Square tiling

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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) CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png = CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png = CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.png = CDel nodes 11.pngCDel iaib.pngCDel nodes.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 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
Square tiling
4.4.4.4 (or 44)

In geometry, the square tiling 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, with 5 of them as kaleidoscopic constructions with corresponding Coxeter diagrams. (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.)

1111 1212 1213 1122 1234
Square tiling uniform coloring 1.png Square tiling uniform coloring 7.png Square tiling uniform coloring 8.png Square tiling uniform coloring 4.png Square tiling uniform coloring 9.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png
p4m
[4,4]
(*442)
pmm
[1+,4,4,1+] = [∞,2,∞]
(*2222)
1112(i) 1112(ii) 1123(ii) 1123(i)
Square tiling uniform coloring 2.png Square tiling uniform coloring 3.png Square tiling uniform coloring 6.png Square tiling uniform coloring 5.png
p4m
[4,4]
(*442)
c2
[∞,2+,∞]
(2*22)
pmm
[∞,2,∞]
(*2222)

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...

Finite Euclidean Compact hyperbolic Paracompact
Uniform polyhedron-43-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0.png
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 45-t0.png
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 46-t0.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 47-t0.png
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 48-t0.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.

Spherical Euclidean Hyperbolic tilings
Spherical square hosohedron.png
{2,4}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
Spherical square bipyramid.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.png
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 245-1.png
{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 246-1.png
{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 247-1.png
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 248-1.png
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
... H2 tiling 24i-1.png
{∞,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
Dimensional family of quasiregular polyhedra and tilings: 4.n.4.n
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]
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 4.pngCDel node.png
Quasiregular
figures
configuration
Uniform tiling 432-t1.png
4.3.4.3
Uniform tiling 44-t1.png
4.4.4.4
Uniform tiling 54-t1.png
4.5.4.5
Uniform tiling 64-t1.png
4.6.4.6
Uniform tiling 74-t1.png
4.7.4.7
Uniform tiling 84-t1.png
4.8.4.8
H2 tiling 24i-2.png
4.∞.4.∞
4.∞.4.∞
Dual figures
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 4.pngCDel node.png
Dual
(rhombic)
figures
configuration
Rhombicdodecahedron.jpg
V4.3.4.3
Uniform tiling 44-t0.png
V4.4.4.4
Order-5-4 quasiregular rhombic tiling.png
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.∞
Dimensional family of expanded polyhedra and tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Uniform polyhedron-43-t02.png Uniform tiling 44-t02.png Uniform tiling 54-t02.png Uniform tiling 64-t02.png Uniform tiling 74-t02.png Uniform tiling 84-t02.png H2 tiling 24i-5.png
Coxeter
Schläfli
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{4,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{5,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{6,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{7,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{8,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{∞,4}
Dual
(rhombic)
figures
configuration
Deltoidalicositetrahedron.jpg
V3.4.4.4
Uniform tiling 44-t1.png
V4.4.4.4
Deltoidal tetrapentagonal tiling.png
5.4.4.4
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
Coxeter CDel node f1.pngCDel 3.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 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png

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.

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.png Uniform tiling 44-t01.png Uniform tiling 44-t1.png Uniform tiling 44-t12.png 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

Quadrilateral tiling variations[edit]

Quadrilateral tilings can be made with the identical {4,4} topology as the square tiling (4 quads around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 17 variations, with the first 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
Square
p4m
Isohedral tiling p4-54.png
Rectangle
pmm
Isohedral tiling p4-51.png
Parallelogram
pmg
Isohedral tiling p4-51c.png
Rhombus
pmg
Isohedral tiling p4-53.png
Kite
pmg
Isohedral tiling p3-7.png
Isosceles triangle
pmg
Isohedral tiling p4-46.png
Quadrilateral
pgg
Isohedral tiling p4-47.png
Quadrilateral
pgg
Isohedral tiling p3-4.png
Isosceles triangle
pgg
Isohedral tiling p3-5.png
Isosceles triangle
pgg
Isohedral tiling p3-3.png
Scalene triangle
pgg
Isohedral tiling p3-6.png
Scalene triangle
pgg
Isohedral tiling p4-55.png
Rhombus
cmm
Isohedral tiling p4-50.png
Parallelogram
p2
Isohedral tiling p4-52.png
Trapezoid
cmm
Isohedral tiling p4-52b.png
Trapezoid
cmm
Isohedral tiling p3-2.png
Scalene triangle
p2
Isohedral tiling p4-43.png
Quadrilateral
p2

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). 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[edit]

References[edit]

  1. ^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481

External links[edit]