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Elongated triangular tiling

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Title: Elongated triangular tiling  
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Subject: Demiregular tiling, Wythoff construction, Circle packing, Uniform tiling, Gilbert tessellation
Collection: Euclidean Tilings, Isogonal Tilings, Semiregular Tilings
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Elongated triangular tiling

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
Symmetry cmm, [∞,2+,∞], (2*22)
Rotation symmetry p2, [∞,2,∞]+, (2222)
Bowers acronym Etrat
Dual Prismatic pentagonal tiling
Properties Vertex-transitive

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.

Conway calls it a isosnub quadrille.[1]

There are 3 regular and 8 semiregular tilings in the plane. This tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order.

Contents

  • Construction 1
  • Uniform colorings 2
  • Circle packing 3
  • Related tilings 4
    • Prismatic pentagonal tiling 4.1
      • Geometric variations 4.1.1
      • Related 2-uniform dual tilings 4.1.2
  • See also 5
  • Notes 6
  • References 7
  • External links 8

Construction

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

There is one uniform colorings of an elongated triangular tiling. Two 2-uniform colorings have a single vertex figure, 11123, with two colors of squares, but are not 1-uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently. The 2-uniform tilings are also called Archimedean colorings. There are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings.

11122 (1-uniform) 11123 (2-uniform or 1-Archimedean)
cmm (2*22) pmg (22*) pgg (22×)

Circle packing

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).[2]

Related tilings

It is first in a series of symmetry mutations[3] with hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n.4.3.3.3, and Coxeter diagram . Their duals have hexagonal faces in the hyperbolic plane, with face configuration V4.n.4.3.3.3.

Symmetry mutation 2*n2 of uniform tilings: 4.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
or or

There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares.[4][5]

Double elongated Triple elongated Half elongated One third elongated

Prismatic pentagonal tiling

Prismatic pentagonal tiling
Type Dual uniform tiling
Coxeter diagram
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 15 known isohedral pentagon tilings. It can be seen as a stretched hexagonal tiling with a set of parallel bisecting lines through the hexagons.

Conway calls it a iso(4-)pentille.[1] Each of its pentagonal faces has three 120° and two 90° angles.

It is related to the Cairo pentagonal tiling with face configuration V3.3.4.3.4.

Geometric variations

Monohedral pentagonal tiling type 6 has the same topology, but two edge lengths and a lower p2 (2222) wallpaper group symmetry:


a=d=e, b=c
B+D=180°, 2B=E

Related 2-uniform dual tilings

There are four related 2-uniform dual tilings, mixing in rows of squares or hexagons.

See also

Notes

  1. ^ a b Conway, 2008, p.288 table
  2. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern F
  3. ^ by Daniel HusonTwo Dimensional symmetry Mutations
  4. ^ Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications 17: 147–165.  
  5. ^ http://www.uwgb.edu/dutchs/symmetry/uniftil.htm

References

  • , p. 58-65) Regular and uniform tilings (Chapter 2.1:  
  • p37  
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [2]
  • Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern Q2, Dual p. 77-76, pattern 6
  • Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56

External links

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