Square tiling

Square tiling
Type	Regular tiling
Vertex configuration	4.4.4.4 (or 4^4)
Face configuration	V4.4.4.4 (or V4^4)
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 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: 1112_i from 1213, 1123_i from 1234, and 1112_ii reduced from 1123_ii.

9 uniform colorings
1111	1212	1213	1112i	1122
p4m (*442)		p4m (*442)		pmm (*2222)
1234	1123i	1123ii	1112ii
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} v t e
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} v t e
Spherical		Euclidean	Hyperbolic tilings
2^4	3^4	4^4	5^4	6^4	7^4	8^4	...∞^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.	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 v t e
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
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 v t e
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, 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

Square p4m, (*442)	Quadrilateral p4g, (4*2)	Rectangle pmm, (*2222)	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.

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
4{4}4 or	2{8}4 or	4{8}2 or

See also

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

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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]

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 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E^10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
E^n-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21