Elongated triangular tiling
|Elongated triangular tiling|
|Wythoff symbol||2 | 2 (2 2)|
|Symmetry||cmm, [∞,2+,∞], (2*22)
pgm, [(∞,2)+,∞], (22*)
pgg, [(∞,2)+,∞+], (22×)
|Rotation symmetry||p2, [∞,2,∞]+, (2222)|
|Dual||Prismatic pentagonal tiling|
Vertex figure: 188.8.131.52.4
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.
There are two uniform colorings of an elongated triangular tiling. (Naming the colors by indices around a vertex (184.108.40.206.4): 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)|
|cmm (2*22)||pgm (22*)||pgg (22×)|
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).
It is related to similarly constructed hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n.220.127.116.11, and Coxeter diagram . These duals have hexagonal faces, with face configuration V4.n.18.104.22.168.
Prismatic pentagonal tiling
|Prismatic pentagonal tiling|
|Type||Dual uniform tiling|
|Symmetry group||cmm, [∞,2+,∞], (2*22)|
|Rotation group||p2, [∞,2,∞]+, (2222)|
|Dual||Elongated triangular tiling|
It is the dual of the elongated triangular tiling.
- Tilings of regular polygons
- Elongated triangular prismatic honeycomb
- Gyroelongated triangular prismatic honeycomb
|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 
- Weisstein, Eric W., "Uniform tessellation", MathWorld.
- Weisstein, Eric W., "Semiregular tessellation", MathWorld.
- Richard Klitzing, 2D Euclidean tilings, elong( x3o6o ) - etrat - O4