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OPTIMAL SPHERICAL PACKING OF CIRCLES 
AND HILBERT'S 14TH PROBLEM 

JAMES R. VAN DYKE

jvandyke@slip.net

Abstract:   We discover a system of mathematics that provides positive examples for the 14th problem of Hilbert, two new families of polytopes, a master Abelian rotational group Gf'PnXGfPnfor the polytopes, twelve unique solution sets for the optimal spherical packing of circles, and the foundations for a sexagesimal analytic geometry. Over the surface of a hyper-complex sphere, 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 their coordinates, (Z2,...,Z2), provide integer solutions sets for polynomial equations with integer coefficients. Using unitary diagonal matrices as elements (labels), the closed geometrically finite groups of the Tetrahedral, the Octahedral, and the  Icosahedral (and their dual groups) are combined into the master Abelian rotational group, Gf'P3XGfP3, when n = 3. The group geometry defines the polytopes over a spherical projection plane with the matrix algebra and demonstrates that the 14th problem of Hilbert has positive examples over 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 isomorphic to the symmetric group S6, which is solvable when defined by Gf'P3XGfP3 and labeled with matrices.  The group is then extended to include  semi-regular polytopes when n = 6, 12

OPTIMAL SPHERICAL PACKING OF CIRCLES AND HILBERT'S 14TH PROBLEM

INTRODUCTION.1 A geometry of circles, geodesics, and spheres has existed for the last five millennia. Unfortunately, the knowledge that the ancient Sumerians used to define their analytic geometry was lost during the millennium preceding the great mathematical achievements of the Greek civilization.  The following research did not start with the search for a Sumerian analytic geometry, but instead, for a mathematical method to deal with the fact that there exists two systems of coordinates (left-handed and right-handed). The solution to their unification is very fruitful and is the portal to the discovery of many interesting aspects of mathematics that includes the foundation for a Sino-Sumerian analytic geometry based on spherical polytopes.  The investigation into the invariant nature of objects possessing a sense or handedness leads to discoveries about the structure of curved space and it leads to Hilbert's 14th problem. Hilbert asks, does an example exist for the finiteness of certain systems of relative integral functions? He then stipulates that the solutions and the coefficients of the polynomial equations are to be integers. 

We use a geometric approach to the 14th problem, where positive finite extension fields are developed over a unit sphere by modeling larger and larger polytopes. The 14th problem of Hilbert has not yet been completely solved, however, a geometric program has been found that will lead to its solution and the reconstruction of the Sino-Sumerian sexagesimal analytical geometry.  In the following examples, algebra, geometry, group theory, linear algebra, and number theory are woven together to demonstrate positive geometric examples to Hilbert's 14th problem based on spherical polytopes.  In the process of dividing the surface of the sphere into smaller and smaller congruent parts, we discover a master Abelian rotational group Gf'PnXGfPnfor the polytopes, two new families of polytopes, twelve unique solution sets for the optimal spherical packing of circles, and the foundation for a sexagesimal analytic geometry.

At the International Congress in Paris in 1900, Hilbert [4a] delivered the opening lecture and offered samples of 23 problems to be solved during the new century. The 14th problem remained unsolved until 1959, 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 Gf8. 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 Gf’8XGf8 and/or Gf64 creates ten tetrahedra [3b].  The invariant midpoints of the edges for all of the regular polytopes are defined by these ten tetrahedra. The polytopes are contained as subgroups in the group  Gf64, of order Pn and type (Z2,...,Z2), where Z2 = ±1, P = 2 (which is prime), and the dimension of n = 6. The Abelian rotational group Gf64 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 Gf64 describes handedness with five 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, solution sets for algebraic systems of integral functions exist when n = 3, 6, 12, which provide integer solutions for the coefficients of polynomial equations of degree n that are defined by known finite geometric sets of polytopes. The coordinates, (Z2,...,Z2), define a regular system of points that are the integer solutions sets for the polynomial equations with integer coefficients. A polynomial over a field F in the indeterminate value x is expressed by an equation, such that

f(x) = c0xn + c1xn-1 + ... + cn-1x + cn = 0,

where c0, c1, c2,...,cn 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. The solution sets are restricted to the surface of a sphere when the last coefficient is equal to one, cn = ±1.  [2]

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 1.1. 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 1.2. Any finite region of a regular system of points is to contain only a finite number of points. 

