Elongated triangular tiling

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Elongated triangular tiling
Elongated triangular tiling
Type Semiregular tiling
Vertex configuration 3.3.3.4.4
Schläfli symbol {3,6}:e
s{∞}h1{∞}
Wythoff symbol 2 | 2 (2 2)
Coxeter diagram CDel node.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
Symmetry cmm, [∞,2+,∞], (2*22)
pgm, [(∞,2)+,∞], (22*)
pgg, [(∞,2)+,∞+], (22×)
Rotation symmetry p2, [∞,2,∞]+, (2222)
Bowers acronym Etrat
Dual Prismatic pentagonal tiling
Properties Vertex-transitive
Tiling 33344-vertfig.png
Vertex figure: 3.3.3.4.4

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex.

Conway calls it a isosnub quadrille.[1]

There are 3 regular and 8 semiregular tilings in the plane. This tiling is related to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order. It is also the only uniform tiling that can't be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal prisms and apeirogonal antiprisms.

Uniform colorings[edit]

There are two uniform colorings of an elongated triangular tiling. (Naming the colors by indices around a vertex (3.3.3.4.4): 11122 and 11123.) Two colorings have a single vertex figure, 11123, with two colors of squares, but are not uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently.

11122 (Uniform) 11123 (not uniform)
Elongated triangular tiling 1.png Elongated triangular tiling 3.png Elongated triangular tiling 2.png
cmm (2*22) pgm (22*) pgg (22×)

Circle packing[edit]

The elongated triangular tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).

Elongated triangular tiling circle packing.png

Related tilings[edit]

It is related to similarly constructed hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n.4.3.3.3, and Coxeter diagram CDel node.pngCDel ultra.pngCDel node h.pngCDel n.pngCDel node h.pngCDel ultra.pngCDel node 1.png. These duals have hexagonal faces, with face configuration V4.n.4.3.3.3.

4.2.4.3.3.3 4.3.4.3.3.3 4.4.4.3.3.3
2*22 2*32 2*42
Elongated triangular tiling 4.2.4.3.3.3.png Uniform tiling 4.3.4.3.3.3.png Hyper 4.4.4.3.3.3a.png
CDel node.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png CDel node.pngCDel ultra.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel ultra.pngCDel node 1.png or CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png CDel node.pngCDel ultra.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel ultra.pngCDel node 1.png or CDel label4.pngCDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png

Prismatic pentagonal tiling[edit]

Prismatic pentagonal tiling
Tiling Dual Semiregular V3-3-3-4-4 Prismatic Pentagonal.svg
Type Dual uniform tiling
Coxeter diagram CDel node.pngCDel infin.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel infin.pngCDel node f1.png
CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel infin.pngCDel node f1.png
Faces irregular pentagons
Face configuration V3.3.3.4.4
Symmetry group cmm, [∞,2+,∞], (2*22)
Rotation group p2, [∞,2,∞]+, (2222)
Dual Elongated triangular tiling
Properties face-transitive

The prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 14 known isohedral pentagon tilings.

Conway calls it a iso(4-)pentille.[2] Each of its pentagonal faces has three 120° and two 90° angles.

It is the dual of the elongated triangular tiling.[3]

See also[edit]

Notes[edit]

  1. ^ Conway, 2008, p.288 table
  2. ^ Conway, 2008, p.288 table
  3. ^ Weisstein, Eric W., "Dual tessellation", MathWorld.

References[edit]

External links[edit]