Square tiling

Square tiling

Type Regular tiling
Vertex configuration 4.4.4.4 (or 44)
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.

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, with 5 of them as kaleidoscopic constructions with corresponding Coxeter diagrams. (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.)

1111 1212 1213 1122 1234

p4m
[4,4]
(*442)
pmm
[1+,4,4,1+] = [∞,2,∞]
(*2222)
1112(i) 1112(ii) 1123(ii) 1123(i)
p4m
[4,4]
(*442)
c2
[∞,2+,∞]
(2*22)
pmm
[∞,2,∞]
(*2222)

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...

Finite Euclidean 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.

Spherical Euclidean Hyperbolic tilings

{2,4}

{3,4}

{4,4}

{5,4}

{6,4}

{7,4}

{8,4}
...
{∞,4}
Dimensional family of quasiregular polyhedra and tilings: 4.n.4.n
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]

[iπ/λ,4]
Coxeter
Quasiregular
figures
configuration

4.3.4.3

4.4.4.4

4.5.4.5

4.6.4.6

4.7.4.7

4.8.4.8

4.∞.4.∞
4.∞.4.∞
Dual figures
Coxeter
Dual
(rhombic)
figures
configuration

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.∞
Dimensional family of expanded polyhedra and tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Coxeter
Schläfli

rr{3,4}

rr{4,4}

rr{5,4}

rr{6,4}

rr{7,4}

rr{8,4}

rr{∞,4}
Dual
(rhombic)
figures
configuration

V3.4.4.4

V4.4.4.4

5.4.4.4

V6.4.4.4

V7.4.4.4

V8.4.4.4

V∞.4.4.4
Coxeter

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
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

An isogonal variation with two types of faces

Quadrilateral tilings can be made with the identical {4,4} topology as the square tiling (4 quads around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 17 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two colinear edges. Symmetry given assumes all faces are the same color.[1]

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.