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).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E 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.
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Related polyhedra and tilings
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.
<span class="anchor" id="Floret pentagonal tiling"></span> 6-fold pentille tiling
In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008,   (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) It is one of 15 known isohedral pentagon tilings.
Its six pentagonal tiles radiate out from a central point, like petals on a flower.Five space-filling polyhedra by Guy Inchbald Each of its pentagonal faces has four 120° and one 60° angle.
It is the dual of the uniform tiling, snub trihexagonal tiling, and has rotational symmetry of orders 6-3-2 symmetry.
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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.
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 (F for V34.6, C for V32.4.3.4, B for V33.42, H for V36):
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 tiling|Rhombitrihexagonal]], [[Truncated hexagonal tiling|Truncated Hexagonal]] and [[Truncated trihexagonal tiling|Truncated Trihexagonal]] Tilings
Related tilings
See also
Tilings of regular polygons
List of uniform tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008,
(Chapter 2.1: Regular and uniform tilings, p. 58-65)
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, , pp.
50–56, dual rosette tiling p. 96, p. 114
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