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Snub hexahexagonal tiling

In today's world, Snub hexahexagonal tiling has become a topic of great relevance and interest to a wide spectrum of society. From its impact on the global economy to its influence on people's daily lives, Snub hexahexagonal tiling has sparked debates and discussions in different areas. In order to understand this phenomenon more deeply, it is essential to analyze its different dimensions and repercussions. In this article, we will explore the many facets of Snub hexahexagonal tiling and its relevance today, as well as the future perspectives it raises.

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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 +, (662)
, (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 symmetry as s{6,4}, . In this construction there is only one color of hexagon.

Uniform hexahexagonal tilings
Symmetry: , (*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

(*663)
(6*3)
(*3232)
(6*3)
(*663)
(2*33) 		+
(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: , (*642)
(with (*662), (*443) , (*3222) index 2 subsymmetries)
(And (*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

(*443)
(6*2)
(*3222)
(4*3)
(*662)
(2*32) 		+
(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.
Wikious
Boobota
Sagapedia