More combinatorial threads – Collapsoids
Peter Hilton, Jean J. Pedersen, Sylvie Donmoyer · Cambridge University Press eBooks · 2010
What is a collapsoid? There is an interesting class of polyhedra having the property that all faces are congruent parallelograms. Since all faces are parallelograms, the polyhedra in this class have the property that every edge determines a zone of faces such that each face in the zone has two sides parallel to the given edge. Polyhedra having this latter property are called zonohedra ; we may speak of an n -zonohedron to emphasize that the polyhedron in question has n zones. As interesting examples of polyhedra in this class, the rhombic dodecahedron (which has 12 faces and 4 zones) and the rhombic triacontahedron (which has 30 faces and 6 zones) appear in Figure 12.1, which is based on illustrations by H. S. M. Coxeter (1907–2003). In [8], Coxeter describes the general theory of zonohedra and states that the angles on the faces of the rhombic dodecahedron are 70°32′ and 109°28′, while the angles on the faces of the rhombic triacontahedron are 63°26′ and 116°34′. You will readily believe that these angles are not ones that we get easily by folding paper (though we could get them in principle!). However, these are beautiful models and we can construct polyhedra like them by replacing each of the rhombic faces with a 4-faced pyramid without its base, which is composed of 4 equilateral triangles. We call this a cell and refer to each of the triangles as a triangular face of the cell.