Order-8 octagonal tiling

Order-8 octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure88
Schläfli symbol{8,8}
Wythoff symbol8 | 8 2
Coxeter diagram
Symmetry group[8,8], (*882)
Dualself dual
PropertiesVertex-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} and is self-dual.

Symmetry

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.

Related polyhedra and tiling

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.

{8,2}

{8,3}

{8,4}

{8,5}

{8,6}

{8,7}

{8,8}
...
{8,}
Spherical Hyperbolic tilings

{2,8}

{3,8}

{4,8}

{5,8}

{6,8}

{7,8}

{8,8}
...
{,8}
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
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{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

See also

Wikimedia Commons has media related to Order-8 octagonal tiling.

References

External links