Order-8 square tiling

Order-8 square tiling
Order-8 square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 48
Schläfli symbol {4,8}
Wythoff symbol 8 | 4 2
Coxeter diagram
Symmetry group [8,4], (*842)
Dual Order-4 octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as [1+,8,8,1+], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.

This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4[3]}, :

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact

{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}...
{4,∞}
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
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444) 		[(1+,4,4,4)]
(*4242) 		[(4+,4,4)]
(4*22)










t0(4,4,4)
h{8,4} 		t0,1(4,4,4)
h2{8,4} 		t1(4,4,4)
{4,8}1/2 		t1,2(4,4,4)
h2{8,4} 		t2(4,4,4)
h{8,4} 		t0,2(4,4,4)
r{4,8}1/2 		t0,1,2(4,4,4)
t{4,8}1/2 		s(4,4,4)
s{4,8}1/2 		h(4,4,4)
h{4,8}1/2 		hr(4,4,4)
hr{4,8}1/2
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

See also

Wikimedia Commons has media related to Order-8 square 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.