Square tiling

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Square tiling
Type	Regular tiling
Vertex figure	4.4.4
Schläfli symbol	{4,4}
Wythoff symbol	4 | 2 4
Coxeter-Dynkin
Symmetry	p4m
Dual	self-dual
Properties	Vertex-transitive, edge-transitive, face-transitive
4.4.4

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}.

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.

Contents 1 Uniform colorings 2 Wythoff constructions from square tiling 3 See also 4 References 5 External links

[edit] 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.)

[edit] 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

[edit] See also

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

[edit] 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)

[edit] External links

• Eric W. Weisstein, Square Grid at MathWorld.