Hi,

1. Consider bitruncated cubic honeycomb in Euclidean space (filling by truncated octahedra). It is uniform, all edges loops have even length, so we can build alternated honeycomb for it. Cells of the original honeycomb are transforming to icosahedra (and we can make them regular), but what will be in space between them? Maze of non-regular tetrahedra? Or something else? What is the vertex figure of this honeycomb? It has four pentagonal faces... or not?

2. Is there any good way to enumerate cells of regular hyperbolic honeycombs (with finite cells and vertices) by exact mathematical objects - like we can use triples of integers for enumeration of Euclidean honeycombs? We can use real numbers and represent cells as coordinates of thier centers in some Poincare model, but I afraid that we'll quickly loose accuracy when go from the center. By "good way" I mean that we can easily check that two cells are the same (given their representation), and calculate action of transitions and rotations of the honeycomb. For {6,3,3} we can use elements of field Q[x]/[x^2+x+1] (or better matrices 2x2 over Z[x]/[x^2+x+1] - they give all movements of the honeycomb), and I hope that for other honeycombs with affine cells or vertices there is something similar. But what can we do with {4,3,5} and three others?