Triangular tiling honeycomb Triangular tiling honeycomb The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called "paracompact" because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling. It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, ↔ , and as from , which
24-cell honeycomb 24-cell honeycomb In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}. The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space. If a 3-sphere is inscribed in each hypercell of this tessellation, the resulting arrangement is the densest possible regular sphere packing in four dimensions, with the kissing number 24. The packing density of this arrangement is The 24-cell