Snub trihexagonal tiling
Type	Semiregular tiling
Vertex configuration	3.3.3.3.6
Schläfli symbol	sr{6,3} or${\displaystyle s\left\{\begin{array}{c}6\\ 3\end{array}\right\}}$
Wythoff symbol	| 6 3 2
Coxeter diagram
Symmetry	p6, [6,3]+, (632)
Rotation symmetry	p6, [6,3]+, (632)
Bowers acronym	Snathat
Dual	Floret pentagonal tiling
Properties	Vertex-transitivechiral

Ingeometry, thesnub hexagonal tiling (orsnub trihexagonal tiling) is asemiregular tiling of the Euclidean plane. There are four triangles and one hexagon on eachvertex. It hasSchläfli symbolsr{3,6}. Thesnub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbolsr{4,6}.

Conway calls it asnub hextille, constructed as asnub operation applied to ahexagonal tiling (hextille).

There are threeregular and eightsemiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.

There is only oneuniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)

The snub trihexagonal tiling can be used as acircle 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).^[1] The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from thetriangular tiling.

Uniform hexagonal/triangular tilings
Fundamental domains	Symmetry: [6,3], (*632)							[6,3]+, (632)
	{6,3}	t{6,3}	r{6,3}	t{3,6}	{3,6}	rr{6,3}	tr{6,3}	sr{6,3}
Config.	63	3.12.12	(6.3)2	6.6.6	36	3.4.6.4	4.6.12	3.3.3.3.6

This semiregular tiling is a member of a sequence ofsnubbed polyhedra and tilings with vertex figure (3.3.3.3.n) andCoxeter–Dynkin diagram. These figures and their duals have (n32) rotationalsymmetry, 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 intodigons.

n32 symmetry mutations of snub tilings: 3.3.3.3.n v t e
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
	232	332	432	532	632	732	832	∞32
Snub figures
Config.	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.∞
Gyro figures
Config.	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.∞

Floret pentagonal tiling
Type	Dual semiregular tiling
Coxeter diagram
Wallpaper group	p6, [6,3]+, (632)
Rotation group	p6, [6,3]+, (632)
Dual	Snub trihexagonal tiling
Face configuration	V3.3.3.3.6 Face figure:
Properties	face-transitive,chiral

Ingeometry, the6-fold pentille orfloret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.^[2] It is one of the 15 knownisohedralpentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on aflower.^[3] Each of its pentagonalfaces has four 120° and one 60° angle.

It is the dual of the uniform snub trihexagonal tiling,^[4] and hasrotational symmetries of orders 6-3-2 symmetry.

The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedralpentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes adeltoidal trihexagonal tiling.

General	Zero length degenerate	Special cases
(See animation)	Deltoidal trihexagonal tiling
a=b, d=e A=60°, D=120°	a=b, d=e,c=0 A=60°, 90°, 90°, D=120°	a=b=2c=2d=2e A=60°, B=C=D=E=120°	a=b=d=e A=60°, D=120°, E=150°	2a=2b=c=2d=2e 0°, A=60°, D=120°	a=b=c=d=e 0°, A=60°, D=120°

There are manyk-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labelingF for V3⁴.6,C forV3².4.3.4,B forV3³.4²,H for V3⁶:

uniform (snub trihexagonal)	2-uniform		3-uniform
F, p6 (t=3, e=3)	FH, p6 (t=5, e=7)	FH, p6m (t=3, e=3)	FCB, p6m (t=5, e=6)	FH2, p6m (t=3, e=4)	FH2, p6m (t=5, e=5)
dual uniform (floret pentagonal)	dual 2-uniform		dual 3-uniform
3-uniform		4-uniform
FH2, p6 (t=7, e=9)	F2H, cmm (t=4, e=6)	F2H2, p6 (t=6, e=9)	F3H, p2 (t=7, e=12)	FH3, p6 (t=7, e=10)	FH3, p6m (t=7, e=8)
dual 3-uniform		dual 4-uniform

Replacing every V3⁶ hexagon by arhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.

Replacing every V3⁶ hexagon by atruncated hexagon furnishes a 8-uniform tiling, five vertices of 3².12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.

Replacing every V3⁶ hexagon by atruncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4².6, and one vertex of 3.4.6.4.

In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since there is no chiral symmetry (the domains have 3:2 side lengths of${\displaystyle 1+\frac{1}{\sqrt{3}}:2+\frac{2}{\sqrt{3}}}$ in the rhombitrihexagonal;${\displaystyle 1+\frac{2}{\sqrt{3}}:2+\frac{4}{\sqrt{3}}}$ in the truncated hexagonal; and${\displaystyle 1+\sqrt{3}:2+2\sqrt{3}}$ in the truncated trihexagonal).

Fractalizing the Snub Trihexagonal Tiling using theRhombitrihexagonal,Truncated Hexagonal andTruncated Trihexagonal Tilings

Rhombitrihexagonal	Truncated Hexagonal	Truncated Trihexagonal

Dual uniform hexagonal/triangular tilings

Symmetry: [6,3], (*632)						[6,3]+, (632)
V63	V3.122	V(3.6)2	V36	V3.4.6.4	V.4.6.12	V34.6

Wikimedia Commons has media related toUniform tiling 3-3-3-3-6 (snub trihexagonal tiling).

• Tilings of regular polygons
• List of uniform tilings

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,The Symmetries of Things 2008,ISBN 978-1-56881-220-5[1]
• Grünbaum, Branko; 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
• Keith Critchlow,Order in Space: A design source book, 1970, p. 69-61, Pattern R, Dual p. 77-76, pattern 5
• Dale Seymour andJill Britton,Introduction to Tessellations, 1989,ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114

• Weisstein, Eric W."Uniform tessellation".MathWorld.
• Weisstein, Eric W."Semiregular tessellation".MathWorld.
• Klitzing, Richard."2D Euclidean tilings s3s6s - snathat - O11".

• Chiral figures
• Euclidean tilings
• Isogonal tilings
• Semiregular tilings
• Snub tilings

• Commons category link is on Wikidata