Equiprojective polyhedra

Equiprojective Polyhedra

A convex polyhedron P is equiprojective if, for some k, the orthogonal projection (or “shadow”) of P in every direction, except those directions parallel to faces of P , is a k-gon. We address an open question posed by Shepherd [11], and reported in [5], by characterizing equiprojective polyhedra, and giving an O(n log n) time recognition algorithm.

Some New Equiprojective Polyhedra

A convex polyhedron P is k-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of P are k-gon for some fixed value of k. Since 1968 it is an open problem to construct all equiprojective polyhedra. Recently, Hasan and Lubiw [8] have given a characterization of equiprojective polyhedra. Based on their characterization, in this paper we discover s...

polyhedra ∗

Graphs of polyhedra; polyhedra as graphs

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedr...

Cubic Polyhedra

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears many (but not all) of them can be relaxed to smooth minimal surfaces (under an appropriate smoothing flow, keeping their symmetries). Here we give a complete classification of the cubic polyhedra. Among these are five new infi...