Order-4-4 pentagonal honeycomb
From Wikipedia, the free encyclopedia
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.
| 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 |
Geometry
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 |
Related polytopes and honeycombs
Summarize
Perspective
It is a part of a series of regular polytopes and honeycombs with {p,4,4} Schläfli symbol, and square tiling vertex figures:
Order-4-4 hexagonal honeycomb
| 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
| 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 |
See also
References
External links
Wikiwand - on
Seamless Wikipedia browsing. On steroids.