# Square tiling

Square tiling :
| Type | Regular tiling |
| Vertex configuration | 4.4.4.4 (or 4$^4$) |
| Face configuration | V4.4.4.4 (or V4$^4$) |
| Schläfli symbol(s) | {4,4} {∞}×{∞} |
| Wythoff symbol(s) | 4 | 2 4 |
| Coxeter diagram(s) | |
| Symmetry | p4m, [4,4], (*442) |
| Rotation symmetry | p4, [4,4]$^+$, (442) |
| Dual | self-dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a quadrille.
The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling.

## Uniform colorings

There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112$_i$ from 1213, 1123$_i$ from 1234, and 1112$_{ii}$ reduced from 1123$_{ii}$.
9 uniform colorings :
| 1111 | 1212 | 1213 | 1112$_i$ | 1122 |
| p4m (*442) | p4m (*442) | p4m (*442) | p4m (*442) | pmm (*2222) |
| 1234 | 1123$_i$ | 1123$_{ii}$ | 1112$_{ii}$ |
| pmm (*2222) | pmm (*2222) | cmm (2*22) | cmm (2*22) |

## Related polyhedra and tilings

This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
*n42 symmetry mutation of regular tilings: {4,n} vte :
| Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| {4,3} | {4,4} | {4,5} | {4,6} | {4,7} | {4,8}... | {4,∞} |
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)$^2$ :
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | [iπ/λ,4] |
| Tiling Conf. | V4.3.4.3 | V4.4.4.4 | V4.5.4.5 | V4.6.4.6 | V4.7.4.7 | V4.8.4.8 | V4.∞.4.∞ | V4.∞.4.∞ |
*n42 symmetry mutation of expanded tilings: n.4.4.4 vte :
| Symmetry [n,4], (*n42) | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry [n,4], (*n42) | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4] | *∞42 [∞,4] |
| Expanded figures | | | | | | | |
| Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4 |
| Rhombic figures config. | V3.4.4.4 | V4.4.4.4 | V5.4.4.4 | V6.4.4.4 | V7.4.4.4 | V8.4.4.4 | V∞.4.4.4 |

## Wythoff constructions from square tiling

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling, truncated square tiling, snub square tiling.
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 |

## Topologically equivalent tilings

Other quadrilateral tilings can be made which are topologically equivalent to the square tiling (4 quads around every vertex).
Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity, there are 18 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.
Isohedral quadrilateral tilings :
| Square p4m, (*442) | Quadrilateral p4g, (4*2) | Rectangle pmm, (*2222) | Parallelogram p2, (2222) | Parallelogram pmg, (22*) | Rhombus cmm, (2*22) | Rhombus pmg, (22*) |
| Trapezoid cmm, (2*22) | Trapezoid cmm, (2*22) | Quadrilateral pgg, (22×) | Kite pmg, (22*) | Quadrilateral pgg, (22×) | Quadrilateral p2, (2222) |
Degenerate quadrilaterals or non-edge-to-edge triangles :
| Isosceles pmg, (22*) | Isosceles pgg, (22×) | Isosceles pgg, (22×) | Scalene pgg, (22×) | Scalene pgg, (22×) | Scalene p2, (2222) |

## Circle packing

The 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 4 other circles in the packing (kissing number). The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.

## Related regular complex apeirogons

There are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.
| Self-dual | Duals | Duals |
| 4{4}4 or | 2{8}4 or | 4{8}2 or |