), HB = (+++- - -), HC = (- - - - - -), and HD = (- - -+++),

the four identity operators, and when they are in composition with the alternate group, HA, they regenerate Gf64. We note (HA)f = G, such that

HA(++++++) u HA(+++- - -) u HA(- - - - - -) u HA(- - -+++) = Gf64,

is valid by provision (c), just proved, and the laws of multiplication and/or addition provided by the laws of finite fields. We prove the first and last provisions together, with an understanding of the meaning of a transitive group or subgroup. A transitive group has the property of possessing an element, which replaces any given element by any other given element. (32) Axiom A-1 provides closure and the meaning for relating the elements of a transitive group. Now using the provisions proved above. we demonstrate the existence of, such that

(a) (HA u HB)f u (HA u HC)f u (HA u HD)f Gf64f  and

(e) (HC u HB) f u (HD u HB)f u (HC u HD)f Gf64f.       

In the case of (a), f is the identity, (++++++), and in the case of (e), f is the complex conjugate identity,      (- - -+++), the mirror image inversion generator, i.e.; if (a) and (- - -+++) are multiplied together, the result is (e), and (e)*(- - -+++) = (a), which is all that is required to finish the proof of the theorem. Being triply transitive under addition regenerates the group, as both (a) and (e) indicate. Statements (a) and (e) are valid as a bijective isomorphism, with transformations being one to one in both directions. Both (a) and (e) are again valid, when f is equal to the conjugates, (- - - - - -) and (+++- - -) respectfully. 

For P6, the order of transitivity is six, as just proved. These groups are important in the definition of geodesic lines, which are then used to reconstruct the regular polytopes in a field of hyper-complex numbers, where they are all mapped to the outside surface of the hyper-complex sphere. A new extended field is constructed in this manner by the addition of the left-handed array, which is six-dimensional, and the right-handed array, which is also six-dimensional. Both are then mapped to the outside surface, resulting in a twelve-dimensional array that unifies chirality and invariance over a single surface of the unitary hyper-complex sphere.

The following process of extension

Gf'64XGf64 Gf(64+64) Gf 128

is the field of geodesic lines, and in this process two additional fields are generated. When extended by addition, we consider a field of double numbers (P,Q) with the coma indicating addition. We determine if P or Q is real or imaginary by evaluating specific bits in their array, [(0,1,2,3,4,5),(0,1,2,3,4,5)]. If the products of bits 2 and 5 are positive, the element is considered real and, if the products are negative, the element is considered imaginary. We define a finite quaternionic field by its four subfields of hyper-complex numbers; (real, real) on the outside surface; (imaginary, real) on the outside surface; (real, imaginary) on the inside surface; and (imaginary, imaginary) on the inside surface. Hyper-complex numbers represent all four of these possible subfields modeled over a hyper-complex sphere, with the two additional fields mapped to the inside surface. The complex numbers only represent the fields of (real, imaginary), real, or imaginary modeled over the Euclidean plain. The variations of subfields (P,Q) are all defined as subgroups of the Abelian group Gf'64XGf64 of order pn, and type (Z2,..,Z2), where Z2 = 1,
P = 2, and n = 12.

8. CONCLUSIONS AND QUESTIONS. The use of the identity element to introduce an observer into the mathematical structure of this analysis is the most important aspect of the paper. Half of the axioms introduced, A-3, A-5, A-6, and A-7, all state a geometric interpretation for an observer's viewpoint of an object and express this viewpoint in an algebraic fashion. Theorem 5.2 provides the basis to map the point field of Gf'64XGf64 to the two surfaces of the hyper-complex sphere and gives validity to the algebra. The mathematics is significant because it is able to keep track of an object's orientation and the orientation of the observer viewing the object. The object's final orientation, as seen from the viewpoint of the observer, is described by the algebra after moving the object, the observer, or both.

We accomplished the goal of constructing a cube, Gf'8XGf8, geometrically and algebraically with a pair of chiral tetrahedrons over a hyper-complex spherical surface. By the combination of simple geometric forms, which create higher forms, the remaining polytopes may be constructed geometrically in a similar fashion. Using the ideas of a field and a group, these polytopes are constructed algebraically by the addition of various subgroups of Gf64. We now note that the group Gf64 is the master group of the regular polytopes. When the group is extended by multiplication, Gf'64XGf64 defines geodesic lines. Sets of these geodesic lines are then used to reconstruct these polytopes on the same surface. The group of 64 hyper-complex numbers defines dual orthogonal subgroups. When these subgroups are mapped to a spherical tessellation of an icosahedron, 32 of these numbers are found on each side of the membrane. Thirty are found at the midpoints of the tessellated edges and two additional pole points define the observer's axis.

