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
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]