# Snub square tiling

Snub square tiling :
| Type | Semiregular tiling |
| Vertex configuration | 3.3.4.3.4 |
| Schläfli symbol | s{4,4} sr{4,4} or $s{\begin{Bmatrix}4\\4\end{Bmatrix}}$ |
| Wythoff symbol | | 4 4 2 |
| Coxeter diagram | or |
| Symmetry | p4g, [4$^+$,4], (4*2) |
| Rotation symmetry | p4, [4,4]$^+$, (442) |
| Bowers acronym | Snasquat |
| Dual | Cairo pentagonal tiling |
| Properties | Vertex-transitive |
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.
Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).
There are 3 regular and 8 semiregular tilings in the plane.

## Uniform colorings

There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)
| Coloring | 11212 | 11213 |
| Symmetry | 4*2, [4$^+$,4], (p4g) | 442, [4,4]$^+$, (p4) |
| Schläfli symbol | s{4,4} | sr{4,4} |
| Wythoff symbol | | | 4 4 2 |
| Coxeter diagram | | |

## Circle packing

The snub square 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).

## Wythoff construction

The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.
An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.
If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.
Example:
| Regular octagons alternately truncated | →(Alternate truncation) | Isosceles triangles (Nonuniform tiling) |
| Nonregular octagons alternately truncated | →(Alternate truncation) | Equilateral triangles |

## Related tilings

* A snub operator applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.
* A related isogonal tiling that combines pairs of triangles into rhombi
* A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.
* The Cairo pentagonal tiling is dual to the snub square tiling.

### Related k-uniform tilings

This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many k-uniform tilings.
Related tilings of triangles and squares :
| snub square | elongated triangular | 2-uniform | 2-uniform | 3-uniform | 3-uniform | 3-uniform |
| p4g, (4*2) | p2, (2222) | p2, (2222) | cmm, (2*22) | p2, (2222) | p2, (2222) |
| [3$^2$434] | [3$^3$4$^2$] | [3$^3$4$^2$; 3$^2$434] | [3$^3$4$^2$; 3$^2$434] | [2: 3$^3$4$^2$; 3$^2$434] | [3$^3$4$^2$; 2: 3$^2$434] |

### Related topological series of polyhedra and tiling

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
4n2 symmetry mutations of snub tilings: 3.3.4.3.n vte :
| Symmetry 4n2 | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry 4n2 | 242 | 342 | 442 | 542 | 642 | 742 | 842 | ∞42 |
| Snub figures | | | | | | | | |
| Config. | 3.3.4.3.2 | 3.3.4.3.3 | 3.3.4.3.4 | 3.3.4.3.5 | 3.3.4.3.6 | 3.3.4.3.7 | 3.3.4.3.8 | 3.3.4.3.∞ |
| Gyro figures | | | | | | | | |
| Config. | V3.3.4.3.2 | V3.3.4.3.3 | V3.3.4.3.4 | V3.3.4.3.5 | V3.3.4.3.6 | V3.3.4.3.7 | V3.3.4.3.8 | V3.3.4.3.∞ |
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
4n2 symmetry mutations of snub tilings: 3.3.n.3.n :
| Symmetry 4n2 | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| Symmetry 4n2 | 222 | 322 | 442 | 552 | 662 | 772 | 882 | ∞∞2 |
| Snub figures | | | | | | | | |
| Config. | 3.3.2.3.2 | 3.3.3.3.3 | 3.3.4.3.4 | 3.3.5.3.5 | 3.3.6.3.6 | 3.3.7.3.7 | 3.3.8.3.8 | 3.3.∞.3.∞ |
| Gyro figures | | | | | | | | |
| Config. | V3.3.2.3.2 | V3.3.3.3.3 | V3.3.4.3.4 | V3.3.5.3.5 | V3.3.6.3.6 | V3.3.7.3.7 | V3.3.8.3.8 | V3.3.∞.3.∞ |
Uniform tilings based on square tiling symmetry vte :
| Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | Symmetry: [4,4], (*442) | [4,4]$^+$, (442) | [4,4$^+$], (4*2) |
| {4,4} | t{4,4} | r{4,4} | t{4,4} | {4,4} | rr{4,4} | tr{4,4} | sr{4,4} | s{4,4} |
| Uniform duals |||||||||
| V4.4.4.4 | V4.8.8 | V4.4.4.4 | V4.8.8 | V4.4.4.4 | V4.4.4.4 | V4.8.8 | V3.3.4.3.4 | V3.3.4.3.4 |