Square tiling

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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
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

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


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...

{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	Hyperbolic...
Symmetry *4n2 [n,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]	*∞42 [∞,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.∞
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.∞

Dimensional family of expanded polyhedra and tilings: *n.4.4.4*
Symmetry [n,4], (*n42)	Spherical	Euclidean	Hyperbolic tiling
Symmetry [n,4], (*n42)	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]	*∞42 [∞,4]
Quasiregular 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

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	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.

References

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

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.

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–11
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

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