In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). 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.
| Order-3-6 heptagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,6} {7,3[3]} |
| Coxeter diagram |       =   |
| Cells | {7,3}  |
| Faces | {7} |
| Vertex figure | {3,6} |
| Dual | {6,3,7} |
| Coxeter group | [7,3,6] [7,3[3]] |
| Properties | Regular |
Geometry
The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
 Poincaré disk model |
 Ideal surface |
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.
Order-3-6 octagonal honeycomb
| Order-3-6 octagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,6} {8,3[3]} |
| Coxeter diagram |  = |
| Cells | {8,3}  |
| Faces | Octagon {8} |
| Vertex figure | triangular tiling {3,6} |
| Dual | {6,3,8} |
| Coxeter group | [8,3,6] [8,3[3]] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
 Poincaré disk model |
Order-3-6 apeirogonal honeycomb
| Order-3-6 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,6} {∞,3[3]} |
| Coxeter diagram |  = |
| Cells | {∞,3}  |
| Faces | Apeirogon {∞} |
| Vertex figure | triangular tiling {3,6} |
| Dual | {6,3,∞} |
| Coxeter group | [∞,3,6] [∞,3[3]] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 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 order-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
 Poincaré disk model |
 Ideal surface |
It has a quasiregular construction, , which can be seen as alternately colored cells.
See also
References
External links
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