Snub trihexagonal tiling

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

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 sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

Snub trihexagonal tiling

Type	Semiregular tiling
Vertex configuration	3.3.3.3.6
Schläfli symbol	sr{6,3} or $s{\begin{Bmatrix}6\\3\end{Bmatrix}}$
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-transitive chiral

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

There are three regular and eight 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. (Labeling the colors by numbers, "3.3.3.3.6" gives "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).[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 the triangular tiling.

Related polyhedra and tilings

There is one related 2-uniform tiling, which mixes the vertex configurations 3.3.3.3.6 of the snub trihexagonal tiling and 3.3.3.3.3.3 of the triangular 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

Symmetry mutations

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.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
Symmetry n32	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.∞

6-fold pentille tiling

In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.[2] It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower.[3] Each of its pentagonal faces has four 120° and one 60° angle.

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

Variations

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

Related k-uniform and dual k-uniform tilings

There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labeling F for V3⁴.6, C for V3².4.3.4, B for V3³.4², 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

Fractalization

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

Replacing every V3⁶ hexagon by a truncated 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 a truncated 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 $1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}$ in the rhombitrihexagonal; $1+{\frac {2}{\sqrt {3}}}:2+{\frac {4}{\sqrt {3}}}$ in the truncated hexagonal; and $1+{\sqrt {3}}:2+2{\sqrt {3}}$ in the truncated trihexagonal).

Fractalizing the Snub Trihexagonal Tiling using the Rhombitrihexagonal, Truncated Hexagonal and Truncated Trihexagonal Tilings
Rhombitrihexagonal	Truncated Hexagonal	Truncated Trihexagonal

Related tilings

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

References

Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things, 2008, ISBN 978-1-56881-220-5, "A K Peters, LTD. - The Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)
Five space-filling polyhedra by Guy Inchbald
Weisstein, Eric W. "Dual tessellation". MathWorld.

John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
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 and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114

External links

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

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[Critchlow-1] Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E

[2] John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things, 2008, ISBN 978-1-56881-220-5, "A K Peters, LTD. - The Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p. 288, table)

[3] Five space-filling polyhedra by Guy Inchbald

[4] Weisstein, Eric W. "Dual tessellation". MathWorld.