Snub trihexagonal tiling

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 trihexagonal tiling. (Naming the colors by indices (3.3.3.3.6): 11213.)

Circle packing

The snub trihexagonal 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.

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 Compact hyperbolic Paracompact
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

Floret pentagonal tiling

Floret pentagonal tiling
TypeDual semiregular tiling
Coxeter diagram
Facesirregular pentagons
Face configurationV3.3.3.3.6
Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
DualSnub trihexagonal tiling
Propertiesface-transitive, chiral

In geometry, the floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 14 known isohedral pentagon tilings. It is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower.[1] Conway calls it a 6-fold pentille.[2] Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform tiling, snub trihexagonal tiling,[3] and has rotational symmetry of orders 6-3-2 symmetry.

See also

Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6.

References

  1. ↑ Five space-filling polyhedra by Guy Inchbald
  2. ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  3. ↑ Weisstein, Eric W., "Dual tessellation", MathWorld.

External links