ABSTRACT
The periodic order of shell layers for the elements is geometrically analyzed based on the vertices, faces, and edges for the primary six-dimensional polyhedrons: tetrahedron, octahedron, cuboctahedron, truncated octahedron and truncated rhombic dodecahedron. The atoms of the Periodic Table of the Elements include the number of electrons per shell, where Robert Mulliken counted the number of orbitals of electrons per shell as a one-electron orbital wave function, called electron orbital theory. All of the crystal systems are a synergetic combination of three and six dimensions. There is also ligand bonding found in the transition elements which presents evidence of a dual coordinate system of three axes of symmetry and six dimensions. The self-assembly process justifies the existence of a based on forces radiation from the nucleus on six dimensions and returning to the nucleus on three dimensions.