The theorem of Lagrange is considered as a theorem of composition, and an Abelian group has this property, as we have just shown. The group Gf64 possesses this property and the property of decomposition. Various subgroups, into which Gf64 decomposes, are used to define the platonic solids. The first question is: What are these subgroups, including their identity operators? Define the right-handed tetrahedra, (there are five) the octahedra, (there are five) and the icosahedron on the outside surface. Then define the dual for each of the above polytopes on the inside surface of the hyper-complex sphere. A more difficult question is: How are all the perpendicular point pairs, (there are 128 pairs) determined from the 4096 possible permutations? The final question is: How are sets of these 128 geodesic lines defined in order to reconstruct the regular polytopes over the field of points, which are defined by the group Gf'64XGf64 on the outside surface of the hyper-complex sphere? We present the questions as a challenge, too again stimulate research into hyper-complex numbers. The correct answer to these questions should follow the format presented in this paper. For example,

T1 (++++++)[(++++++),(-++-++),(+-++-+),(- -+- -+),
                         (- - - - - -),(+- -+- -),(-+- -+-),(++-++-)].

In addition, there are four more tetrahedrons, T2, T3, T4, and T5.

                                             T'1(- - -+++)[(- - -+++),(+- - -++),(-+-+-+),(++- - -+),
                                                                    (+++- - -),(-+++- -),(+-+-+-),(- -+++-)].

In addition. there are four more tetrahedrons, T'2, T'3, T'4, and T'5. When H stands for the hexahedron or cube, we have

H1 (- - -+++)[(++++++),(-++-++),(+-++-+),(--+--+),
                         (- - - - - -),(+- -+- -),(-+- -+-),(++-++-),
                           (- - -+++),(+- - -++),(-+-+-+),(++- - -+),
                           (+++- - -),(-+++- -),(+-+-+-),(- -+++-)].

In addition, there are four more hexahedra, H2, H3, H4, and H5. When O stands for the octahedron, we have O1 ....

In addition, there are four more hexahedrons, H2, H3, H4, and H5. When O stands for the octahedron, we have ...

9. ACKNOWLEDGMENTS. I would like to thank Earl Halverson, of Billings, Montana, who taught complex numbers, by having the class imagine the existence of the imaginary axis on the backside of the blackboard. I thank Richard Crandall, at Reed College, Portland, Oregon, for his time, patience, and critical review, which turned my work into understandable articles. I thank Chris Radcliffe, the co-author of Appendix A, and Michael Ryals for their helpful suggestions. Finally, I thank Welcome Lindsey for her applied expertise in technical writing.

REFERENCES

1. A. Einstein, The Meaning Of Relativity, Dover Publications, Inc., New York, 1922, 10.

2. C. Reid, Hilbert, Springer-Verlag, New York, 1970, 247-252.

3. D. Hilbert, Collected Papers, International Mathematical Congress, 1893; vol. 2, item 23.

4. R. Carmichael, Introduction to the Theory of Groups of Finite Order, Dover Publications, Inc., New York, 1937, 242-261.

5. Ibid.

6. Ibid.

7. Ibid.

8. D. Rowe, Klein, Lie, and the "Erlanger Programm", 1830-1930 A Century of Geometry, L. Boi, D. Flament, & J. Salanskis (Eds.), Springer-Verlag, New York, 1992, 43-62.

9. P. J. Ryan, Euclidean And Non-Euclidian Geometry, An Analytical Approach, Cambridge University Press, New York, 1986, 12.

10. M. J. Crowe, A History Of Vector Analysis, Dover Publications, Inc., New York, 1967.

11. L. Boi, D. Flament, & J. Salanskis (Eds.), 1830-1930 A Century of Geometry, Springer-Verlag, New York, 1992.

12. Op. cit., Crowe, 155.

13. F. M. Falicov, Group Theory and Its Physical Applications, Univ. Chicago Press, Chicago, Midway Reprint, 1989, 1-19.

14. D. Hilbert and S.Cohn-Vossen, Geometry And The Imagination, (Chelsea Pub. Co., New York, 1983), 1-93.

15. Ibid.

16. Op. cit., Ryan, 84-123.

17. P. Yale, Geometry and Symmetry, Dover Publications, Inc., New York, 1968, 1-35.

18. Ibid.

19. Ibid.

20. S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, New York, 1993, 15-95.

21. Op. cit., Yale, 56.

22. Op. cit., Carmichael, 138-158. This theorem is not stated in this reference, but as elsewhere in my work, I have used the ideas of others, modified to this paper's needs.

23. I. M. Yaglom, Felix Klein And Sophus Lie, Birkhauser, Boston, 1988, 13-21.

24. H. Weyl, The Theory of Groups and Quantum Mechanics, New York, Dover Publications, Inc., 1950, 1-40.

25. Ibid.

26. Ibid.

27. Appendix A, computer program.

28. Op. cit., Falicov.

29. Op. cit., Ryan, 84-123.

30. Ibid.

31. ...

32. Op. cit., Carmichael, 54.

 

 

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