Uniform Honeycombs in 3-Space
Here is a collection of uniform regular-polyhedral honeycombs. George Olshevsky defines a uniform honeycomb in three-space as "one all of whose cells are uniform polyhedra and whose vertices are transitive on its symmetry group, and which covers three-space exactly once". Banko Grünbaum lists all 28 of these in his 1994 paper: "Uniform tilings of 3-space," Geombinatorics 4, 49-56. I have not read seen the paper yet, so any errors with the figures are my own. Gurnbaum's paper does not represent the original discovery of these figures, as they appear to have been known for some time (dating back at least to the 1950's). But in any case, here is a simple presentation of these figures for those who are interested.
Note: Each figure shows all of the polyhedra surrounding a single vertex, with the exceptions of figures 24, 25, 27 and 28. See the explanation below the table for those figures.
Note: Figures labeled with "=x", (numbers 24, 25, 27 and 28), include the polyhedra around 2 vertices instead of 1, so as to differentiate between related pairs. Thanks to George Olshevsky and Jonathan Bowers for introducing me to this collection, and the names associated with them. Thanks also to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).
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