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 | 4 | 5 2 2 | 5 5 |
| Coxeter diagram | File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node.pngFile:CDel 5.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1-55.pngFile:CDel nodes.png |
| 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
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 File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel n.pngFile:CDel node.png, 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 File:CDel node 1.pngFile:CDel n.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, with n progressing to infinity.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | |||||
| File:Spherical square hosohedron.svg | File:Spherical square bipyramid.svg | File:Uniform tiling 44-t0.svg | File:H2-5-4-dual.svg | File:H2 tiling 246-1.png | File:H2 tiling 247-1.png | File:H2 tiling 248-1.png | File:H2 tiling 24i-1.png |
| 24 | 34 | 44 | 54 | 64 | 74 | 84 | ...∞4 |
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
| *5n2 symmetry mutations of quasiregular tilings: (5.n)2 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *5n2 [n,5] |
Spherical | Hyperbolic | Paracompact | Noncompact | ||||
| *352 [3,5] |
*452 [4,5] |
*552 [5,5] |
*652 [6,5] |
*752 [7,5] |
*852 [8,5]... |
*∞52 [∞,5] |
[ni,5] | |
| Figures | File:Uniform tiling 532-t1.png | File:H2-5-4-rectified.svg | File:H2 tiling 255-2.png | File:H2 tiling 256-2.png | File:H2 tiling 257-2.png | File:H2 tiling 258-2.png | File:H2 tiling 25i-2.png | |
| Config. | (5.3)2 | (5.4)2 | (5.5)2 | (5.6)2 | (5.7)2 | (5.8)2 | (5.∞)2 | (5.ni)2 |
| Rhombic figures |
File:Rhombictriacontahedron.svg | File:H2-5-4-rhombic.svg | File:H2-5-4-primal.svg | File:Order-6-5 quasiregular rhombic tiling.png | ||||
| Config. | V(5.3)2 | V(5.4)2 | V(5.5)2 | V(5.6)2 | V(5.7)2 | V(5.8)2 | V(5.∞)2 | V(5.∞)2 |
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)
- Coxeter, H. S. M. (1999), Chapter 10: Regular honeycombs in hyperbolic space (PDF), The Beauty of Geometry: Twelve Essays, Dover Publications, ISBN 0-486-40919-8, LCCN 99035678, invited lecture, ICM, Amsterdam, 1954.