DEFINITION 1.3. 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. 

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. [4b]

DEFINITION 1.4. 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

Group theory links the geometric ideas together with an isomorphism (a one to one relationship) between  sets of point fields for spherical analogs of the polyhedron collectively termed polytopes, sets of point fields for the optimal packing of circles, and solution sets for algebraic systems of integral functions. The optimal spherical packing of circles for the smallest group is the cuboctahedron. The representation fields for the optimal packing of circles are old and new families of polytopes. The algebra of the field is that of hyper-complex numbers of the form (Z2,...,Z2), where Z2 = ± 1, and they have an order of Pn with P = 2. When n= 3, 6, 12, these hyper-complex numbers represent points on the surfaces of a unitary hyper-complex sphere and the systems of extension fields, Gf'PnXGfPn, provides positive examples for Hilbert's 14th problem. 

Hilbert's 14th problem is linked by the master group Gf'PnXGfPn to the hyper-complex unitary solution fields of regular and semi-regular spherical polytopes. The largest polytope, modulo 60, is a truncated rhombic triacontahedron with 44,132 faces and 43,200 vertices (centers of an optimal packing of circles that are one degree in diameter).  The polytope has three different types of regular spherical polygons for faces, 12 pentagonal, 42,240 rhombic, and 1880 triangular, all of which have edges of one degree.  A little mathematics 
                                     12*5 +42,240*4+1880*3 = 60+168,960+5640 = 174,660/2 = 87,330 edges

shows that the polytope obeys Euler's Formula for Polyhedra ,V - E + F = 2, (43,200 -87,330 +44,132 = 2). The sets of the optimal spherical packing of circles, modulo 60, on the surface of a unit sphere are 

12(60); 48(30); 108(20); 192(15); 300(12); 432(10); 
1,200(6); 1,728(5); 2,700(4); 4,800(3); 10,800(2); and 43,200(1)

The integers in parentheses are the diameters of the circles in degrees.  The twelve solution sets are positive geometric examples that satisfy Hilbert's 14th problem where the polynomial equations have integers for coefficients and integers for solutions. Each solution set represents a polytope presented in Table 1.5. First is the three polytope family of the truncated rhombic triacontahedron and second is the ten polytope family of the truncated rhombic dodecahedron

POLYTOPE #EDGES #VERTICES #FACES #TRIANGLE #RHOMBIC #PENTAGON
43,200 (1)  87,330 43,200 44,132 1,880 42,240 12
10,800 (2) 21,810 10,800 11,012 440 10,560 12
2,700 (4) 5,430 2,700 2,732 80 2,640 12

 


POLYTOPE #EDGES #VERTICES  #FACES  #TRIANGLE #RHOMBIC #SQUARE
4,800 (3) 13,932 4,800  9,134 8,672 462
1,728 (5) 4,908 1,728 3,182 2,912 270
1,200 (6) 3,372 1,200  2,174 1,952 222
432 (10) 1,164 432 734 608 126
300 (12) 792 300 494 392 102
192 (15) 492 192 302  224 78
108 (20) 264 108 158 104 54
48 (30) 108 48  62 32 30
48 (30) 96 48 50 8 12 30
12 (60) 24 12  14  6

Table 1.5

The twelve solution sets for the polytopes are generated by the optimal spherical packing of circles, note that set 48(30) has two forms. Each of the thirteen cases for the optimal spherical packing of circles has a density of 0.75. There are only twelve unique solution sets for the optimal spherical packing of circles upon the surface of a sphere. They are determined by a unit arc-radius of sixty degrees and the twelve factors of sixty. The  solution sets that are derived from the twelve factors of sixty are the geometric reason a circle is divided into 360 degrees and for the Sumerians' choice of a base sixty system.  When the radius of the circle is one degree, 10,800 circles are an optimal spherical packing of circles upon the surface of the sphere and each of the thirty faces of a truncated spherical rhombic triacontahedron has an optimal spherical packing of 360 circles per face. 

