User:Tomruen/Uniform honeycombs Johnson

Original copy, without wikilinks and formatting. = Uniform honeycombs in three-space = by Norman Johnson, 2003

Besides the generic term polytope, we have the more specific terms polygon, polyhedron, and polychoron for polytopes of two, three, and four dimensions. Likewise, honeycomb is the generic term for a polytopal covering of n-space, but one can speak more specifically of a partition of a line, a tessellation of a plane, or a cellulation of 3-space.

There is a standard name, a little simpler than George Olshevsky's panoploid honeycomb, for a honeycomb that fills n-space without gaps or overlaps, namely, a tiling. A tiling is convex if all the tiles are convex.

A number of uniform tilings of Euclidean 3-space were described by Alfredo Andreini (1905). Others were discovered by J. C. P. Miller and H. S. M. Coxeter (see Coxeter 1932, 1940, 1985). In 1959 I found another one (No. 23 in George's list), which I described in my 1966 Ph.D. thesis. So far as I know, the first appearance in the literature of the presumably complete list of 28 uniform tilings was a note by Branko Grunbaum (1994).

To my knowledge, there is as yet no proof that there are exactly 28 uniform tilings of Euclidean 3-space, but it seems highly likely that this is the case. All of these tilings have counterparts in higher-dimensional space.

Most of the uniform tilings of 3-space can be derived by Wythoff's construction either from the regular tiling of cubes or from the eleven uniform tilings of the plane, but there are also a few anomalous ones.

Here I describe each of the twenty-eight, giving an extended Schlafli symbol, the name I use, the number of cell polyhedra of each type at a vertex, and a cross reference to the list recently by George. Two tilings with alternative derivations are listed twice.

Cells
The thirteen uniform solids listed below occur as cell polyhedra of uniform tilings of 3-space. Some of these may occur with less than their full symmetry and are given in alternative forms.