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 leads to a circle packing, each vertex becoming the center of a circle of fixed diameter. 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 CoxeterâÂÂDynkin 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 120ð and one 60ð 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 V3<sup>4</sup>.6, C for V3<sup>2</sup>.4.3.4, B for V3<sup>3</sup>.4<sup>2</sup>, H for V3<sup>6</sup>:
Replacing every V3<sup>6</sup> 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 V3<sup>6</sup> hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 3<sup>2</sup>.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V3<sup>6</sup> hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4<sup>2</sup>.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 in the rhombitrihexagonal; in the truncated hexagonal; and in the truncated trihexagonal).