Eulerian Stratification of 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...

polyhedra ∗

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...

Uniform Polyhedra

Uniform polyhedra consist of regular faces and congruent vertices. Allowing for nonconvex faces and vertex figures, there are 75 such polyhedra, as well as 2 infinite families of prisms and antiprisms. A recently discovered uniform way of computing their vertex coordinates is the basis for our program to display all of these solids, among which are many beautiful and stunning shapes. We will al...

Eulerian polynomials

A short report on my lecture at the Conference in Honor of W. Killing in Münster on December 7, 2007, together with some remarks on the preceding paper by ArjehM. Cohen. I reported on Euler’s famous paper Remarques sur un beau rapport entre les series des puissances tant direct que reciproques, Académie des sciences de Berlin, Lu en 1749, Opera Omnia Serie I, Bd. 15, pp. 70-90. In modern termin...