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Snub hexahexagonal tiling

Snub hexahexagonal tiling
Snub hexahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.3.6.3.6
Schläfli symbol s{6,4}
sr{6,6}
Wythoff symbol | 6 6 2
Coxeter diagram
Symmetry group [6,6]+, (662)
[6+,4], (6*2)
Dual Order-6-6 floret hexagonal tiling
Properties Vertex-transitive

In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}.

Images

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

Symmetry

A higher symmetry coloring can be constructed from [6,4] symmetry as s{6,4}, . In this construction there is only one color of hexagon.

Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{6,6}
= h{4,6} 		t{6,6}
= h2{4,6} 		r{6,6}
{6,4} 		t{6,6}
= h2{4,6} 		{6,6}
= h{4,6} 		rr{6,6}
r{6,4} 		tr{6,6}
t{6,4}
Uniform duals
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663) 		[6+,6]
(6*3) 		[6,1+,6]
(*3232) 		[6,6+]
(6*3) 		[6,6,1+]
(*663) 		[(6,6,2+)]
(2*33) 		[6,6]+
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)

=

=
=
=
=
=

=

=

=
=
=


=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443) 		[6+,4]
(6*2) 		[6,1+,4]
(*3222) 		[6,4+]
(4*3) 		[6,4,1+]
(*662) 		[(6,4,2+)]
(2*32) 		[6,4]+
(642)

=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2 		Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞

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 3-3-6-3-6.
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