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.

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

• Tilings of regular polygons
• Elongated triangular prismatic honeycomb
• Gyroelongated triangular prismatic honeycomb

Notes

References

Wikimedia Commons has media related to Uniform tiling 3-3-3-4-4.

Wikimedia Commons has media related to Prismatic pentagonal tiling.

• 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
• 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

• Weisstein, Eric W., "Uniform tessellation", MathWorld.
• Weisstein, Eric W., "Semiregular tessellation", MathWorld.
• Richard Klitzing, 2D Euclidean tilings, elong( x3o6o ) - etrat - O4

Tessellation   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet   Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg   By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82