Triangular tiling honeycomb
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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.
Triangular tiling honeycomb | |
---|---|
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbol | {3,6,3} h{6,3,6} h{6,3[3]} ↔ {3[3,3]} |
Coxeter-Dynkin diagrams | ↔ ↔ ↔ |
Cells | {3,6} |
Faces | triangle {3} |
Edge figure | triangle {3} |
Vertex figure | hexagonal tiling |
Dual | Self-dual |
Coxeter groups | , [3,6,3] , [6,3[3]] , [3[3,3]] |
Properties | Regular |
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.