The Architecture and Growth of Extended Platonic Polyhedra

The formation of polyhedra has attracted much interest as an attractive research topic that is connected with chemistry. In this paper, we focus on the grow law of so-called Goldberg method based on Platonic polyhedra. There are four classes of extended Platonic polyhedra we can construct: the extended tetrahedra; the extended hexahedra; the extended octahedra; the extended dodecahedra. The extended tetrahedra, extended hexahedra, and extended dodecahedra are, respectively, assembled by using the method of adding hexagons, whereas the extended octahedra are made by means of adding squares. We also prove that this method fails to be applied to icosahedra. The study of the architecture and growth of extended Platonic polyhedra provides further insight into the molecular design and theoretical characterization of chemical molecules.