The Sumerians used a counting system of base ten for their daily commerce transactions, just as we do today [6].  This fact and the manner in which the sexagesimal system is presently used, suggests the following hypothesis.  The sexagesimal system is unique to the geometry of the sphere developed by the Sumerians from their knowledge of the twelve unique solution sets for the optimal spherical packing of circles.After five thousand years, portions of the Sumerians' sexagesimal system are still in use today.  We use their base sixty system for astronomy, cartography, navigation, surveying, and timekeeping.  When time is thought of as a transformation of the earth's orientation with respect to the sun, then all of the above arts are aspects of their geometry.  Hilbert's 14th problem is considered next and later, the first half of the problem is solved for systems with integer solutions.  The solutions for fractional parts must wait for a future paper.

2. THE 14th PROBLEM OF HILBERT. We now state the 14th 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 functionsX1, X2,...,Xm of the n variables x1, x2,...,xn be given,

X1 = f1(x1,...,xn), 
X2= f2(x1,...,xn),
(S)    ............................, 
............................, 
Xm = fm(x1,...,xn). 

Every rational integral combination of X1,...,Xm must evidently always become, after substitution of the above expressions, a rational integral function of x1,...,xn. [Nevertheless, there may well be rational fractional functions of X1,...,Xm which, by the operation of the substitution S, become integral functions in x1,...,xn.] Every such rational functions of X1,...,Xm, which becomes integral in x1,...,xn after the application of the substitution S, I propose to call a relatively integral function of X1,...,Xm. Every integral function of X1,...,Xmis 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 GfPn, with the form Z2,...,Zn, where Z2 = ±1, and p = 2. A positive example result when Xm = 8, 64, 4096 and with xn = 3, 6, 12 respectfully.

The problem is to now demonstrate the existence of such "a finite system of relatively integral functions X1,...,Xm, by which every other relatively integral function of X1,...,Xm 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,...,fm have coefficients that are integers and included among the relatively integral functions of X1,...,Xm only such rational functions of these arguments as they become, by the application of the substitution S, rational integral function of x1,...,xn 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. [3a]

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,...,Xn-1 be a collection of unitary, orthogonal Tetrahedral-vectors, represented by a diagonal matrix, of the Abelian rotational group Z2,Z2,Z2, or Gf8.  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 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 Gf’PnXGfPn and/or GfPn+n, with the form Z2,...,Zn, where Z2 = ±1, P = 2, and n+n = 2n, by the action of multiplication and/or addition when the value for n = 3, 6, 12.  Figure 3.2 illustrates the idea of a tetrahedron projected upon a hyper-complex sphere created by theorem 3.1.

tetre.gif (41258 bytes)

Figure 3.2

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

COROLLARY 3.4. 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-complex plane incident to the given central point.

COROLLARY 3.5. Diametrically opposite 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 3.6. 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 their pole point, another perpendicular point, separated from the first two points by arcs of 90 degrees.

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. 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 Gf8 over the outside surface of a three-sphere. Augmented by Gf’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 Gf8 and the dual vector field Gf’8 is created. The left-handed system of coordinates Gf’8 is defined by the tetrahedron in the mirror. The right-handed system of coordinates Gf8 is defined by the original tetrahedron Gf8. The familiar vector field is now labeled with unitary diagonal matrices that commute. The spherical tetrahedra are illustrated in Figure 4.1

mirrortetra.GIF (71551 bytes)

Figure 4.1

In 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 Gf’8XGf8, and/or Gf64, 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 Gf’8XGf8, demonstrates the first example of extension into the new group Gf64. We now prove theorem 3.1 for the algebraic extension field  in the general case.

