Order-5-3 square honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 square honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [4,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 square honeycomb or
4,5,3 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.
Geometry
The Schläfli symbol of the order-5-3 square honeycomb is, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and dodecahedral vertex figures:
Order-5-3 pentagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 pentagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [5,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 pentagonal honeycomb or
5,5,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an
order-5 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-5-3 pentagonal honeycomb is, with three order-5 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
Order-5-3 hexagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 hexagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [6,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 hexagonal honeycomb or
6,5,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an
order-5 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-5-3 hexagonal honeycomb is, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
Order-5-3 heptagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 heptagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [7,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 heptagonal honeycomb or
7,5,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an order-5 heptagonal 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-5-3 heptagonal honeycomb is, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
Order-5-3 octagonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 octagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [8,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 octagonal honeycomb or
8,5,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an order-5 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-5-3 octagonal honeycomb is, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
Order-5-3 apeirogonal honeycomb
| bgcolor=#e7dcc3 colspan=2 | Order-5-3 apeirogonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [∞,5,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
order-5-3 apeirogonal honeycomb or
∞,5,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an
order-5 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, with three order-5 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, .
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.
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
