Order-3-6 heptagonal honeycomb
In the geometry of hyperbolic 3-space, the order-3-6 heptagonal 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-3-6 heptagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {7,3,6} {7,3[3]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {7,3} ![]() |
Faces | {7} |
Vertex figure | {3,6} |
Dual | {6,3,7} |
Coxeter group | [7,3,6] [7,3[3]] |
Properties | Regular |
Geometry
The Schläfli symbol of the order-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an triangular tiling, {3,6}.
It has a quasiregular construction,
![]() Poincaré disk model |
![]() Ideal surface |
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.
Form | Paracompact | Noncompact | |||||
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Name | {3,3,6} {3,3[3]} |
{4,3,6} {4,3[3]} |
{5,3,6} {5,3[3]} |
{6,3,6} {6,3[3]} |
{7,3,6} {7,3[3]} |
{8,3,6} {8,3[3]} |
... {∞,3,6} {∞,3[3]} |
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Image | ![]() |
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Cells | ![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() |
Order-3-6 octagonal honeycomb
Order-3-6 octagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {8,3,6} {8,3[3]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {8,3} ![]() |
Faces | Octagon {8} |
Vertex figure | triangular tiling {3,6} |
Dual | {6,3,8} |
Coxeter group | [8,3,6] [8,3[3]] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction,
![]() Poincaré disk model |
Order-3-6 apeirogonal honeycomb
Order-3-6 apeirogonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {∞,3,6} {∞,3[3]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {∞,3} ![]() |
Faces | Apeirogon {∞} |
Vertex figure | triangular tiling {3,6} |
Dual | {6,3,∞} |
Coxeter group | [∞,3,6] [∞,3[3]] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal 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-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
![]() Poincaré disk model |
![]() Ideal surface |
It has a quasiregular construction,
See also
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (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 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
- George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
- 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.