Square tiling

Square tiling
Type	Regular tiling
Vertex figure	4.4.4.4 (or 44)
Schläfli symbol(s)	{4,4}
Wythoff symbol(s)	4 | 2 4
Coxeter-Dynkin(s)
Symmetry	*442
Dual	self-dual
Properties	Vertex-transitive, edge-transitive, face-transitive
4.4.4.4 (or 44)

In geometry, the square tiling 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.)

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}

...

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 unique topologically forms: square tiling, truncated square tiling, snub square tiling.

Operation	Schläfli symbol	Wythoff Symbol	Vertex figure	Image
Parent	t0{4,4}	4 | 2 4	44
Truncation	t0,1{4,4}	2 4 | 4	4.8.8
Rectification	t1{4,4}	2 | 4 4	(4.4)2
Bitruncation	t1,2{4,4}	2 4 | 4	4.8.8
Dual	t2{4,4}	4 | 2 4	44
Cantellation	t0,2{4,4}	4 4 | 2	4.4.4.4
Omnitruncation	t0,1,2{4,4}	2 4 4 |	4.8.8
Snubbing	s{4,4}	| 2 4 4	3.3.4.3.4

See also

• 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
• Williams, Robert The Geometrical Foundation of Natural Structure: A Source Book of Design New York: Dover, 1979. p36
• Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1.  (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 [1]

External links

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