# Elongated triangular tiling

Elongated triangular tiling
Rank3
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymEtrat
Elements
Faces2N triangles, N squares
EdgesN+2N+2N
Vertices2N
Vertex figureIrregular Pentagon, edge lengths 1, 1, 1, 2, 2
Related polytopes
ArmyEtrat
RegimentEtrat
DualPrismatic pentagonal tiling
ConjugateRetroelongated triangular tiling
Abstract & topological properties
Flag count20N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryW2❘W2+
ConvexYes
NatureTame

The elongated triangular tiling, or etrat, is one of the eleven convex uniform tilings of the Euclidean plane. 3 triangles and 2 squares join at each vertex of this tiling.

It is the only one of the 11 regular and uniform convex tilings of the plane to not be derivable from the regulars by truncation operations or alternation. It can be thought of as being constructed from the triangular tiling with layers of squares being inserted between adjacent layers of triangles. It can also be considered a blend of infinitely many apeirogonal antiprisms and apeirogonal prisms.

## Vertex coordinates

The vertices of an elongated triangular tiling of edge length 1 are given by

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

where i and j range over the integers.