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
Order-6 hexagonal tiling honeycomb fundamental domain. It has an octahedral Coxeter diagram with 6 order-3 branches, and 3 infinite-order branches in the shape of a triangular prism, . It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures. There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form. It has a related alternation honeycomb, represented by ↔ , having alternating triangular tiling cells, and a regular form as , called a triangular tiling honeycomb. This honeycomb is a part of a sequence of polychora and