Square tiling

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.

Square tiling

TypeRegular tiling
Vertex configuration4.4.4.4 (or 44)
Face configurationV4.4.4.4 (or V44)
Schläfli symbol(s){4,4}
{∞}×{∞}
Wythoff symbol(s)2 4
Coxeter diagram(s)




Symmetryp4m, [4,4], (*442)
Rotation symmetryp4, [4,4]+, (442)
Dualself-dual
PropertiesVertex-transitive, edge-transitive, face-transitive

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: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii.

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...

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.

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.

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]

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)		 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×)		 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).[2] The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.

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-dualDuals
4{4}4 or 2{8}4 or 4{8}2 or

See also

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
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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