| 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-Dynkin(s) | |
| Symmetry | p4m, [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.
Contents |
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.)
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} |
... |
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 |