Order-7 hexagonal tiling honeycomb
| Order-7 hexagonal tiling honeycomb | |
|---|---|
 Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbol | {6,3,7} |
| Coxeter diagrams |      |
| Cells | {6,3}  |
| Faces | {6} |
| Edge figure | {7} |
| Vertex figure | {3,8} |
| Dual | {7,3,6} |
| Coxeter group | [6,3,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 hexagonal tiling 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.
Closeup
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.
Infinite-order hexagonal tiling honeycomb
| Infinite-order hexagonal tiling honeycomb | |
|---|---|
 Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model | |
| Type | Hyperbolic 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 infinite-order hexagonal tiling 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.
Symmetry constructions
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
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
- 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)