Clusters Produced by Placing Rhombic Triacontahedra at the Vertices of Polyhedra

Abstract

In this article we explore possible clusters of rhombic triacontahedra (RTs), usually by connecting them face to face, which happens when they are placed at the vertices of certain polyhedra. The edge length of such polyhedra is set to be twice the distance of a face of an RT from the origin (about 2.7527). The clusters thus produced can be used to build further clusters using an RT and a rhombic hexecontahedron (RH), the logo of Wolfram|Alpha. We briefly look at other kinds of connections and produce new clusters from old by using matching polyhedra instead of RTs.