| Order-5-3 square honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {4,5,3} |
| Coxeter diagram |      |
| Cells | {4,5}  |
| Faces | {4} |
| Vertex figure | {5,3} |
| Dual | {3,5,4} |
| Coxeter group | [4,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 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-5-3 square honeycomb is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |  Ideal surface |
It is a part of a series of regular polytopes and honeycombs with {p,5,3} Schläfli symbol, and dodecahedral vertex figures:
| Order-5-3 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,5,3} |
| Coxeter diagram | |
| Cells | {5,5}  |
| Faces | {5} |
| Vertex figure | {5,3} |
| Dual | {3,5,5} |
| Coxeter group | [5,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 pentagonal honeycomb or 5,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 pentagonal honeycomb is {5,5,3}, with three order-5 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |  Ideal surface |
| Order-5-3 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {6,5,3} |
| Coxeter diagram |  |
| Cells | {6,5}  |
| Faces | {6} |
| Vertex figure | {5,3} |
| Dual | {3,5,6} |
| Coxeter group | [6,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 hexagonal honeycomb or 6,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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 order-5-3 hexagonal honeycomb is {6,5,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |  Ideal surface |
| Order-5-3 heptagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,5,3} |
| Coxeter diagram |  |
| Cells | {7,5}  |
| Faces | {7} |
| Vertex figure | {5,3} |
| Dual | {3,5,7} |
| Coxeter group | [7,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 heptagonal 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-5-3 heptagonal honeycomb is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |  Ideal surface |
| Order-5-3 octagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,5,3} |
| Coxeter diagram |  |
| Cells | {8,5}  |
| Faces | {8} |
| Vertex figure | {5,3} |
| Dual | {3,5,8} |
| Coxeter group | [8,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 octagonal 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-5-3 octagonal honeycomb is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
| Order-5-3 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,5,3} |
| Coxeter diagram |  |
| Cells | {∞,5}  |
| Faces | Apeirogon {∞} |
| Vertex figure | {5,3} |
| Dual | {3,5,∞} |
| Coxeter group | [∞,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 apeirogonal honeycomb or ∞,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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 {∞,5,3}, with three order-5 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
 Poincaré disk model (Vertex centered) |  Ideal surface |
The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.
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. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.
In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.
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-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-4 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,5,4}.
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-4-4 pentagonal 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}.