HILBERT'S 14^TH PROBLEM AND THE REGULAR POLYTOPES

Copyright © June 1998

BY

JAMES R. VAN DYKE

ABSTRACT: Using unitary diagonal matrices as elements (labels), the closed geometrically finite groups of the Icosahedral, the Octahedral, and the Tetrahedral (and their dual groups) are combined into one master Abelian rotational group, G_f'PⁿXG_fPⁿ, over a six-dimensional space. The research demonstrates that the 14^th problem of Hilbert has positive examples 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 (handedness) 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 G_f'P³XG_fP³ and labeled with matrices.

HILBERT'S 14^TH PROBLEM AND THE REGULAR POLYTOPES

1. INTRODUCTION. The 14^th problem of Hilbert asks, does an example exist for the finiteness of certain systems of relative integral functions? At the International Congress in Paris in 1900, Hilbert [4] delivered the opening lecture and offered samples of 23 problems to be solved during the new century. The 14^th problem remained unsolved until 1958, when Nagata [5] presented counter examples to the problem over the projection plane. A tetrahedron provides a closed finite geometric system, in which a positive example is shown to exists. The problem is solvable when the projection plane is the surface of a sphere. A specialization of linear algebra and group theory combine to form a closed finite geometric system. The rotational group for the invariant midpoints of the tetrahedron’s edges is defined by G_f8. The finite system is defined over an orthogonal vector field in dimension three, which was left as a remaining open problem in Nagata’s paper.

Hilbert then asks, if such a system exists, may the system be extended to a more general case?. We demonstrate that the answer is yes. The geometric system of the tetrahedron is extendable to a more general case that includes all the regular polytopes. The first extension considers the dual vector space of proper and improper rotations, of the tetrahedron and its dual in the mirror. The extension to G_f’8XG_f8 and/or G_f64 creates ten tetrahedra. The invariant midpoints of the edges of all of the regular polytopes are defined by these ten tetrahedra. The polytopes are contained as subgroups in the Abelian group  G_f64, of order Pⁿ and type (Z_2,...Z₂), where Z₂ = ±1, P = 2 (which is prime), and the dimension of n = 6. The Abelian rotational group G_f64 is the master rotational group for the regular polytopes.

A complete system of functions incorporates the concept of handedness in the system. The invariant property of the group G_f64 describes handedness with a pair of chiral tetrahedra. They provide the geometric foundation over a dual vector field for the algebraic proofs of the existence and the finiteness of certain complete systems of functions. Specifically, systems of functions exist when n is equal to three, six, and twelve, which provide integer solutions for the coefficients of polynomial equations of degree n.

2. THE 14^th PROBLEM OF HILBERT. We now state the 14^th problem in the form of a theorem using Hilbert’s own words in order to clarify what is to be proved.

THEOREM 2.1.  "Let a number m of integral rational functions X₁, X₂,...,X_m of the n variables x_1, x₂,...,x_n be given,

X₁ = f₁(x₁,...,x_n),
  (S)           X₂ = f₂(x₁,...,x_n),                 
................,
................,
X_m = f_m(x₁,...,x_n).

Every rational integral combination of X₁,...,X_m must evidently always become, after substitution of the above expressions, a rational integral function of x₁,...,x_n.   [Nevertheless, there may well be rational fractional functions of X₁,...,X_m which, by the operation of the substitution S, become integral functions in x₁,...,x_n.]               Every such rational functions of X₁,...,X_m, which becomes integral in x₁,...,x_n after the application of the substitution S, I propose to call a relatively integral function of X₁,...,X_m. Every integral function of X₁,...,X_m is evidently also relatively integral; further the sum, difference and product of relative integral functions are themselves relatively integral. " [4]  Let the existence of such a finite system be provided by the group G_fPⁿ, with the form Z₂,...,Z_n, where Z₂ = ±1, and p = 2. A positive example result when X_m = 8, 64, 4096 and with x_n = 3, 6, 12 respectfully.

The problem is to now demonstrate the existence of such "a finite system of relatively integral functions X₁,...,X_m, by which every other relatively integral function of X₁,...,X_m may be expressed rationally and integrally."  The idea of a finite field of integrality is expressed by a system of functions from which a finite number of functions can be chosen, in terms of which all other functions of the system are rationally and integrally expressible.  Further, we desire that these integral functions f₁,...,f_m have coefficients that are integers and included among the relatively integral functions of X₁,...,X_m only such rational functions of these arguments as they become, by the application of the substitution S, rational integral function of x₁,...,x_n with rational and integral coefficients. [4]

In order to prove this theorem, we must first construct the tetrahedral example, demonstrate its ability to be extended, and then show that the extension field is Abelian.   The proof that the system is finite is inherent in the geometric choice of the representations (the tetrahedron and its extended family).  The groups that define the regular polytopes are well known closed finite systems. [3]

3. THE CONSTRUCTION OF THE EXAMPLE.  We build the first point field from a tetrahedron and its complex of unitary orthogonal vectors that compose the group Z₂,Z₂,Z₂.

