| Order-4-4 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,4,4} {5,41,1} |
| Coxeter diagram |       |
| Cells | {5,4}  |
| 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  |    |    |  |  |    |    |
| Image |  |  |  |  |  |  |
| Cells |  {2,4} |  {3,4} |  {4,4} |  {5,4} |  {6,4} |  {∞,4} |
| Order-4-4 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {6,4,4} {6,41,1} |
| Coxeter diagram | |
| Cells | {6,4}  |
| 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 | |
| Cells | {∞,4}  |
| 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 |
In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation.
In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation. With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.
In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
In the geometry of hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation with Schläfli symbol {7,3,7}.
In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.
In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,5,4}.
In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,6,4}.
In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.
In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation. 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.
In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation. 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.
In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation with Schläfli symbol {5,4,5}.
In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,5,4}.
In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,7,3}.
In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation. Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,6,4}.
In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,8,3}.
In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,∞,3}.