Order-6 hexagonal tiling honeycomb

Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.

[1] 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.

They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs.

There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.

The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells: It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:

It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces:

Subgroup relations:
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