In the geometry of hyperbolic 3-space, the order-4-4 pentagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Geometry.
The Schläfli symbol of the "order-4-4 pentagonal honeycomb" is {5,4,4}, with four order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
Related polytopes and honeycombs.
It is a part of a series of regular polytopes and honeycombs with {p,4,4} Schläfli symbol, and square tiling vertex figures:
Order-4-4 hexagonal honeycomb.
In the geometry of hyperbolic 3-space, the order-4-4 hexagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the octagonal tiling honeycomb is {6,4,4}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
Order-4-4 apeirogonal honeycomb.
In the geometry of hyperbolic 3-space, the order-4-4 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,4,4}, with three order-4 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.
References.
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