Order-3-4 heptagonal honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Order-3-4 heptagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram |
= |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [7,3,4] |
| Properties | Regular | |
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.
Geometry
The Schläfli symbol of the order-3-4 heptagonal honeycomb is, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, .
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and octahedral vertex figures:
Order-3-4 octagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-3-4 octagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram |
=
|
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [8,3,4]
[8,3<sup>1,1</sup>] |
| 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, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, .
Order-3-4 apeirogonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-3-4 apeirogonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram |
=
|
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [∞,3,4]
[∞,3<sup>1,1</sup>] |
| 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, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, .
See also
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapters 16–17: Geometries on Three-manifolds I, II)
- George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) http://www.sciencedirect.com/science/article/pii/0021869382903180
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)https://arxiv.org/abs/1310.8608
- Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
