Square tiling

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Square tiling
Square tiling
Type Regular tiling
Vertex configuration 4.4.4.4 (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
Symmetry p4m, [4,4], (*442)
Rotation symmetry p4, [4,4]+, (442)
Dual self-dual
Properties Vertex-transitive, edge-transitive, face-transitive
Square tiling
4.4.4.4 (or 44)

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

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.

1111 1212 1213 1112i 1122
Square tiling uniform coloring 1.png Square tiling uniform coloring 7.png Square tiling uniform coloring 8.png Square tiling uniform coloring 2.png Square tiling uniform coloring 4.png
p4m (*442) p4m (*442) pmm (*2222)
1234 1123i 1123ii 1112ii
Square tiling uniform coloring 9.png Square tiling uniform coloring 5.png Square tiling uniform coloring 6.png Square tiling uniform coloring 3.png
pmm (*2222) 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...

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

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

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 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]

Isohedral tiling p4-56.png Isohedral tiling p4-54.png Isohedral tiling p4-50.png Isohedral tiling p4-51.png 90px 90px
Square
p4m, (*442)		 Rectangle
pmm, (*2222)		 Parallelogram
p2, (2222)		 Parallelogram
pmg, (22*)		 Rhombus
cmm, (2*22)		 Rhombus
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
Trapezoid
cmm, (2*22)		 Quadrilateral
pgg, (22×)		 Kite
pmg, (22*)		 Quadrilateral
pgg, (22×)		 Quadrilateral
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
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.

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

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
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • Richard Klitzing, 2D Euclidean tilings, o4o4x - squat - O1
  • Lua error in package.lua at line 80: module 'strict' not found. p36
  • Lua error in package.lua at line 80: module 'strict' not found. (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]

External links

Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family {\tilde{A}}_{n-1} {\tilde{C}}_{n-1} {\tilde{B}}_{n-1} {\tilde{D}}_{n-1} {\tilde{G}}_2 / {\tilde{F}}_4 / {\tilde{E}}_{n-1}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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