Order-4 octahedral honeycomb
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The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.[1]
Order-4 octahedral honeycomb | |
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Perspective projection view within Poincaré disk model | |
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {3,4,4} {3,41,1} |
Coxeter diagrams | ↔ ↔ ↔ |
Cells | {3,4} |
Faces | triangle {3} |
Edge figure | square {4} |
Vertex figure | square tiling, {4,4} |
Dual | Square tiling honeycomb, {4,4,3} |
Coxeter groups | , [3,4,4] , [3,41,1] |
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.