Realizability of Polyhedra

Abstract

We address ourselves to three types of combinatorial and projective problems, all of which concern the patterns of faces, edges and vertices of polyhedra. These patterns, as combinatorial structures, we call combinatorial oriented polyhedra. Which patterns can be realized in space with plane faces, bent along every edge, and how can these patterns be generated topologlcally? Which polyhedra are constructed in space by a series of single or double truncations on the smallest polyhedron of the type (for example from the tetrahedron for spherical polyhedra)? Which plane line drawings portraying the edge graph of a combinatorial polyhedron are actually the projection of the edges of a plane-faced polyhedron in space? Wherever possible known results and specific conjectures are given.