Order 4 square tiling honeycomb

Updated on Jan 06, 2024

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In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb, is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, has four square tilings, {4,4} around each edge, and infinite square tilings around each vertex in an square tiling {4,4} vertex arrangement.

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.

Symmetry

It has many reflective symmetry constructions, as a regular honeycomb, ↔ with alternate types (colors) of square tilings, and with 3 types (colors) of square tilings, with a ratio of 2:1:1. Two more half symmetry construction with pyramidal domains have [4,4,1⁺,4] symmetry: ↔ , and ↔ .

There are two high index subgroups, both index 8: [4,4,4^*] ↔ [(4,4,4,4,1⁺)] exists with a pyramidal fundamental domain, [((4,∞,4)),((4,∞,4))] or , and secondly [4,4^*,4], with 4 orthogonal sets of ultraparallel mirrors in an octahedral fundamental domain: .

Images

It is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞} with infinite apeirogonal faces and all vertices are on the ideal surface.

This honeycomb contains , that tile 2-hypercycle surfaces, similar to these paracompact tilings:

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.

It is a part of a sequence of honeycombs with a square tiling vertex figure:

It is a part of a sequence of honeycombs with square tiling cells:

Rectified order-4 square tiling honeycomb

The rectified order-4 hexagonal tiling honeycomb, t₁{4,4,4}, has square tiling facets, with a square prism, , vertex figure. It is the same as the regular square tiling honeycomb, {4,4,3}, , with a cube vertex figure, .

Truncated order-4 square tiling honeycomb

The truncated order-4 square tiling honeycomb, t_0,1{4,4,4}, has square tiling and truncated square tiling facets, with a square pyramid vertex figure.

Bitruncated order-4 square tiling honeycomb

The bitruncated order-4 square tiling honeycomb, t_1,2{4,4,4}, has truncated square tiling facets, with a tetragonal disphenoid vertex figure.

Cantellated order-4 square tiling honeycomb

The cantellated order-4 square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, .

Cantitruncated order-4 square tiling honeycomb

The cantitruncated order-4 square tiling honeycomb, is the same thing as the truncated square tiling honeycomb, .

Runcinated order-4 square tiling honeycomb

The runcinated order-4 square tiling honeycomb, t_0,3{4,4,4}, has square tiling and cube facets, with a square antiprism vertex figure.

Runcitruncated order-4 square tiling honeycomb

The runcitruncated order-4 square tiling honeycomb, t_0,1,3{4,4,4}, has square tiling, truncated square tiling and cube facets, with a square pyramid vertex figure.

Omnitruncated order-4 square tiling honeycomb

The omnitruncated order-4 square tiling honeycomb, t_0,1,2,3{4,4,4}, has truncated square tiling and octagonal prism facets, with a tetrahedron vertex figure.

Quarter order-4 square tiling honeycomb

The quarter order-4 square tiling honeycomb, q{4,4,4}, has truncated square tiling and octagonal prism facets, with a tetrahedron vertex figure.

References

Order-4 square tiling honeycomb Wikipedia

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