# 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 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 three regular and eight 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. (Labeling the colors by numbers, "3.3.3.3.6" gives "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

Uniform hexagonal/triangular tilings :
| Fundamental domains | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | [6,3]$^+$, (632) |
| Fundamental domains | {6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} |
| Config. | 6$^3$ | 3.12.12 | (6.3)$^2$ | 6.6.6 | 3$^6$ | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |

### 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.
n32 symmetry mutations of snub tilings: 3.3.3.3.n vte :
| Symmetry n32 | Spherical | Spherical | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry n32 | 232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 |
| Snub figures | | | | | | | | |
| Config. | 3.3.3.3.2 | 3.3.3.3.3 | 3.3.3.3.4 | 3.3.3.3.5 | 3.3.3.3.6 | 3.3.3.3.7 | 3.3.3.3.8 | 3.3.3.3.∞ |
| Gyro figures | | | | | | | | |
| Config. | V3.3.3.3.2 | V3.3.3.3.3 | V3.3.3.3.4 | V3.3.3.3.5 | V3.3.3.3.6 | V3.3.3.3.7 | V3.3.3.3.8 | V3.3.3.3.∞ |

## 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.
| General | Zero length degenerate | Special cases | Special cases | Special cases | Special cases |
| (See animation) | Deltoidal trihexagonal tiling | | | | |
| a=b, d=e A=60°, D=120° | a=b, d=e, c=0 A=60°, 90°, 90°, D=120° | a=b=2c=2d=2e A=60°, B=C=D=E=120° | a=b=d=e A=60°, D=120°, E=150° | 2a=2b=c=2d=2e 0°, A=60°, D=120° | a=b=c=d=e 0°, A=60°, D=120° |

### 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 V3$^4$.6, C for V3$^2$.4.3.4, B for V3$^3$.4$^2$, H for V3$^6$:
| uniform (snub trihexagonal) | 2-uniform | 2-uniform | 3-uniform | 3-uniform | 3-uniform |
| F, p6 (t=3, e=3) | FH, p6 (t=5, e=7) | FH, p6m (t=3, e=3) | FCB, p6m (t=5, e=6) | FH$^2$, p6m (t=3, e=4) | FH$^2$, p6m (t=5, e=5) |
| dual uniform (floret pentagonal) | dual 2-uniform | dual 2-uniform | dual 3-uniform | dual 3-uniform | dual 3-uniform |
| 3-uniform | 3-uniform | 4-uniform | 4-uniform | 4-uniform | 4-uniform |
| FH$^2$, p6 (t=7, e=9) | F$^2$H, cmm (t=4, e=6) | F$^2$H$^2$, p6 (t=6, e=9) | F$^3$H, p2 (t=7, e=12) | FH$^3$, p6 (t=7, e=10) | FH$^3$, p6m (t=7, e=8) |
| dual 3-uniform | dual 3-uniform | dual 4-uniform | dual 4-uniform | dual 4-uniform | dual 4-uniform |

### Fractalization

Replacing every V3$^6$ 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 V3$^6$ hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 3$^2$.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V3$^6$ hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.4$^2$.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).
Fractalizing the Snub Trihexagonal Tiling using the Rhombitrihexagonal, Truncated Hexagonal and Truncated Trihexagonal Tilings :
| Rhombitrihexagonal | Truncated Hexagonal | Truncated Trihexagonal |

#### Related tilings

Dual uniform hexagonal/triangular tilings :
| Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | Symmetry: [6,3], (*632) | [6,3]$^+$, (632) |
| V6$^3$ | V3.12$^2$ | V(3.6)$^2$ | V3$^6$ | V3.4.6.4 | V.4.6.12 | V3$^4$.6 |