Snub tetraoctagonal tiling

Snub tetraoctagonal tiling
Snub tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.3.4.3.8
Schläfli symbol sr{8,4} or
Wythoff symbol | 8 4 2
Coxeter diagram
Symmetry group [8,4]+, (842)
Dual Order-8-4 floret pentagonal tiling
Properties Vertex-transitive Chiral

In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.

Images

Drawn in chiral pairs, with edges missing between black triangles:

The snub tetraoctagonal tiling is seventh in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2 		Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)

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{8,4} t{8,4}
 r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444) 		[8+,4]
(8*2) 		[8,1+,4]
(*4222) 		[8,4+]
(4*4) 		[8,4,1+]
(*882) 		[(8,4,2+)]
(2*42) 		[8,4]+
(842)

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h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8

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.

See also

Wikimedia Commons has media related to Uniform tiling 3-3-4-3-8.