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Square tiling

 

Square tiling

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.

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.

Contents

  • Uniform colorings 1
  • Related polyhedra and tilings 2
  • Wythoff constructions from square tiling 3
  • Topologically equivalent tilings 4
  • Circle packing 5
  • See also 6
  • References 7
  • External links 8

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
p4m (*442) p4m (*442) pmm (*2222)
1234 1123i 1123ii 1112ii
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...

*n42 symmetry mutation of regular tilings: 4n
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: n4 or {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

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]

Square
p4m, (*442)
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.

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
  • p36  
  • , p. 58-65) Regular and uniform tilings (Chapter 2.1:  
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [2]

External links

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