Proof of theorem 3.1: Following Weyl's [8] 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         (4.2
we also have 

A(r) = a1 x21e21 + a2x22e22 + ... + anx2ne2n.        (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,... , an are called the characteristic numbers of the form A, and e1, e2,...,en are the corresponding characteristic vectors

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. Then 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 = a1e1. 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         (4.5

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 

wpe14.jpg (12727 bytes)

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 x2,x3,...,xn. 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- an).         (4.7

Thus it follows that the characteristic numbers, a1,a2,...,an, are uniquely determined, and their sum is the trace of A. [8]

Since these fields are discrete, we again verify theorem 3.1 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. [7a]

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 Gf’8XGf8. Using linear algebra and group theory, we develop a form of algebraic geometry that allows the problem to be solved over a discrete spherically symmetric point field when 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 + c1xn-1 + ... + cn-1x + cn = 0,

where c0, c1, c2,...,cn 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. Unitary solution sets result if the last coefficient is equal to one, cn = ±1, which restricts the problem to the surfaces of a hyper-complex sphere. [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 Gf8 and the extensions Gf’PnXGfPn, with the form Z2,...,Zn, where Z2 = ±1, P = 2, and n = 3, 6, 12. 

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

GfPn = GnÉ Gn-1É Gn-2É...É G0= I,

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

Since the full symmetric group S6 is isomorphic to Z2,Z2,Z2,Z2,Z2,Z2, all that remains is to show that Gf26 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

                                                                 Gf26 = 64 É 32 É 16 É 8 É 4 É 2 É 1.

Figure 5.3 is a mirror symmetric representation of the group Gf26 and/or Gf64, 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. 

cross.gif (47393 bytes)

Figure 5.3

Table 5.4 displays the subgroups that Gf26 decomposes into and the various regular polytopes that they represent. 


ORDER n EQUAL REPRESENTATION Gf2n
64 6 The group Gf64
32 5 Icosahedron-Dodecahedron dual space 

[the icosahedron is represented by the blue elements 
and  the dodecahedron by the red elements in Figure 5.3.]

16 4 Octahedron-Hexahedron dual space 

[(++++++),(-++-++),(+-++-+),(- -+- -+), 
  (- - - - - -),(+- -+- -),(-+- -+-),(++-++-), 
(- - -+++),(+- - -++),(-+-+-+),(++- - -+), 
  (+++- - -),(-+++- -),(+-+-+-),(- -+++-)]

8 3 Tetrahedron dual space 

[(++++++),(-++-++),(+-++-+),(- -+- -+),
 (- - - - - -),(+- -+- -),(-+- -+-),(++-++-)]

(- - -+++),(+- - -++),(-+-+-+),(++- - -+), 
(+++- - -),(-+++- -),(+-+-+-),(- -+++-)]

4 2 Kline four group (+1, -1, +i, -i) 

[(++++++),(- - - - - -),(- - -+++),(+++- - -)] 

Alternate group 

[(++++++),(-++-++),(+-++-+),(- -+- -+)]

2 1 Identity and conjugate (+1,-1) 

[(++++++),(- - - - - -)]

1 0 Identity (+1) 

[(++++++)]


Table 5.4

We now use these solvable groups to prove theorem 2.1 (for the integer solutions to the 14-th problem of Hilbert) by demonstrating that all the coefficients to the polynomial equations are integers.

Proof of theorem 2.1 for n = 6, 12: Let the coefficient of the polynomial c0, c1, c2,...,c6 be quantities, which are algebraically independent over k. Set K = k(c0, c1, c2,...,c6) and as before, define the f(x), such that

f(x) = c0x6 + c1x5 + ... + c5x + c6Î Gf26

is the general equation of degree 6 over k. Suppose, f(x) = (x - x1)(x - x2)...(x - x6) [isomorphic with equation (4.7)] is in some extension field of F[Gf26]. It should be clear that the Xm are permutations of the solutions to the equation, when Xm = fm(x1,...,x6). It is not difficult to show that the x1,...,x6 are algebraically independent over k. The proof of this establishes theorem 3.1 as shown above. Finally, the coefficients ci are elementary symmetric functions of the xi, which are related to the polynomials by the following equations, 

                                                                c0 = 1 (a monic polynomial),
                                                                c1 = x1 + x2 + ...x6 (the trace), 
                                                                c2 = x1x2 + x2x3 + ... + x5x6
                                                                c3 = x1x2x3 + x2x3x4 + ... +x4x5x6,
                                                                   ..., 
                                                                c6 = x1x2...x6.                          [2]

The proof is completed by substituting the various permutations of Xm = fm(x1,...,x6), for the unitary values of x1,...,x6, into the above elementary symmetric functions of the xi and generating the coefficient of all the equations of degree six in the solution set. The result is a demonstration that the coefficients are all integers and that each possible permutation satisfies the theorem. The 64 examples for degree six and the 4096 examples for degree twelve are presented in  Appendix A .[7a] The group of permutations for degree twelve is demonstrated in the same manner and represented geometrically by the semi-regular polytopes. The example for degree three is now presented.

