# Triangular tiling

Triangular tiling Rank3
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymTrat
Coxeter diagramo6o3x (     )
Schläfli symbol{3,6}
Elements
Faces2N triangles
Edges3N
VerticesN
Vertex figureHexagon, edge length 1
Measures (edge length 1)
Vertex density$\frac{2\sqrt3}3 \approx 1.15470$ Related polytopes
ArmyTrat
RegimentTrat
DualHexagonal tiling
ConjugateNone
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes

The triangular tiling, or trat, is one of the three regular tilings of the Euclidean plane. 6 triangles join at each vertex of this tiling. It is also the 2-dimensional simplectic honeycomb. It is also the alternation of the hexagonal tiling.

## Vertex coordinates

The vertices of a triangular tiling of edge length 1 are given by

• $\left(i\frac{\sqrt3}{2},\,j+\frac{i}{2}\right),$ where i and j range over the integers.

## Representations

A triangular tiling has the following Coxeter diagrams:

• o6o3x (full symmetry)
• x3o3o3*a (P3 symmetry, triangles considered of two types)
• s6o3o (alternated hexagonal tilling)
• o6s3s
• s3s3s3*a
• xdoo3xodo3xood&#zx (as hull of hexagonal tiling and three larger triangular tilings)

## Related tiling

The triangular tiling is the colonel of a two-member regiment that also includes the ditrigonary triangular-hemiapeirogonal tiling. Also in this regiment is a compound of three hexagonal tilings.