Square Tiling
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Square Tiling
Square tiling
Square tiling
Type Regular tiling
Vertex configuration 4.4.4.4 (or 44)
Tiling 4a vertfig.svg
Face configuration V4.4.4.4 (or V44)
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 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.

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
Uniform tiling 432-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.svg
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H2-5-4-primal.svg
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 246-4.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 247-4.png
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 248-4.png
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4.png
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png

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.

*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
Spherical square hosohedron.png Spherical square bipyramid.png Uniform tiling 44-t0.svg H2-5-4-dual.svg H2 tiling 246-1.png H2 tiling 247-1.png H2 tiling 248-1.png H2 tiling 24i-1.png
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.
Spherical rhombic dodecahedron.png
V4.3.4.3
Uniform tiling 44-t0.svg
V4.4.4.4
H2-5-4-rhombic.svg
V4.5.4.5
Ord64 qreg rhombic til.png
V4.6.4.6
Ord74 qreg rhombic til.png
V4.7.4.7
Ord84 qreg rhombic til.png
V4.8.4.8
Ord4infin qreg rhombic til.png
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
Uniform tiling 432-t02.png Uniform tiling 44-t02.png H2-5-4-cantellated.svg Uniform tiling 64-t02.png Uniform tiling 74-t02.png Uniform tiling 84-t02.png H2 tiling 24i-5.png
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.
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Uniform tiling 44-t0.svg
V4.4.4.4
H2-5-4-deltoidal.svg
V5.4.4.4
Deltoidal tetrahexagonal til.png
V6.4.4.4
Deltoidal tetraheptagonal til.png
V7.4.4.4
Deltoidal tetraoctagonal til.png
V8.4.4.4
Deltoidal tetraapeirogonal tiling.png
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)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 44-t0.svg Uniform tiling 44-t01.png Uniform tiling 44-t1.png Uniform tiling 44-t12.svg Uniform tiling 44-t2.png Uniform tiling 44-t02.png Uniform tiling 44-t012.png Uniform tiling 44-snub.png Uniform tiling 44-h01.png
{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
CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Uniform tiling 44-t0.png Tetrakis square tiling.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
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]

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
Square
p4m, (*442)
Quadrilateral
p4g, (4*2)
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 ?/4=78.54% coverage. There are 4 uniform colorings of the circle packings.

1-uniform-5-circlepack.svg

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
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

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]

External links

Space Family / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133 o 331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152 o 251 o 521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k2 o 2k1 o k21

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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