Order-6 apeirogonal tiling
| Order-6 apeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞6 |
| Schläfli symbol | {∞,6} |
| Wythoff symbol | 6 | ∞ 2 |
| Coxeter diagram |    
|
| Symmetry group | [∞,6], (*∞62) |
| Dual | Infinite-order hexagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.
Symmetry
[edit]The dual to this tiling represents the fundamental domains of [∞,6*] symmetry, orbifold notation *∞∞∞∞∞∞ symmetry, a hexagonal domain with five ideal vertices.
The order-6 apeirogonal tiling can be uniformly colored with 6 colored apeirogons around each vertex, and coxeter diagram: 
, except ultraparallel branches on the diagonals.
Related polyhedra and tiling
[edit]This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with six faces per vertex, starting with the triangular tiling, with Schläfli symbol {n,6}, and Coxeter diagram 
, with n progressing to infinity.
| Regular tilings {n,6} | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
 {2,6} 
|
 {3,6} 
|
 {4,6} 
|
 {5,6} 
|
 {6,6}
|
 {7,6} 
|
 {8,6} 
|
... | {∞,6}
|
See also
[edit]
Wikimedia Commons has media related to Order-6 apeirogonal tiling.
References
[edit]- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
[edit]- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
