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. 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: 1112_i from 1213, 1123_i from 1234, and 1112_ii reduced from 1123_ii.

9 uniform colorings
1111	1212	1213	1112_i	1122

p4m (*442)		p4m (*442)		pmm (*2222)
1234	1123_i	1123_ii	1112_ii

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: {4,n}
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: {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
Symmetry *4n2 [n,4]	*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.
Symmetry [n,4], (*n42)	*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

A 2-isohedral variation with rhombic 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 quadrilateral tilings
Square p4m, (*442)	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.

References

↑ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481
↑ 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
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	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
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]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

Tessellation

Periodic

Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff

Aperiodic

Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal 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

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