Snub trihexagonal tiling

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. : == Related polyhedra and tilings ==

, 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. 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. == 6-fold pentille tiling ==

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. : 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, labeling 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). Related tilings ==See also==