Elongated triangular tiling

Elongated triangular tiling
Rank	3
Type	Uniform
Space	Euclidean
Notation
Bowers style acronym	Etrat
Elements
Faces	2N triangles, N squares
Edges	N+2N+2N
Vertices	2N
Vertex figure	Irregular Pentagon, edge lengths 1, 1, 1, √2, √2
Related polytopes
Army	Etrat
Regiment	Etrat
Dual	Prismatic pentagonal tiling
Conjugate	Retroelongated triangular tiling
Abstract & topological properties
Flag count	20N
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	W2❘W2+
Convex	Yes
Nature	Tame

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.

Gallery

Vertex coordinates

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

• $\left(i\frac{2+\sqrt3}{2}\pm\frac12,\,j+\frac{i}{2}\right),$

where i and j range over the integers.

External links

• Klitzing, Richard. "etrat".

• Wikipedia Contributors. "Elongated triangular tiling".
• McNeill, Jim. The 'traditional' eleven tessellations.
• Complex Uniform Tessellations on the Euclid Plane.