| Order-4-4 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,4,4} {5,41,1} |
| Coxeter diagram |      
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| Cells | {5,4} 
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| Faces | {5} |
| Vertex figure | {4,4} |
| Dual | {4,4,5} |
| Coxeter group | [5,4,4] [5,41,1] |
| Properties | Regular |
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.
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}.
 Poincaré disk model |
 Ideal surface |
It is a part of a series of regular polytopes and honeycombs with {p,4,4} Schläfli symbol, and square tiling vertex figures:
| {p,4,4} honeycombs | ||||||
|---|---|---|---|---|---|---|
| Space | E3 | H3 | ||||
| Form | Affine | Paracompact | Noncompact | |||
| Name | {2,4,4} | {3,4,4} | {4,4,4} | {5,4,4} | {6,4,4} | ..{∞,4,4} |
Coxeter 
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| Cells |  {2,4}
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 {3,4}
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 {4,4}
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{5,4}
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 {6,4}
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 {∞,4}
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| Order-4-4 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {6,4,4} {6,41,1} |
| Coxeter diagram |
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| Cells | {6,4} 
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| Faces | {6} |
| Vertex figure | {4,4} |
| Dual | {4,4,6} |
| Coxeter group | [6,4,4] [6,41,1] |
| Properties | Regular |
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}.
Poincaré disk model |
 Ideal surface |
| Order-4-4 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,4,4} {∞,41,1} |
| Coxeter diagram |
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| Cells | {∞,4}
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| Faces | {∞} |
| Vertex figure | {4,4} |
| Dual | {4,4,∞} |
| Coxeter group | [∞,4,4] [∞,41,1] |
| Properties | Regular |
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}.
Poincaré disk model |
 Ideal surface |