A. THE FOUR AXIOMS.

The axioms of group theory allow an incident geometry to be defined by physical objects (the regular and semi-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 point M lies on the line L, or L passes through point M. (9)  What is a hyper-complex sphere?  It is a sphere that accommodates simultaneous mapping of interrelated systems of coordinates.   The hyper-complex sphere is modeled by a spherical bubble or a two-sided membrane surface.  A group is a collection 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 P³ I = (+1,+1,+1)].

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

B. THE HYPER-COMPLEX SPHERE.

We develop an algebraic space with the following theorem and corollaries.

THEOREM 1.1. The hyper-complex sphere G_f’PⁿXG_fPⁿ, on whose surfaces an incident geometry is defined, is determined over the hyper-complex sphere's finite point field by the following condition,

G_f’PⁿXG_fPⁿ = [x e G_f’PⁿXG_fPⁿ] | x² = I,

with the surfaces of the hyper-complex sphere the loci of all points, the distance I (+1,...,+1) an arc-radius of 60 degrees, from a given central point.

COROLLARY 1.2. A great circle of 360 degrees is the locus of all points, the distance I an arc-radius of 60 degrees, from a given central point and coplanar with a hyper-plane incident to the given central point.

COROLLARY 1.3. Two points that are separated by an arc of 180 degrees are conjugate points. When conjugate points are multiplied together, the result has the form (-1,...,-1).

COROLLARY 1.4. 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. When perpendicular points are multiplied together, the result is another perpendicular point, their pole point, separated from the first two points by arcs of 90 degrees.

With a theorem for algebraic extension, we build polytopes from a regular system of points defined by a tetrahedron and its complex of six unitary orthogonal vectors that compose the group Z₂,Z₂,Z₂.

II. THEOREM OF ALGEBRAIC-GEOMETRIC EXTENSION.

The foundation for a sexagesimal analytical geometry is based on group theory that is defined by a regular system of points constructed from the orthogonal structure of spherical polytopes. For every dual polytope pair, the midpoints of their edges are coincident and their edges cross each other in space at 90 degree angles (orthogonal).  The problem of determining the possible arrangements of the polytopes projected on the surface of a sphere is purely geometrical.   The objects define an arrangement of points and such an arrangement is a regular system of points.  A regular system of points is defined by three properties:

DEFINITION 2.1. A regular system of points in space is to contain infinitely many points, and the number of points of the system contained inside a hyper-complex sphere is to go to infinity as the cube of the radius.

DEFINITION 2.2. Any finite region of a regular system of points is to contain only a finite number of points.

DEFINITION 2.3. 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. [4 b]

DEFINITION 2.4. The optimal spherical packing of circles are defined by a regular system of points that are an integer number of degrees from their closest neighbors and the point's coordinates, (Z₂,...,Z₂), provide integer solutions sets for polynomial equations with integer coefficients.

Using a theorem of extension, we construct a point field that unites the conventional right-handed and left-handed orthogonal fields over a single hyper-complex field.

THEOREM 2.5. Let X₀, X₁, X₂,...,X_n-1 be a collection of unitary orthogonal Tetrahedral-vectors, that are represented by a unitary diagonal matrix, of the Abelian rotational group Z₂,Z₂,Z₂, or G_f8. Then, these six T-vectors define the rotational field for the six midpoints of the tetrahedron's edges. The matrix represents an orientation of the tetrahedron with respect to I, an observer. The observer is represented by the identity operator, I (+1,+1,+1), and its conjugate identity operator, -I (-1,-1,-1) that together define an axis of orientation for the observer. The identity element defines an absolute direction for up, upon which all observers must agree. Furthermore, let each matrix represent a permutation of this collection of six orthogonal T-vectors that 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, the group of elements Z₂,Z₂,Z₂ and the hyper-complex sphere may be extended to G_f’PⁿXG_fPⁿ and/or G_fPⁿ⁺ⁿ, with the form Z₂,...,Z_n, where Z₂ = ±1 and P = 2, by the action of multiplication and/or addition, when n = 3, 6, 12.

Figure 2.6 illustrates the idea of a spherical plane created by Theorem 1.1 with a tetrahedron projected upon the surface. The familiar vector field is labeled with unitary diagonal matrices that commute.

Figure 2.6

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).

