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

1. 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

• 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