Order-6 hexagonal tiling
| Order-6 hexagonal tiling | |
|---|---|
| Order-6 hexagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 66 |
| Schläfli symbol | {6,6} |
| Wythoff symbol | 6 | 6 2 |
| Coxeter diagram | Template:CDD |
| Symmetry group | [6,6], (*662) |
| Dual | self dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the Template:CDD tiling:
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram Template:CDD, progressing to infinity.
| Regular tilings {n,6} Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found. | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
| File:Spherical hexagonal hosohedron.svg {2,6} Template:CDD |
File:Uniform tiling 63-t2.svg {3,6} Template:CDD |
File:H2 tiling 246-4.png {4,6} Template:CDD |
File:H2 tiling 256-4.png {5,6} Template:CDD |
File:H2 tiling 266-4.png {6,6} Template:CDD |
File:H2 tiling 267-1.png {7,6} Template:CDD |
File:H2 tiling 268-1.png {8,6} Template:CDD |
... | File:H2 tiling 26i-1.png {∞,6} Template:CDD |
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram Template:CDD, progressing to infinity.
| *n62 symmetry mutation of regular tilings: {6,n} Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found. | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
| File:Hexagonal dihedron.svg {6,2} |
File:Uniform tiling 63-t0.svg {6,3} |
File:H2 tiling 246-1.png {6,4} |
File:H2 tiling 256-1.png {6,5} |
File:H2 tiling 266-4.png {6,6} |
File:H2 tiling 267-4.png {6,7} |
File:H2 tiling 268-4.png {6,8} |
... | File:H2 tiling 26i-4.png {6,∞} |
| Similar H2 tilings in *3232 symmetry Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found. | ||||||||
|---|---|---|---|---|---|---|---|---|
| Coxeter diagrams |
Template:CDD | Template:CDD | Template:CDD | Template:CDD | ||||
| Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD | |
| Template:CDD | Template:CDD | Template:CDD | Template:CDD | |||||
| Vertex figure |
66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
| Image | File:Uniform tiling verf 666666.png | File:Uniform tiling verf 34343434.png | File:Uniform tiling verf 34664.png | File:3222-uniform tiling-verf4646.png | ||||
| Dual | File:Uniform tiling verf 666666b.png | File:H2chess 246a.png | ||||||
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Page Module:Citation/CS1/styles.css has no content."Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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
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