Proof for n = 3: let n equal three and substitute the permutations of Xm = fm(x1,x2,x3), into the given equations for the coefficients,

  c0 = 1, 
c1 = x1 + x2 + x3 (the trace), 
  c2 = x1x2 + x2x3
c3 = x1x2x3, 

and solve the equation for the polynomial functions, 

  f(xi) = c0x3 + c1x2 + c2x + c3 Î Gf8

using the unitary permutations of Xm = fm(x1,x2,x3), such that

   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). 

The following solution set of polynomial equations are generated when Xmis eight and xn is three, 

              X0 =(+1,+1,+1) = f(x0) = x3 + 3x2 + 2x + 1 = 0,
    X1 =(+1,+1,-1) = f(x1) = x3 + x2 - 1 = 0, 
           X2 =(+1,-1,+1) = f(x2) = x3 + x2 - 2x - 1 = 0,
  X3 =(+1,-1,-1) = f(x3) = x3 - x2 + 1= 0,
    X4 =(-1,+1,+1) = f(x4) = x3 + x2 - 1 = 0, 
           X5 =(-1,+1,-1) = f(x5) = x3 - x2 - 2x + 1 = 0, 
    X6 =(-1,-1,+1) = f(x6) = x3 - x2 + 1 = 0,
                     X7 =(-1,-1,-1) = f(x7) = x3 - 3x2 + 2x - 1 = 0.

In dimension three, the algebraic and geometric system furnishes a positive example for the existence of a finite system of functions having integers for coefficients.

6. THE IDENTITY ELEMENT AND 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 by theorem 3.1 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. 

When n = 2, the polynomial equations have the form

c0x2 + c1x + c2 = 0,

they define a great circle around a hyper-complex sphere, which has only four solutions. The solutions are the four invariant Gaussian numbers, +1, -1, +i, -i, and defined respectfully for dimensions two and six by

(+1,+1), (-1,-1), (-1,+1), (+1,-1); 
(+1,+1,+1+1,+1,+1), (-1,-1,-1,-1,-1,-1), (-1,-1,-1,+1,+1,+1), (+1,+1,+1,-1-1,-1,).

Theorem 3.1 physically place an observer into the mathematics by defining an orientation of an object from the observer's perspective. The kernel identity subgroup of Gf64 is 

(++++++), (- - -+++), (- - - - - -), (+++- - -)

the four identity operators. The circle of the solution set defines a hyper-plane that divides the hyper-complex sphere into four sections. The observer, who is defined by the operator (++++++), sees the circle with a vertical set of diametrically opposite points, (++++++) and (- - - - - -). The observer, who is defined by the operator (- - -+++),sees the circle with a horizontal set of diametrically opposite points, (- - -+++) and (+++- - -).

The mirror image of a tetrahedron is defined on the inside of the hyper-complex sphere. An observer defined on the outside of the  sphere 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. 

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. A geometric interpretation for the identity element is an algebraic expression for an observer's viewpoint of an object and the ability to express this viewpoint in relation to the observer’s long axis. 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 rotating the object, the observer, or both over the surfaces of the hyper-complex sphere.

7. THE ALGORITHM FOR THE NUMBER OF CIRCLES IN THE OPTIMAL SPHERICAL PACKING OF CIRCLES AND THE RESULTING POLYTOPES FOR THE SOLUTION SETS. We desire an algorithm that generates different values for the number of circles in a solution set for the optimal spherical packing of circles. The different numbers represent larger and larger regular systems of points on the surface of a sphere. With the knowledge that a circular covering problem is related to the formula for the surface area of the sphere, we start with the formula A = 4 pr2.   For almost three thousand years, the Sumerians and their decendents used two values for Pi, one of which was three.   It seems absurd that they would use such a poor value for such a long period of time, unless one considers our beginning hypothesis.  The sexagesimal system is unique to the geometry of the sphere developed by the Sumerians from their knowledge of the twelve unique solution sets for the optimal spherical packing of circles. On the surface of a sphere, straight lines are arcs of great circles and the ratio of twice the arc-radius to the circumference of the great circle is three.  A plane illustration of a system without straight lines is shown in Figure 7.1.

