Square tiling

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Square tiling
1-uniform n5.svg
Bowers style acronymSquat
Coxeter diagramx4o4o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png)
Schläfli symbol{4,4}
FacesN squares
Vertex figureSquare, edge length 2
Measures (edge length 1)
Vertex density
Related polytopes
DualSquare tiling
Petrie dualPetrial square tiling
Abstract & topological properties

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]

Square tilings in vertex figures
Name Picture Schläfli symbol
Octahedral honeycomb
H3 344 CC center.png
Order-4 square tiling honeycomb
H3 444 FC boundary.png

Related tilings[edit | edit source]

The square tiling is the colonel of a two-member regiment that also includes the square-hemiapeirogonal tiling.

o4o4o truncations
Name OBSA Schläfli symbol CD diagram Picture
Square tiling squat {4,4} x4o4o
Uniform tiling 44-t0.png
Truncated square tiling tosquat t{4,4} x4x4o
Uniform tiling 44-t01.png
Rectified square tiling = Square tiling squat r{4,4} o4x4o
Uniform tiling 44-t1.png
Truncated square tiling tosquat t{4,4} o4x4x
Uniform tiling 44-t12.png
Square tiling squat {4,4} o4o4x
Uniform tiling 44-t2.png
Cantellated square tiling = Square tiling squat rr{4,4} x4o4x
Uniform tiling 44-t02.png
Omnitruncated square tiling = Truncated square tiling tosquat tr{4,4} x4x4x
Uniform tiling 44-t012.png
Snub square tiling snasquat sr{4,4} s4s4s
Uniform tiling 44-snub.png

External links[edit | edit source]