Triangular tiling

(Redirected from Trat)
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
Petrie polygonsZigzags
HolesHexagons
Measures (edge length 1)
Vertex density${\displaystyle {\frac {2{\sqrt {3}}}{3}}\approx 1.15470}$
Related polytopes
ArmyTrat
RegimentTrat
DualHexagonal tiling
Petrie dualPetrial triangular tiling
φ 2 Hexagonal tiling
ConjugateNone
Abstract & topological properties
Flag count12N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes
NatureTame

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 simplicial 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

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

where i  and j  range over the integers.

Integral vertex coordinates for the triangular tiling can be given in 3D as:

• ${\displaystyle \left(i+j,\,i,\,j\right)}$,

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)

In vertex figures

Triangular tilings in vertex figures
Name Picture Schläfli symbol
Tetrahedral honeycomb {3,3,6}
Order-6 cubic honeycomb {4,3,6}
Order-6 dodecahedral honeycomb {5,3,6}
Order-6 hexagonal tiling honeycomb {6,3,6}

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.