Elongated triangular tiling

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Elongated triangular tiling
Elongated triangular tiling
Type Semiregular tiling
Vertex configuration Tiling 33344-vertfig.png
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)
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.

Construction[edit]

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 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)
Elongated triangular tiling 1.png Elongated triangular tiling 3.png Elongated triangular tiling 2.png
cmm (2*22) pmg (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).[2]

Elongated triangular tiling circle packing.png

Related tilings[edit]

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 CDel node.pngCDel ultra.pngCDel node h.pngCDel n.pngCDel node h.pngCDel ultra.pngCDel node 1.png. 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
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

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
2-uniform n4.svg 2-uniform n3.svg 2-uniform n14.svg 2-uniform n15.svg

Prismatic pentagonal tiling[edit]

Elongated triangular tiling
1-uniform 8 dual.svg
Type Dual uniform tiling
Faces irregular pentagons
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
Symmetry group cmm, [∞,2+,∞], (2*22)
Rotation group p2, [∞,2,∞]+, (2222)
Dual polyhedron Elongated triangular tiling
Face configuration V3.3.3.4.4
Tiling face 3-3-3-4-4.svg
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[edit]

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

P5-type6.png Prototile p5-type6.png
a=d=e, b=c
B+D=180°, 2B=E

Related 2-uniform dual tilings[edit]

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

2-uniform 4 dual.svg 2-uniform 3 dual.svg 2-uniform 14 dual.svg 2-uniform 15 dual.svg

See also[edit]

Notes[edit]

  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. ^ Two Dimensional symmetry Mutations by Daniel Huson
  4. ^ Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9. 
  5. ^ http://www.uwgb.edu/dutchs/symmetry/uniftil.htm

References[edit]

  • Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1.  (Chapter 2.1: Regular and uniform tilings, p. 58-65)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p37
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
  • 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[edit]