Duality of the platonic polyhedrons and isomorphism of the regular deployable polyhedral mechanisms (DPMs)

Abstract

This paper investigates the duality of the five Platonic polyhedrons via the ray and axis coordinates of Line geometry. Then, based on the five Platonic polyhedrons, regular deployable polyhedral mechanisms are constructed by implanting the plane-symmetric eight-bar linkages into the edges, faces and vertices of the polyhedrons. Due to the duality of the five regular Platonic polyhedrons, it is revealed in this paper that the dual regular deployable polyhedral mechanisms constructed based on the dual Platonic polyhedrons are isomorphic and the isomorphism of the dual regular deployable polyhedral mechanisms is then for the first time presented via topology graphs and their corresponding adjacency matrices. © Springer-Verlag London 2012.