Heptagonal tiling honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Heptagonal tiling honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol | |
| Coxeter diagram | |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [7,3,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
heptagonal tiling honeycomb or
7,3,3 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 heptagonal tiling honeycomb is, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, .
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with Schläfli symbol, and tetrahedral vertex figures:
It is a part of a series of regular honeycombs, .
It is a part of a series of regular honeycombs, with .
Octagonal tiling honeycomb
| bgcolor=#e7dcc3 colspan=2 | Octagonal tiling honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol |
t
2t
t |
| Coxeter diagram |
(all 4s) |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [8,3,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
octagonal tiling honeycomb or
8,3,3 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 octagonal tiling honeycomb is, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, .
Apeirogonal tiling honeycomb
| bgcolor=#e7dcc3 colspan=2 | Apeirogonal tiling honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbol |
t
2t
t |
| Coxeter diagram |
(all ∞) |
| Cells | |
| Faces | |
| Vertex figure | |
| Dual | |
| Coxeter group | [∞,3,3] |
| Properties | Regular | |
In the
geometry of
hyperbolic 3-space, the
apeirogonal tiling honeycomb or
∞,3,3 honeycomb a regular space-filling
tessellation (or
honeycomb). Each infinite cell consists of an
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 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, .
The "ideal surface" projection below is a plane-at-infinity, in the Poincare 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
