Grasshopper formulation for the snub dodecahedron and the pentagonal hexecontahedron

Abstract

The purpose of the present paper is to demonstrate how the script components of Grasshopper, the user-programming platform of the geometry program Rhinoceros 7/8, can be used to create a software tool for the generation of two complex semi-regular polyhedra, namely, the snub dodecahedron and its dual, the pentagonal hexecontahedron. The snub dodecahedron is an Archimedean polyhedron whose geometric construction is not as simple or direct as that of other polyhedra. The dual of this polyhedron is the Catalan polyhedron known as the pentagonal hexecontahedron, a convex polyhedron with 60 identical, axially symmetric, semi-regular, pentagonal faces. The intricacy of deriving and specifying the geometric particulars of these two polyhedra might be a reason for their infrequent application in dome-like architectural envelopes, such as the “Amazon Spheres” in Seattle (2018). A numerical method for the geometric specification of the snub dodecahedron based on the regular icosahedron is used in this article to illustrate its formulation in Grasshopper (GH). The successive GH implementation of the duality principle to obtain the pentagonal hexecontahedron completes the formulation exercise. Here, the geometric construction method and its algorithm, as well as the data organization and its handling within the data tree structure in Grasshopper, pose the actual challenge for an efficient formulation. The practical outcome of the present discussion is provided by the GH-script that generates the mentioned polyhedra, which can be used in further design and construction processes.