# Snub trihexagonal tiling

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}.

Snub trihexagonal tiling

TypeSemiregular tiling
Vertex configuration
3.3.3.3.6
Schläfli symbolsr{6,3} or ${\displaystyle s{\begin{Bmatrix}6\\3\end{Bmatrix}}}$
Wythoff symbol| 6 3 2
Coxeter diagram
Symmetryp6, [6,3]+, (632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymSnathat
DualFloret pentagonal tiling
PropertiesVertex-transitive chiral

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).[1] 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.

### 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.[2] Conway calls it a 6-fold pentille.[3] Each of its pentagonal faces has four 120° and one 60° angle.

It is the dual of the uniform tiling, snub trihexagonal tiling,[4] 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.

 (See animation) a=b, d=eA=60°, D=120° Deltoidal trihexagonal tiling a=b, d=e, c=060°, 90°, 90°, 120°

There are many duals to k-uniform tiling, which mixes the 6-fold florets with other tiles, for example:

2-uniform dual 3-uniform dual 4-uniform dual

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

Fractalizing the Snub Trihexagonal Tiling using the Truncated Hexagonal and Truncated Trihexagonal Tilings
Truncated Hexagonal Truncated Trihexagonal
Dual Fractalization Dual Fractalization
Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6