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Square tiling

## Summary

Square tiling

Type Regular tiling
Vertex configuration 4.4.4.4 (or 44)
Face configuration V4.4.4.4 (or V44)
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. 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: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii.

## 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}
Spherical 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.

*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings

24 34 44 54 64 74 84 ...4
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)2
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]
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
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*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
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

## Topologically equivalent tilings

An isogonal variation with two types of faces, seen as a snub square tiling with trangle pairs combined into rhombi.

Topological square tilings can be made with concave faces and more than one edge shared between two faces. This variation has 3 edges shared.

Other quadrilateral tilings can be made which are topologically equivalent to the square tiling (4 quads around every vertex).

A 2-isohedral variation with rhombic faces

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.[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).[2] 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.[3]

Self-dual Duals

4{4}4 or     2{8}4 or     4{8}2 or

## References

1. ^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481
2. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern 3
3. ^ Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Klitzing, Richard. "2D Euclidean tilings o4o4x - squat - O1".
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p36
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]