Heptagonal tiling honeycomb
| Heptagonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,3} |
| Coxeter diagram |    
|
| Cells | {7,3} 
|
| Faces | Heptagon {7} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,7} |
| 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 {7,3,3}, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, {3,3}.
 Poincaré disk model (vertex centered) |
 Rotating |
 Ideal surface |
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures: Template:Tetrahedral vertex figure tessellations
It is a part of a series of regular honeycombs, {7,3,p}.
It is a part of a series of regular honeycombs, with {7,p,3}.
| {7,3,3} | {7,4,3} | {7,5,3}... |
|---|---|---|
| Error creating thumbnail: | File:Hyperbolic honeycomb 7-4-3 poincare vc.png | File:Hyperbolic honeycomb 7-5-3 poincare vc.png |
Octagonal tiling honeycomb
| Octagonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,3} t{8,4,3} 2t{4,8,4} t{4[3,3]} |
| Coxeter diagram | File:CDel 8.pngError creating thumbnail: Error creating thumbnail: File:CDel 4.pngFile:CDel 8.pngError creating thumbnail: File:CDel 8.pngFile:CDel 4.pngFile:CDel 8.png File:CDel branch 11.pngFile:CDel split2-44.png File:CDel 8.pngFile:CDel label4.pngFile:CDel branch 11.pngFile:CDel splitcross.pngFile:CDel branch 11.pngFile:CDel label4.png (all 4s) |
| Cells | {8,3} File:H2-8-3-dual.svg |
| Faces | Octagon {8} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,8} |
| 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 {8,3,3}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
| File:Hyperbolic honeycomb 8-3-3 poincare vc.png Poincaré disk model (vertex centered) |
File:Hyperbolic subgroup tree 338-direct.png Direct subgroups of [8,3,3] |
Apeirogonal tiling honeycomb
| Apeirogonal tiling honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,3} t{∞,3,3} 2t{∞,∞,∞} t{∞[3,3]} |
| Coxeter diagram | File:CDel infin.pngError creating thumbnail: Error creating thumbnail: File:CDel infin.pngFile:CDel infin.pngError creating thumbnail: File:CDel infin.pngFile:CDel infin.pngFile:CDel infin.png File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2-ii.png File:CDel infin.pngFile:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel splitcross.pngFile:CDel branch 11.pngFile:CDel labelinfin.png (all ∞) |
| Cells | {∞,3} File:H2-I-3-dual.svg |
| Faces | Apeirogon {∞} |
| Vertex figure | tetrahedron {3,3} |
| Dual | {3,3,∞} |
| 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 {∞,3,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.
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.
| File:Hyperbolic honeycomb i-3-3 poincare vc.png Poincaré disk model (vertex centered) |
File:H3 i33 UHS plane at infinity.png Ideal surface |
See also
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8 Search this book on
.. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) - The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Search this book on . (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 Search this book on . (Chapters 16–17: Geometries on Three-manifolds I, II)
- George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
- Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
- John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
- Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]
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