On the surface of a three-sphere, a point is determined by three parameters, Z₂,Z₂,Z₂. [4b] When any element defining a point, is multiplied by itself, the result is equal to the identity element, I. The unitary three-sphere encompasses an invariant volume of space. One must now confront the chiral property of space, because modeling objects in space results in both left-handed and right-handed variations (hyper-complex). Our next task is to establish the mathematical method to model both fields with one unified chiral field that demonstrates the technique of extension presented in the above theorem. In nature there exist objects which, in all respects, are identical, except for their six-dimensional spatial orientations. This chiral property is known as handedness or enantiomorphous, and the difference between a person's two hands best represents the concept. We model this idea mathematically over the hyper-complex sphere, which accommodates simultaneous mapping of interrelated systems of coordinates. The hyper-complex sphere is defined by G_f8 over the outside surface of a three-sphere. Augmented by G_f’8 that is defined on the inside surface, a spherical bubble models the six-dimensional hyper-complex sphere, as a two-sided membrane.

The dual vector field G_f’8 is created by the mirror and demonstrates the inversion of G_f8. The left-handed system of coordinates G_f’8 is defined by the tetrahedron in the mirror or on the inside surface of the hyper-complex sphere. The right-handed system of coordinates G_f8 is defined by the original tetrahedron G_f8. This is illustrated in figure 2.7, where the spherical tetrahedra are labeled with matrices.

Figure 2.7

The dual vector field, the left-handed tetrahedron, is isomorphic to the field of the right-handed tetrahedron. In the field of the left-handed tetrahedron, the matrices are those of the secondary diagonals. The matrices are converted into matrices with main diagonals by exchanging their first and last columns. In all cases, this action produces negative matrices. When multiplied and/or added together, they form the group G_f’P³XG_fP³, and/or G_f64, which has the form (-Z₂,-Z₂,-Z₂ + Z₂,Z₂,Z₂). The group is Abelian, of order Pⁿ⁺ⁿ, and type (Z₂,..,Z₂), where Z₂ = ±1, P = 2, and n+n = 6. The group G_f’8XG_f8, demonstrates the first example of extension into the new group G_f64. Before we complete our definition of a point, we need to introduce polynomial equations and an idea about their solution fields.

III. POLYNOMIAL EQUATIONS.

The following theorems use the linear algebraic extension of matrices to provide the building blocks for the finite point field of G’_f8XG_f8. By using matrices as labels, an algebraic geometry demonstrates the construction of discrete spherically symmetric point fields for n = 3, 6, 12. These solutions, however, do not provide a counter example to Galois theory of equations because they are not a radical form and they do not provide a general solution. We simply eliminate the need to use the radical form for the following existence proofs. Collection of algebraic systems of functions exist for n = 3, 6, 12, which furnish integer solutions for the coefficients of polynomial equations of degree n that are defined by known finite geometric collection of polytopes.

DEFINITION 3.1.. A polynomial over a field F in the indeterminate value x is expressed by an equation, such that

f(x) = c₀xⁿ + c₁x^n-1 + ... + c_n-1x + c_n = 0,

where c₀, c₁, c₂,...,c_n are elements of F called coefficients of the polynomial.

Integral functions, as polynomials are sometimes referred to, are completely determined by their coefficients.  Polynomials are monic when the leading coefficient is equal to one, c₀ = 1 and the solution sets are restricted to the surfaces of a hyper-complex sphere if the last coefficient is equal to one, c_n = 1.   [2]

THEOREM 3.2. A function f(x) = 0 is solvable by algebraic extension, if and only if G is a solvable group. Furthermore, let the group G be equal to G_f8 and the extensions G_f’PⁿXG_fPⁿ, with the form Z₂,...,Z_n, where Z₂ = ±1, P = 2, and n = 3, 6, 12.

Proof: The finite group G_fPⁿ is solvable if there is a sequence of consecutive subgroups, which start with the full group G_fPⁿ and ends with a subgroup that contains only the identity I. In the decomposition chain, each subsequent subgroup is contained (É means "contains") in the preceding one as a subgroup of index two, such that

G_fPⁿ = G_n É G_n-1 É G_n-2É ... É G₀ = I,

for a, such that 1 £ a < n, G_a is normal in G_a+1, and the ratio [G_a+1 : G_a] is prime.

Since the full symmetric group S₆ is isomorphic to Z₂,Z₂,Z₂,Z₂,Z₂,Z₂, all that remains is to show that G_f2⁶ decomposes into various subgroups as required above. The order of each subgroup is 2ⁿ with n = 0, 1, 2, 3, 4, 5, 6 and the decomposition series is given, such that

G_f2⁶ = 64 É 32 É 16 É 8 É 4 É 2 É 1.

Figure 3.3 is a mirror symmetric representation of the group G_f2⁶ and/or G_f64, which illustrates the first decomposition into two thirty-two element subgroups. The original pair of chiral tetrahedra are highlighted. The horizontal field represents the left-handed tetrahedron and the vertical field represents the right-handed tetrahedron. 

Figure 3.3

Table 3.4 displays the subgroups that G_f2⁶ decomposes into and the various regular polytopes that they represent. In all cases, the subgroup ratios are equivalent to two, which is prime.