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
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
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
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Order-6 apeirogonal tiling.
References
- 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
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