Families of polyhedra

Isogonal Kaleidoscopical Polyhedra Families

A new class of isogonal polyhedra is considered. Polyhedra are constructed using a combination of reflections in several symmetry planes of a given symmetry group. The procedure is a generalization of a Wythoff construction used for building uniform polyhedra. Every valid combination of symmetry planes generates an entire family of isogonal polyhedra. There are seven families with tetrahedral s...

Connected Layer Families and Diameter of Polyhedra Semester Project

Given a polyhedron, its diameter is defined as the maximum distance between two of its vertices, where the distance between two vertices u and v is the number of edges contained in the shortest u− v path in the skeleton of the polyhedron. Bounding the diameter of a polyhedron with respect to its dimension d and the number n of its facets is one of the most important open problems in convex geom...

Complexity in Surfaces of Densest Packings for Families of Polyhedra

Elizabeth R. Chen, Daphne Klotsa, Michael Engel, Pablo F. Damasceno, and Sharon C. Glotzer School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Materials Science and...

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...

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