Snub trihexagonal tiling
In geometry, the snub hexagonal 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. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr.
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling.
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.
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. The lattice domain 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
Symmetry mutations
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure and Coxeter–Dynkin diagram. These figures and their duals have 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.Floret pentagonal tiling
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. Conway calls it a 6-fold pentille. 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.
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.:File:Snub_trihexagonal_tiling_variations.gif| | a=b, d=e A=60°, D=120° | Deltoidal trihexagonal tiling | a=b, d=e, c=0 60°, 90°, 90°, 120° |
Related dual k-uniform tilings
There are many duals to k-uniform tiling, which mixes the 6-fold florets with other tiles, for example:Fractalization
Replacing every hexagon by a truncated hexagon furnishes a uniform 8 tiling, 5 vertices of configuration 32.12, 2 vertices of configuration 3.4.3.12, and 1 vertex of configuration 3.4.6.4.Replacing every hexagon by a truncated trihexagon furnishes a uniform 15 tiling, 12 vertices of configuration 4.6.12 and 3 vertices of configuration 3.4.6.4.
In both tilings, every vertex is in a different orbit since there is no chiral symmetry; and the uniform count was from the Floret pentagon region of each fractal tiling.
| Truncated Hexagonal | Truncated Trihexagonal |
| Dual Fractalization | Dual Fractalization |