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The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity.
The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling, {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices. The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
Order-6-4 square honeycomb; Order-7-3 triangular honeycomb; Order-8-3 triangular honeycomb; Q. Quaquaversal tiling; Quarter cubic honeycomb; R. Rhombic dodecahedral ...
In the geometry of hyperbolic 3-space, the order-8-4 square honeycomb (or 4,8,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,8,4}. All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 octagonal tiling vertex figure.
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge.
In the geometry of hyperbolic 3-space, the order-7-4 square honeycomb (or 4,7,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,7,4}. All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 heptagonal tiling vertex ...
In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle , each of which has a limiting circle on the ideal sphere.
The birectified order-5 120-cell honeycomb constructed by all rectified 600-cells, with octahedron and icosahedron cells, and triangle faces with a 5-5 duoprism vertex figure and has extended symmetry [[5,3,3,5]].
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