Snub trihexagonal tiling

Snub trihexagonal tiling
Type	Semiregular tiling
Vertex configuration	3.3.3.3.6
Schläfli symbol	sr{6,3}
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

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

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).^[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#circle packing.

Related polyhedra and tilings

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

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.	6^3	3.12.12	(6.3)^2	6.6.6	3^6	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.
	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.∞
Gryro 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

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

In geometry, the floret pentagonal tiling or rosette pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 15 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.^[2] Conway calls it a 6-fold pentille.^[3] Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform tiling, snub trihexagonal tiling,^[4] and has rotational symmetry 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.

(See animation)

a=b, d=e A=60°, D=120°

Deltoidal trihexagonal tiling

a=b, d=e, c=0 60°, 90°, 90°, 120°

Related tilings

Dual uniform hexagonal/triangular tilings

Symmetry: [6,3], (*632)						[6,3]^+, (632)
V6^3	V3.12^2	V(3.6)^2	V3^6	V3.4.12.4	V.4.6.12	V3^4.6

See also

• Tilings of regular polygons
• List of uniform tilings

References

1. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
2. Five space-filling polyhedra by Guy Inchbald
3. 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)
4. 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 ; 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
• 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.
• Richard Klitzing, 2D Euclidean tilings, s3s6s - snathat - O11