What are "Uniform Polyhedra"?
- Eric Hdz
What is a polyhedron?
This innocent looking question is actually hard to answer. Since Euclid first treated polyhedra in the first place, he failed to rigorously define what he was talking about, and this mistake was uncorrected for centuries, leaving the consequences to more modern mathematicians. Eventually, some definitions were created, but some of them were better in certain contexts than others. Here, we will use one very recent definition by Jonathan Bowers (translated into plain English for simplicity):
A polyhedron is a 3-dimensional shape bounded by flat faces. True polyhedra must be such that exactly two faces meet at each edge, no two faces, edges or vertices coincide completely, and every vertex must be connected to each other by a path of edges (i.e. it is not a compound).This definition is actually not as restrictive as it might seem at first glance. Actually, it turns out that one can create many polyhedra with an arbitrarily large number of faces, edges or vertices! Therefore, we might be interested in finding a subset of the set of all polyhedra (plural of polyhedron) with certain interesting characteristics. One obvious subset is the subset of convex polyhedra, where a convex shape is roughly defined as a shape in which every segment connecting two points on its surface is completely contained inside it. Simple examples of these include most prisms and pyramids, but actually most of the "imaginable" polyhedra fall into this category. An interesting property of convex polyhedra is the Euler Formula, which states that for all convex polyhedra with F faces, E edges and V vertices, the following happens: A simple proof of this fact, involving just elementary graph theory is the following: Consider any tree containing as vertices all of the vertices of the polyhedron, and containing as edges some of the polyhedron's edges. Now, let's consider a graph , with a vertex on each face of the polyhedron, and an edge between adjacent faces of each edge of the polyhedron that was not in . We note that must be connected, as if it wasn't, it would be composed of at least two disconnected regions, and they would have to be separated by a cycle in , which does not exist. On the other hand, if contained a cycle, it would mean that a vertex on is in isolation from all of its neighbors, which also does not happen. Therefore, is a tree. However, in any tree, the number of edges is equal to the number of vertices minus one. Since has V vertices, since has F vertices, and since the total number of edges on both trees is equal to E, we have , which is what we wanted. ■ Enough of convex polyhedra, maybe we want a category that also includes concave polyhedra, but is a bit more specific. Another idea would be to consider regular-faced polyhedra, polyhedra