Square tiling
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| Square tiling | |
|---|---|
 | |
| Rank | 3 |
| Type | Regular |
| Space | Euclidean |
| Notation | |
| Bowers style acronym | Squat |
| Coxeter diagram | x4o4o (   ) |
| Schläfli symbol | {4,4} |
| Elements | |
| Faces | N squares |
| Edges | 2N |
| Vertices | N |
| Vertex figure | Square, edge length √2 |
| Measures (edge length 1) | |
| Vertex density | |
| Related polytopes | |
| Army | Squat |
| Regiment | Squat |
| Dual | Square tiling |
| Petrie dual | Petrial square tiling |
| Conjugate | None |
| Topological properties | |
| Surface | Sphere |
| Orientable | Yes |
| Genus | 0 |
| Properties | |
| Symmetry | R3 |
| Convex | Yes |
The square tiling, or squat, is one of the three regular tilings of the Euclidean plane. 4 squares join at each vertex of this tiling. It is the only one of the three regular tilings to be self-dual. It is also the 2D hypercubic honeycomb.
Vertex coordinates[edit | edit source]
Coordinates for the vertices of a square tiling of edge length 1 are given by
where i and j range over the integers.
Representations[edit | edit source]
A square tiling has the following Coxeter diagrams:
- x4o4o (regular)
- o4x4o (as rectified square tiling)
- x4o4x (as small rhombated square tiling)
- xØx xØx (W2|W2 symmetry, as comb product of two apeirogons)
- s4o4o (as alternated square tiling)
- o4s4o
- s4o4s
- s4x4o (as additional alternated facetings)
- x4s4x
- s4x4s
- x4s4o
- s4s4x
- qo4oo4oq&#zx (as hull of two dual square tilings)
- qo4xx4oq&#zx (as hull of two oopposite variant truncated square tilings)
In vertex figures[edit | edit source]
| Name | Picture | Schläfli symbol |
|---|---|---|
| Octahedral honeycomb |  |
{3,4,4} |
| Order-4 square tiling honeycomb |  |
{4,4,4} |
Related tilings[edit | edit source]
The square tiling is the colonel of a two-member regiment that also includes the square-hemiapeirogonal tiling.
| Name | OBSA | Schläfli symbol | CD diagram | Picture |
|---|---|---|---|---|
| Square tiling | squat | {4,4} | x4o4o |  |
| Truncated square tiling | tosquat | t{4,4} | x4x4o |  |
| Rectified square tiling = Square tiling | squat | r{4,4} | o4x4o |  |
| Truncated square tiling | tosquat | t{4,4} | o4x4x |  |
| Square tiling | squat | {4,4} | o4o4x |  |
| Cantellated square tiling = Square tiling | squat | rr{4,4} | x4o4x |  |
| Omnitruncated square tiling = Truncated square tiling | tosquat | tr{4,4} | x4x4x |  |
| Snub square tiling | snasquat | sr{4,4} | s4s4s |  |
External links[edit | edit source]
- Klitzing, Richard. "squat".
- Wikipedia Contributors. "Square tiling".