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
Cubic honeycomb and its dual triangular pyramidille. Four cells exist around each vertex: The "cantitruncated cubic honeycomb" can be orthogonally projected into the euclidean plane with various symmetry arrangements. Cells can be shown in two different symmetries. The linear Coxeter diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating. The dual of the "cantitruncated cubic honeycomb" is called a triangular pyramidille, with Coxeter diagram, . This honeycomb cells represents the fundamental domains of formula_2 symmetry. A cell can be as 1/24 of a