Square tiling

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Square tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymSquat
Coxeter diagramx4o4o ()
Schläfli symbol{4,4}
Elements
FacesN squares
Edges2N
VerticesN
Vertex figureSquare, edge length 2
HolesApeirogons
Measures (edge length 1)
Vertex density
Related polytopes
ArmySquat
RegimentSquat
DualSquare tiling
Petrie dualPetrial square tiling
HalvingSquare tiling, edge length
ConjugateNone
Abstract & topological properties
Flag count8N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryR3
ConvexYes
NatureTame

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.

The square tiling is the honeycomb product of two apeirogons. It can also be seen as the result of stacking infinitely many apeirogonal prisms on top of each other.

The square tiling is also notable because it is equivalent to its own rectification and cantellation.

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)
  • xØx2xØx () (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 opposite variant truncated square tilings)

In vertex figures[edit | edit source]

Square tilings in vertex figures
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.

External links[edit | edit source]