Hexagonal tiling

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Hexagonal tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymHexat
Coxeter diagramx6o3o ()
Schläfli symbol{6,3}
Elements
FacesN hexagons
Edges3N
Vertices2N
Vertex figureEquilateral triangle, edge length 3
Measures (edge length 1)
Vertex density
Related polytopes
ArmyHexat
RegimentHexat
DualTriangular tiling
Petrie dualPetrial hexagonal tiling
ConjugateNone
Abstract & topological properties
Flag count12N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes
NatureTame

The hexagonal tiling, or hexat, is one of the three regular tilings of the Euclidean plane. 3 hexagons join at each vertex of this tiling. It can also be formed as a truncation of the triangular tiling.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a hexagonal tiling of edge length 1 are given by:

  • ,
  • ,

where i  and j  range over the integers.

Representations[edit | edit source]

A hexagonal tiling has the following Coxeter diagrams:

  • x6o3o () (full symmetry)
  • o6x3x () (as truncated triangular tiling)
  • x3x3x3*a () (P3 symmetry, as omnitruncated cyclotriangular tiling)
  • s6x3x () (additional alternated faceting form)
  • x∞s2s∞o () (as alternated faceting from the square tiling)
  • uBxx3uxBx3uxxB&#zx (B = 4)
  • ho3oo3oh&#zx (hull of two opposite triangular tilings)

In vertex figures[edit | edit source]

The hexagonal tiling appears as the vertex figure of the hyperbolic triangular tiling honeycomb.

External links[edit | edit source]