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Figure 7.1

Therefore, we use the surface area formula to generate the integer number of circles on the surface of a sphere when Pi is three and the variables for the circles diameters are divisions of the arc-radius. The divisions of the arc-radius that provide integer solutions are the twelve factors of sixty.   The reason for our ancestors' choice of a sexagesimal system based on the number sixty is the fact that only these twelve numbers, the twelve factors of sixty, provide integer solutions (modulo 60)  for the optimal spherical packing of circles. The twelve sets of the optimal spherical packing of circles, modulo 60, on the surface of a unit sphere are given once again,

12(60); 48(30); 108(20); 192(15); 300(12); 432(10); 
1,200(6); 1,728(5); 2,700(4); 4,800(3); 10,800(2); and 43,200(1)

The integers in parentheses are the diameters of the circles in degrees. Since we define the hyper-complex sphere as a two sided membrane, there would also be twelve solution sets that are on the inside surface for a total of twenty-four sets. Is this the geometric reason for the division of the day into twelve and twenty-four hours?  The algebraic and geometric facts about the optimal spherical packing of circles are the reasons why a circle is divided into 360 degrees and why the  Sumerians and their decendents persisted in their use of three for Pi.

When the centers of the circles are considered as vertices they define two families of polytopes. The representation fields for the optimal packing of circles are old and new families of polytopes. The largest polytope, modulo 60, is a truncated rhombic triacontahedron with 44,132 faces, 43,200 vertices and 87,330 edges. The family of the truncated rhombic triacontahedron has two additional members. The second family is the ten polytope family of the truncated rhombic dodecahedron and the optimal spherical packing of circles for the smallest polytope in the group is the cuboctahedronAppendix B[7b] provides a detailed description of each polytope in descending order based on their size and family. The cuboctahedron is the first polytope in our extension fields to have its vertices defined by rational whole numbers. The polytope is related to the hexahedron (cube) by the fact that its vertices are the midpoints of the cube's edges. The V4 vertices of its dual, the rhombic dodecahedron, are the center of the cube's faces, which are determined by the crossing of the perpendicular edges of a chiral pair of dual tetrahedra that are coincident at their midpoints.

8. CONCLUSIONS AND QUESTIONS. Theorem 2.1 was stated in the exact words of Hilbert [the bracketed section] although not proved, was left in. The reason for this is the extension to our algebra, which proves the bracketed section, also extends the field of the hyper-complex sphere to include the rational fractional functions of X1,...,Xm which, by the operation of the substitution S, become integral functions in x1,...,xn. Therefore, the completion of this proof [the bracketed portion] and the geometric proof for the group Gf212 must wait for additional papers. Additional proofs for the theorems and the generation of the algebraic examples, 64 for Gf26 and 4096 for Gf212, are demonstrated in Appendix A.Since the field is finite, proof is by exhaustion with the algorithm examining all permutations. Corollary 3.6 is proved with the geometric proof for the group Gf212 in a future paper. 

We accomplished the goal of constructing positive examples algebraically and geometrically by the combination of simple yet invariant geometric forms, which create higher forms. Using the ideas of a field and a group, the regular polytopes are constructed in an algebraic fashion with a tetrahedron over Gf8, a pair of chiral tetrahedra over Gf’8XGf8 (which combine to form a hexahedron), and all of the regular polytopes over Gf64 by the addition of the various subgroups. We again note that the group Gf64 is the master rotational group for the regular polytopes. 

