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 HexagonalTruncated Trihexagonal
Dual FractalizationDual Fractalization

Related tilings