THEOREM 3.1. Let X₀, X₁, X₂,...,X_n-1 be a set of unitary, orthogonal Tetrahedral-vectors, represented by a 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 introduced and represented by the identity element, I (+1,+1,+1), and conjugate, -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 set 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 where Z₂ = ±1, P = 2, and n+n = 2n, by the action of multiplication and/or addition when n = 3, 6, 12.  Figure 3.1 illustrates the idea of a tetrahedron projected upon a spherical plane created by theorem 3.1.

Figure 3.

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).

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

THEOREM 3.2. The hyper-complex sphere G_f’PⁿXG_fPⁿ, on whose surfaces an incident geometry is defined, is determined over the 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) of 60 degrees, from a given central point.

COROLLARY 3.3. 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.

COROLLARY 3.4. 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 3.5. 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.

On the surface of a three-sphere, a point is determined by three parameters, Z₂,Z₂,Z₂.   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.   Our next task is to establish the mathematical method to model both fields with one unified chiral field and to demonstrate the concept of extension presented in the following analysis of invariant forms.

4. CHIRAL ISOMORPHISM AND DUAL TETRAHEDRAL GROUPS. 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.

A mirror demonstrates the inversion of G_f8 and the dual vector field G_f’8 is created. The left-handed system of coordinates G_f’8 is defined by the tetrahedron in the mirror. The right-handed system of coordinates G_f8 is defined by the original tetrahedron G_f8. The familiar vector field is now labeled with unitary diagonal matrices that commute. The spherical tetrahedra are illustrated in Figure 4.1.

Figure 4.1

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’8XG_f8, and/or G_f64, which has the form (-Z₂,-Z₂,-Z₂ + Z₂,Z₂,Z₂). The group is Abelian, of order Pⁿ⁺ⁿ, and type (Z_2,...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. We now prove theorem 3.1 for the algebraic extension field  in the general case.

Proof of theorem 3.1: 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 e_i, such that in addition to

r = x₁e₁ + x₂e₂ + ... + x_ne_n

r² = x²₁e²₁ + x²₂e²₂ + ... + x²_ne²_n       (4.2)

we also have

A(r) = a₁ x²₁e²₁ + a₂x²₂e²₂ + ... + a_nx²_ne²_n.      (4.3)

That is, A will be brought into normal form 4.3 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,... , a_n are called the characteristic numbers of the form A, and e₁, e₂,...,e_n are the corresponding characteristic vectors. [7]

We consider the correspondence r - r'= Ar and seek those vectors r ¹ 0, which are transformed into multiples r' = l r of themselves by A. We thus obtain the well known "secular equation"  

f( l ) = det( l 1 - A) = 0,      (4.4)

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, e_{2 ,}e₃ ,..., e_n, in such a manner that these n vectors then constitute a normal coordinate system. In these coordinates, the formula

e_i' = Ae_i = S _k a_kie_k       (4.5)

for the correspondence A requires, in accordance with the definition on e₁, that the following coefficients a₂₁,a₃₁,...,a_n1 must vanish and a₁₁ = a₁, and because of the symmetry conditions a_ki = a_ik, the coefficients a₁₂,a₁₃,...,a_1n must also vanish. Hence, in the new coordinates, the matrix A takes the form

and the modified hermitic form becomes

A(r) = a₁x²₁ + A'(r),     (4.6)

where A' is the modified form containing only the n - 1 variables x_2,x₃,...,x_n. Repeating this process, we establish the validity of theorem 3.1. The characteristic polynomial of equation 4.3 is

det(  l 1 - A) = ( l - a₁)( l - a₂)...( l - a_n).       (4.7)

Thus it follows that the characteristic numbers, a₁,a ₂,...,a_n, are uniquely determined, and their sum is the trace of A. [7]

Since these fields are discrete, we again verify theorem 3.1 with a computer program, which uses an algorithm to examine each possible permutation of Pⁿ that has the form Z₂,...,Z_n, 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. [6]

The elements of the mirror image do not form a group. They do form a semi-group, however, that divides the group in half. The group properties return to the mirrored elements, when their complex conjugate identity operator (-1,-1,-1,+1,+1,+1) is included in the algebraic law of multiplication. In the next section, we use the above extension of liner algebra to demonstrate that the full symmetric group S₆, which is directly related to general equations of the sixth-degree, is solvable over the finite field of the regular polytopes.

5. THE GENERAL N-TH DEGREE EQUATION. According to Galois theory, equations of the n-th degree are not solvable by radicals when n is greater than or equal to five. The following theorems use a linear algebraic extension of matrices to provide the building blocks for the finite point field of G_f’8XG_f8. We develop a form of algebraic geometry that allows the problem to be solved over a discrete spherically symmetric point field 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. Sets of algebraic systems of functions do exist, when n = 3, 6, 12, that furnish integer solutions for the coefficients of polynomial equations of degree n, when defined by a known finite geometric set of polytopes.

DEFINITION 5.1. 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 5.2. 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 5.3 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 red and the blue fields). 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 5.3

Table 5.4 displays the subgroups that G_f2⁶ decomposes into and the various regular polytopes that they represent.