Polyhedra Analogues of the Platonic Solids
In this chapter we investigate polyhedra in Euclidean 3-space, E 3, without self-intersections and with some local and global properties related to those of the Platonic solids. A polyhedron is the geometric realization of a compact 2-manifold in E 3 such that its 2-faces are (not necessarily convex) plane polygons bounded by finitely many line segments. Adjacent faces and edges are not coplanar. A flag of a polyhedron P is any triple consisting of a vertex, an edge, and a face of P, all mutually incident.