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.

Uniform colorings

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
p4m [4,4] (*442)			pmm [1+,4,4,1+] = [∞,2,∞] (*2222)
1112(i)	1112(ii)	1123(ii)	1123(i)
p4m [4,4] (*442)	c2 [∞,2+,∞] (2*22)		pmm [∞,2,∞] (*2222)

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...

Finite	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.

Spherical		Euclidean	Hyperbolic tilings
{2,4}	{3,4}	{4,4}	{5,4}	{6,4}	{7,4}	{8,4}	...	{∞,4}

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
Quasiregular figures configuration	4.3.4.3	4.4.4.4	4.5.4.5	4.6.4.6	4.7.4.7	4.8.4.8	4.∞.4.∞	4.∞.4.∞
Dual figures
Coxeter
Dual (rhombic) figures configuration	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.∞

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
Coxeter Schläfli	rr{3,4}	rr{4,4}	rr{5,4}	rr{6,4}	rr{7,4}	rr{8,4}	rr{∞,4}
Dual (rhombic) figures configuration	V3.4.4.4	V4.4.4.4	5.4.4.4	V6.4.4.4	V7.4.4.4	V8.4.4.4	V∞.4.4.4
Coxeter

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

Quadrilateral tiling variations

An isogonal variation with two types of faces

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]

Square p4m	Rectangle pmm	Parallelogram pmg	Rhombus pmg	Kite pmg	Isosceles triangle pmg
Quadrilateral pgg	Quadrilateral pgg	Isosceles triangle pgg	Isosceles triangle pgg	Scalene triangle pgg	Scalene triangle pgg
Rhombus cmm	Parallelogram p2	Trapezoid cmm	Trapezoid cmm	Scalene triangle p2	Quadrilateral p2

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). The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.

See also

Wikimedia Commons has media related to Order-4 square tiling.

• Checkerboard
• List of regular polytopes
• List of uniform tilings
• Square lattice
• Tilings of regular polygons

References

• 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
• 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 ; and 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

External links

• Weisstein, Eric W., "Square Grid", MathWorld.
• Weisstein, Eric W., "Regular tessellation", MathWorld.
• Weisstein, Eric W., "Uniform tessellation", MathWorld.

Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family					/ /
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

Tessellation & tiling   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & tessellation Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagon Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet   Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg   By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82