Square tiling
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| Square tiling | |
 |
|
| Type | Regular tiling |
|---|---|
| Vertex figure | 4.4.4.4 |
| Schläfli symbol(s) | {4,4} |
| Wythoff symbol(s) | 4 | 2 4 |
| Coxeter-Dynkin(s) |      |
| Symmetry | p4m |
| Dual | self-dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
 4.4.4.4 |
|
In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}.
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.
Contents |
[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)
