# Trihexagonal tiling

Trihexagonal tiling
Rank3
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymThat
Coxeter diagramo6x3o ()
Elements
Faces2N triangles, N hexagons
Edges6N
Vertices3N
Vertex figureRectangle, edge lengths 1 and 3
Measures (edge length 1)
Vertex density${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Related polytopes
ArmyThat
RegimentThat
DualRhombille tiling
ConjugateNone
Abstract & topological properties
Flag count24N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes
NatureTame

The trihexagonal tiling, or that, is one of the eleven convex uniform tilings of the Euclidean plane. 2 hexagons and 2 triangles join at each vertex of this tiling. It can be formed from the rectification of either the triangular tiling or its dual hexagonal tiling.

## Vertex coordinates

The vertices of a trihexagonal tiling of edge length 1 are given by:

• ${\displaystyle \left({\sqrt {3}}i,\,i+2j+1\right)}$,
• ${\displaystyle \left({\sqrt {3}}i+{\frac {\sqrt {3}}{2}},\,j+{\frac {1}{2}}\right)}$,

where i  and j  range over the integers.

## Representations

A trihexagonal tiling has the following Coxeter diagrams:

• o6x3o () (full symmetry)
• x3x3o3*a () (P3 symmetry, triangles of two types)
• s6x3o () (as alternated faceting)
• s6o3x ()

## Related tilings

The trihexagonal tiling is the colonel of a three-member regiment that also includes the triangular-hemiapeirogonal tiling and the hexagonal-hemiapeirogonal tiling.