Square tiling

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Square tiling
Square tiling
Type Regular tiling
Vertex configuration (or 44)
Square tiling vertfig.png
Schläfli symbol(s) {4,4}
Wythoff symbol(s) 4 | 2 4
Coxeter diagram(s) CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png
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 calls 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[edit]

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[edit]

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 CDel node 1.pngCDel n.pngCDel node.pngCDel 4.pngCDel node.png, with n progressing to infinity.

Wythoff constructions from square tiling[edit]

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[edit]

An isogonal variation with two types of faces
A 2-isohedral variation with rhombic faces

Other quadrilateral tilings can be made with 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 colinear edges. Symmetry given assumes all faces are the same color.[1]

Isohedral quadrilateral tilings
Isohedral tiling p4-56.png Isohedral tiling p4-49.png Isohedral tiling p4-54.png Isohedral tiling p4-50.png Isohedral tiling p4-51.png Isohedral tiling p4-55.png Isohedral tiling p4-51c.png
p4m, (*442)
p4g, (4*2)
pmm, (*2222)
p2, (2222)
pmg, (22*)
cmm, (2*22)
pmg, (22*)
Isohedral tiling p4-52b.png Isohedral tiling p4-52.png Isohedral tiling p4-46.png Isohedral tiling p4-53.png Isohedral tiling p4-47.png Isohedral tiling p4-43.png
cmm, (2*22)
pgg, (22×)
pmg, (22*)
pgg, (22×)
p2, (2222)
Degenerate quadrilaterals or non-edge-to-edge triangles
Isohedral tiling p3-7.png Isohedral tiling p3-4.png Isohedral tiling p3-5.png Isohedral tiling p3-3.png Isohedral tiling p3-6.png Isohedral tiling p3-2.png
pmg, (22*)
pgg, (22×)
pgg, (22×)
p2, (2222)

Circle packing[edit]

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.

Square tiling circle packing.png Rectified square tiling circle packing.png Expanded square tiling circle packing.png Translational square tiling circle packing.png

Related regular complex apeirogons[edit]

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 contrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.[3]

Self-dual Duals
Complex apeirogon 4-4-4.png Complex apeirogon 2-8-4.png Complex apeirogon 4-8-2.png
4{4}4 or CDel 4node 1.pngCDel 4.pngCDel 4node.png 2{8}4 or CDel node 1.pngCDel 8.pngCDel 4node.png 4{8}2 or CDel 4node 1.pngCDel 8.pngCDel node.png

See also[edit]


  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.

External links[edit]

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 5-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 6-honeycomb {3[6]} δ6 6 6
E6 Uniform 7-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 8-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 9-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 10-honeycomb {3[10]} δ10 10 10
En-1 Uniform n-honeycomb {3[n]} δn n n 1k22k1k21