Snub trihexagonal tiling

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}.

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".)

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 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.

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.

In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. 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. Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform snub trihexagonal tiling, and has rotational 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 monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.

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:

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 ${\displaystyle 1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}}$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).

• 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 and Jill 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".