Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles

Abstract

The cupolae with concave polyhedral surfaces consist of: two regular polygons, n-gone and 2n-gone in parallel planes, interconnected by an envelope constituted of series of equilateral triangles. The paper describes cupolae which originate by corrugating of a fourfold strip of equilateral triangles, forming thereby the envelope of a cupola. In this manner, a non-convex polyhedron is emerged. Such a method of corrugating the envelope, allows the solutions for generating cupolae with base polygon which number of sides exceedes n=10, which was the maximal number of sides for cupole with the envelope consisted of twofold strip of equilateral triangles. By analyzing the elements of these polyhedra and by help of their paper models, we find geometric constructions and projection procedures by which it is possible to graphically display the cupolae.