Order-8 octagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 88 |
Schläfli symbol | {8,8} |
Wythoff symbol | 8 | 8 2 |
Coxeter diagram | |
Symmetry group | [8,8], (*882) |
Dual | self dual |
Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} (eight octagons around each vertex) and is self-dual.
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *44444444 with 8 order-4 mirror intersections. In Coxeter notation can be represented as [8,8*], removing two of three mirrors (passing through the octagon center) in the [8,8] symmetry.
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
Space | Spherical | Compact hyperbolic | Paracompact | |||||
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Tiling | ||||||||
Config. | 8.8 | 83 | 84 | 85 | 86 | 87 | 88 ...8∞ |
Regular tilings: {n,8} | |||||||||||
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Spherical | Hyperbolic tilings | ||||||||||
{2,8} |
{3,8} |
{4,8} |
{5,8} |
{6,8} |
{7,8} |
{8,8} ... {∞,8} |
Uniform octaoctagonal tilings | |||||||||||
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Symmetry: [8,8], (*882) | |||||||||||
= = |
= = |
= = |
= = |
= = |
= = |
= = | |||||
{8,8} | t{8,8} |
r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
Uniform duals | |||||||||||
V88 | V8.16.16 | V8.8.8.8 | V8.16.16 | V88 V4.8.4.8 V4.16.16 Alternations [1+,8,8](*884) [8+,8](8*4) [8,1+,8](*4242) [8,8+](8*4) [8,8,1+](*884) [(8,8,2+)](2*44) [8,8]+(882) = = = = = = = | |||||||
h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
Alternation duals | |||||||||||
V(4.8)8 | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)8 | V46 | V3.3.8.3.8 |