Order-3-7 hexagonal honeycomb
Order-3-7 hexagonal honeycomb | |
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Poincaré disk model | |
Type | Regular honeycomb |
Schläfli symbol | {6,3,7} |
Coxeter diagrams |
|
Cells | {6,3} |
Faces | {6} |
Edge figure | {7} |
Vertex figure | {3,7} |
Dual | {7,3,6}
|
Coxeter group |
[6,3,7] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.
Poincaré half-space model
|
Closeup |
Related polytopes and honeycombs
It a part of a sequence of
{6,3,p} honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Paracompact
|
Noncompact | |||||||||
Name | {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {6,3,7}
|
{6,3,8}
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... {6,3,∞}
| ||||
Coxeter |
|||||||||||
Image | |||||||||||
Vertex figure {3,p} |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
Order-3-8 hexagonal honeycomb
Order-3-8 hexagonal honeycomb | |
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Type | Regular honeycomb |
Schläfli symbols | {6,3,8} {6,(3,4,3)} |
Coxeter diagrams |
= |
Cells | {6,3} |
Faces | {6} |
Edge figure | {8} |
Vertex figure | {3,8} {(3,4,3)} |
Dual | {8,3,6}
|
Coxeter group |
[6,3,8] [6,((3,4,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.
Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].
Order-3-infinite hexagonal honeycomb
Order-3-infinite hexagonal honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbols | {6,3,∞} {6,(3,∞,3)} |
Coxeter diagrams |
↔ ↔ |
Cells | {6,3} |
Faces | {6} |
Edge figure | {∞} |
Vertex figure | {3,∞}, {(3,∞,3)} |
Dual | {∞,3,6}
|
Coxeter group |
[6,3,∞] [6,((3,∞,3))] |
Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Poincaré disk model |
Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of hexagonal tiling cells.
See also
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Infinite-order dodecahedral honeycomb
References
- 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, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- 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) [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]