The group of 64 hyper-complex numbers defines dual orthogonal subgroups. When these subgroups are mapped to a spherical tessellation of an icosahedron, thirty-two of the numbers are found on each side of the membrane. Thirty are found at the midpoints of the tessellated edges. Two additional pole points define the observer’s long axis and its relationship to the icosahedron. The dodecahedron, the dual of the former, is mapped to the inside surface, with its thirty edges orthogonal to its dual. The dual polytopes midpoints for the two sets of edges are coincident to each other, but on opposite sides of the membrane. 

When the group Gf64 is extended by multiplication, Gf’64XGf64 defines geodesic lines. Sets of these geodesic lines are then used to reconstruct the regular polytopes on the same surface. Gf’64XGf64 defines 128 pairs of perpendicular points out of 4096 possible permutations. In the proof of corollary 3.6, how are these 128 pairs of perpendicular points determined? The theorem of Lagrange is considered as a theorem of composition. [1] The algebraic extension (composition) to Gf’8XGf8, and/or Gf64 creates ten tetrahedra. The ten tetrahedra are the various finite geometrical subgroups, into which Gf64 decomposes and by which the regular polytopes are constructed (composition). We have just shown by theorems 3.1 and 5.2 that the Abelian group Gf64 manifests both of these properties (composition and decomposition). We ask the following questions: What are the 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. These questions will be answered in a journal published paper. Meanwhile, we offer the above questions as a challenge, to again stimulate research into hyper-complex numbers. The correct answers should follow the format presented in this paper with the unitary value of each sign understood, such as

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 .... 

Hint, Figure 5.3 is mirror symmetric and the group Gf64 decomposes into mirror symmetric subgroups similar to Figure 5.3. Therefore, one should also submit a set of mirror symmetric images that portray the decomposition of the two groups in Figure 5.3 into two groups that have five tetrahedra each as subgroups.

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. R. Carmichael, Introduction to the Theory of Groups of Finite Order, Dover Publications, Inc., New York, 1937, 120-354.

2. A. Clark, Elements Of Abstract Algebra, Dover Publications, Inc., New York, 1971, 67-129.

3a. H. Coxeter, Introduction to Geometry, 2nd Ed., John Wiley & Son, Inc., New York, 1989, 148-159.

3b. H. Coxeter, The Regular Polytopes, Dover Publications, Inc., New York, 1971, 67-129.

4a. D. Hilbert, Mathematical Problems*, Bull. Amer. Math. Soc., 8, 1902, 437-479, *Translated for the Bulletin, by Dr. Mary Winston Newson. The original appeared in the Gottinger Gachrighten, 1900, pp. 253-297, and Archiv der Mathematik and physik, 3d ser., vol. 1 (1901), pp. 44-63 and 213-237.

4b. D. Hilbert and S Cohn-Vossen, Geometry And The Imagination, Chelsea Pub. Co., New York, 1983, 52-93.

5. M. Nagata, On The 14-th Problem Of Hilbert, Am. Journal Of Mathematics, Vol. 81, 1959, 766-792.

6. O. Neugebauer, The Exact Sciences In Antiquity, Dover Publications, Inc., New York, 1957, 1-40.

7a. J. Van Dyke, and C. Radcliffe, Appendix A, pending publication on the Internet, upon publication of OPTIMAL SPHERICAL PACKING OF CIRCLES AND HILBERT'S 14TH PROBLEM.

7b. J. Van Dyke, and C. Radcliffe, Appendix B, pending publication on the Internet, upon publication of OPTIMAL SPHERICAL PACKING OF CIRCLES AND HILBERT'S 14TH PROBLEM.

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

Mathematics Subject Classification:

03C13, 11R20.

KEY WORDS:

master Abelian rotational group, polytopes, Gf'PnXGfPn, new families of polytopes, optimal spherical packing of circles, Mathematical foundations, sexagesimal analytic geometry, unitary diagonal matrices, elements, finite groups, Icosahedral, Octahedral, Tetrahedral, dual groups, six-dimensional space, the 14th problem of Hilbert, nth degree polynomials, spherical projection plane, Z2,Z2,Z2,Z2,Z2,Z2, solvable group S6, matrix algebra, regular polytopes, surface of a hyper-complex sphere, extension fields, chirality, invariance, unified commuting vector field, orthogonal rotational group.
 
 

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