The web site's purpose, using modern mathematical techniques and the computer, is to share the mathematical discoveries that result from the study of unitary spherical polytopes (regular and semi-regular polyhedron projected upon the surface of a unit sphere).  In the process of dividing the surface of the sphere into smaller and smaller congruent parts, 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 G_f'PⁿXG_fPⁿ for the polytopes, twelve unique solution sets for the optimal spherical packing of circles, and the foundation for a sexagesimal analytic geometry.

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.

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.

Two millennia before the Greek civilization created their great mathematical works, there existed a geometry of circles, geodesics, and spheres developed by their ancestors, the Sumerians. 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.  The Sumerians used a counting system of base ten for their daily commerce transactions, just as we do today.  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 that the Sumerians developed from their knowledge of the twelve unique solution sets for the optimal spherical packing of circles.

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 (CHIRAL ISOMORPHISM OF ABELIAN GROUPS OVER G_f'PⁿXG_fPⁿ) 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 examples given in the OPTIMAL SPHERICAL PACKING OF CIRCLES AND HILBERT'S 14TH PROBLEM, 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.  At the International Congress in Paris in 1900, Hilbert 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 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. Ten tetrahedra are created by the extension to G_f'8XG_f8 and/or G_f64.  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 G_f64 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 G_f64 is the master rotational group for the regular polytopes.  When the dimension of n = 12 the Abelian rotational group G_f4096 is the master rotational group for the semi-regular polytopes.

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