Order-4 pentagonal tiling
| Order-4 pentagonal tiling | |
|---|---|
| Order-4 pentagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 54 |
| Schläfli symbol | {5,4} r{5,5} or |
| Wythoff symbol | 5 2 2 | 5 5 |
| Coxeter diagram | Template:CDD Template:CDD or Template:CDD |
| Symmetry group | [5,4], (*542) [5,5], (*552) |
| Dual | Order-5 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as [5*,4], removing two of three mirrors (passing through the pentagon center) in the [5,4] symmetry.
The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{5,5} and as a quasiregular tiling is called a pentapentagonal tiling.
Related polyhedra and tiling
Template:Order 5-4 tiling table Template:Order 5-5 tiling table
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol {5,n}, and Coxeter diagram Template:CDD, progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram Template:CDD, with n progressing to infinity. Template:Order-4 regular tilings
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). Template:Regular square tiling table
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Script error: No such module "citation/CS1"., invited lecture, ICM, Amsterdam, 1954.
See also
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External links
- Script error: No such module "Template wrapper".
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- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch