Order-8 octagonal tiling

Order-8 octagonal tiling
Order-8 octagonal tiling
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.

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.

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.

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8∞
Regular tilings: {n,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)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{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

  • 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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.