Order-3-4 heptagonal honeycomb
From Wikipedia, the free encyclopedia
In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 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-3-4 heptagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,4} |
| Coxeter diagram |       =   |
| Cells | {7,3}  |
| Faces | heptagon {7} |
| Vertex figure | octahedron {3,4} |
| Dual | {4,3,7} |
| Coxeter group | [7,3,4] |
| Properties | Regular |
Geometry
The Schläfli symbol of the order-3-4 heptagonal honeycomb is {7,3,4}, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
 Poincaré disk model (vertex centered) |
 One hyperideal cell limits to a circle on the ideal surface |
 Ideal surface |
Related polytopes and honeycombs
Summarize
Perspective
It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:
Order-3-4 octagonal honeycomb
| Order-3-4 octagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,4} |
| Coxeter diagram |  =       |
| Cells | {8,3}  |
| Faces | octagon {8} |
| Vertex figure | octahedron {3,4} |
| Dual | {4,3,8} |
| Coxeter group | [8,3,4] [8,31,1] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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-3-4 octagonal honeycomb is {8,3,4}, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
 Poincaré disk model (vertex centered) |
Order-3-4 apeirogonal honeycomb
| Order-3-4 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,4} |
| Coxeter diagram |  =   |
| Cells | {∞,3}  |
| Faces | apeirogon {∞} |
| Vertex figure | octahedron {3,4} |
| Dual | {4,3,∞} |
| Coxeter group | [∞,3,4] [∞,31,1] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-4 apeirogonal honeycomb or ∞,3,4 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-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.
 Poincaré disk model (vertex centered) |
 Ideal surface |
See also
References
External links
Wikiwand - on
Seamless Wikipedia browsing. On steroids.