Optimal Spherical Packing of Circles and Hilbert's 14th Problem

James R. Van Dyke

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'P^nXG_fP^n\$ for 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, \$(Z_2,\dots,Z_2), 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, \$G_f'P^3XG_fP^3\$, 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 \$Z_2,Z_2,Z_2,Z_2,Z_2,Z_2\$ is isomorphic to the symmetric group \$S_6\$, which is solvable when defined by \$G_f'P^3XG_fP^3\$ and labeled with matrices. The group is then extended to include semi-regular polytopes when n = 6, 12.