Snub trihexagonal tiling

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Snub trihexagonal tiling

TypeSemiregular tiling
Vertex configuration3.3.3.3.6
Schläfli symbolsr{6,3}
Wythoff symbol| 6 3 2
Coxeter diagram
Symmetryp6, [6,3]+, (632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymSnathat
DualFloret pentagonal tiling
PropertiesVertex-transitive chiral

Vertex figure: 3.3.3.3.6

In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol of sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

Conway calls it a snub hexatille, constructed as a snub operation applied to a hexagonal tiling (hexatille).

There are 3 regular and 8 semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.

There is only one uniform coloring of a snub hexagonal tiling. (Naming the colors by indices (3.3.3.3.6): 11213.)

Related polyhedra and tilings

Wythoff constructions from hexagonal and triangular tilings

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Related polyhedra and tilings

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+		 Spherical Euclidean Hyperbolic
232
[2,3]+
D3		 332
[3,3]+
T		 432
[4,3]+
O		 532
[5,3]+
I		 632
[6,3]+
P6		 732
[7,3]+		 832
[8,3]+		 ∞32
[∞,3]+
Snub
figure
3.3.3.3.2
3.3.3.3.3
3.3.3.3.4
3.3.3.3.5
3.3.3.3.6
3.3.3.3.7
3.3.3.3.8
3.3.3.3.∞
Coxeter
Schläfli
sr{2,3}
sr{3,3}
sr{4,3}
sr{5,3}
sr{6,3}
sr{7,3}
sr{8,3}
sr{∞,3}
Snub
dual
figure
V3.3.3.3.2
V3.3.3.3.3
V3.3.3.3.4
V3.3.3.3.5
V3.3.3.3.6
V3.3.3.3.7		 V3.3.3.3.8 V3.3.3.3.∞
Coxeter
Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)		 [1+,6,3]
(*333)		 [6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}		 t{3,6}
t{3[3]}		 {3,6}
{3[3]}		 rr{6,3}
s2{6,3}		 tr{6,3} sr{6,3} h{6,3}
{3[3]}		 h2{6,3}
r{3[3]}		 s{3,6}
s{3[3]}

=
=
= 		=
or 		=
or
=





Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36

Circle packing

The snub hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number). The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling#circle packing.

See also

Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6.
  • Tilings of regular polygons
  • List of uniform tilings

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
  • Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1.  (Chapter 2.1: Regular and uniform tilings, p.58-65)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p.39

External links

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