# Snub trihexagonal tiling

Snub trihexagonal tiling Type Semiregular tiling
Vertex configuration 3.3.3.3.6
Schläfli symbol sr{6,3} or $s{\begin{Bmatrix}6\\3\end{Bmatrix}}$ Wythoff symbol | 6 3 2
Coxeter diagram     Symmetry p6, [6,3]+, (632)
Rotation symmetry p6, [6,3]+, (632)
Bowers acronym Snathat
Dual Floret pentagonal tiling
Properties Vertex-transitive chiral

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). 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

### 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.

### Floret pentagonal tiling

Floret pentagonal tiling TypeDual semiregular tiling
Facesirregular pentagons
Coxeter diagram     Symmetry groupp6, [6,3]+, (632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronSnub trihexagonal tiling
Face configurationV3.3.3.3.6
Propertiesface-transitive, chiral

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.

#### 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 (3 side lengths of $1+{\frac {2}{\sqrt {3}}}$ and 2 side lengths of $2+{\frac {4}{\sqrt {3}}}$ in the truncated hexagonal; and 3 side lengths of $1+{\sqrt {3}}$ and 2 side lengths of $2+2{\sqrt {3}}$ in the truncated trihexagonal).