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Dear fellow physical scientist and mathematician, The last two centuries of mathematical research demand that we readdress the foundations of our science. Any mathematical reform, which develops the way mathematics is taught, must first consider the two most important factors, the history and the foundation, upon which, the art and science of mathematics is constructed. The incompleteness theorem published by Godel in 1931 demands that our ideas must be very creative. Credibility, of course, is the reason why they are so important. A geometry and group theory that is based on spherical polytopes solves the fundamental problem, but at the same time offers a paradox. The most ancient meaning of the word paradox is a surprise. The solution to the formulation of a correct geometry that models our perceived world is at least five thousand years old and of Sumerian origin! In the pursuit of the proof of this concept, the following discoveries were made. We discover a system of mathematics that provides positive examples for the 14th problem of Hilbert, a master Abelian rotational group Gf'PnXGfPn for the polytopes, two new families of 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 is 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 the semi-regular polytopes when n = 6, 12. As you shall soon see, a five thousand year old system resolves many modern day problems. We have established a non-profit organization to teach the mathematics on the internet. Our web site address is http://sexagesimalsystems.org/. Our first goal is to teach group theory using linear algebra and demonstrate how the fields of algebra, geometry, group theory, linear algebra, and number theory are woven together. The linear algebra approach to group theory eliminates the confusion of lettered arrays. For example, permutations of say (ABCDEF) have only 64 permutation when represented by diagonal matrices, not 6! or 720 permutations of traditional methods! The 64 permutation of the diagonal matrices are easier for the student to understand and to quickly determine the results of any number of permutations taken together. The commutative property of multiplication is retained in the system. And additionally, all the fundamental mathematical concepts may be taught through geometry. We have submitted the OPTIMAL SPHERICAL PACKING OF CIRCLES AND HILBERT'S 14TH PROBLEM, as a rebuttal to Nagata's paper On The 14-th Problem Of Hilbert, Am. Journal Of Mathematics, Vol. 81, 1959, 766-792, to the electronic journal Visual Mathematics, Vol. 2, No. 4, 2000, where it is published on-line. The additional papers at our web site represents a portrait into the development of the research at the time of each paper's copyright. Thank you for your attention and time. Sincerely, James R. Van Dyke
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