| Order-6-3 square honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {4,6,3} |
| Coxeter diagram |      |
| Cells | {4,6}  |
| Faces | {4} |
| Vertex figure | {6,3} |
| Dual | {3,6,4} |
| Coxeter group | [4,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). 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.
The Schläfli symbol of the order-6-3 square honeycomb is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
 Poincaré disk model |  Ideal surface |
It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:
| Order-6-3 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,6,3} |
| Coxeter diagram |  |
| Cells | {5,6}  |
| Faces | {5} |
| Vertex figure | {6,3} |
| Dual | {3,6,5} |
| Coxeter group | [5,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-6-3 pentagonal honeycomb is {5,6,3}, with three order-6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
 Poincaré disk model |  Ideal surface |
| Order-6-3 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {6,6,3} |
| Coxeter diagram | |
| Cells | {6,6}  |
| Faces | {6} |
| Vertex figure | {6,3} |
| Dual | {3,6,6} |
| Coxeter group | [6,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-6-3 hexagonal honeycomb is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
 Poincaré disk model |  Ideal surface |
| Order-6-3 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,6,3} |
| Coxeter diagram |  |
| Cells | {∞,6}  |
| Faces | Apeirogon {∞} |
| Vertex figure | {6,3} |
| Dual | {3,6,∞} |
| Coxeter group | [∞,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
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 |  Ideal surface |
In the field of hyperbolic geometry, the order-4 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 is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
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.
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 field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or a regular space-filling tessellation with Schläfli symbol {6,3,7}.
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-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-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-6-4 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}.