Octagonal tiling

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For other uses, see truncated square tiling.
Octagonal tiling
Octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 83
Schläfli symbol {8,3}
t{4,8}
Wythoff symbol 3 | 8 2
2 8 | 4
4 4 4 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.png
Symmetry group [8,3], (*832)
[8,4], (*842)
[(4,4,4)], (*444)
Dual Order-8 triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex.

Uniform colorings[edit]

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

Regular Truncations
Uniform tiling 83-t0.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t12.png
t{4,8}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
Uniform tiling 444-t012.png
t{4[3]}
CDel node 1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.png
Dual tiling
Uniform tiling 83-t2.png
{3,8}
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 433-t2.png
CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png = CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2checkers 444.png
CDel node f1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node h0.png = CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png

Related polyhedra and tilings[edit]

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling H2 tiling 238-1.png H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png H2 tiling 288-4.png H2 tiling 28i-4.png
Config. 8.8 83 84 85 86 87 88 ...8

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

See also[edit]

References[edit]

External links[edit]