Английская Википедия:Elongated triangular tiling
Шаблон:Uniform tiles db 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 convex uniform tiling that can not 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) | |
|---|---|---|
| Файл:Elongated triangular tiling 1.png | Файл:Elongated triangular tiling 3.png | Файл:Elongated triangular tiling 2.png |
| 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
Sections of stacked triangles and squares can be combined into radial forms. This mixes two vertex configurations, 3.3.3.4.4 and 3.3.4.3.4 on the transitions. Twelve copies are needed to fill the plane with different center arrangements. The duals will mix in cairo pentagonal tiling pentagons.[3]
Symmetry mutations
It is first in a series of symmetry mutations[4] with hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n.4.3.3.3, and Coxeter diagram Шаблон:CDD. Their duals have hexagonal faces in the hyperbolic plane, with face configuration V4.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 |
| Файл:Elongated triangular tiling 4.2.4.3.3.3.png | Файл:Uniform tiling 4.3.4.3.3.3.png | Файл:Hyper 4.4.4.3.3.3a.png |
| Шаблон:CDD | Шаблон:CDD or Шаблон:CDD | Шаблон:CDD or Шаблон:CDD |
There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares.[5][6]
| Double elongated | Triple elongated | Half elongated | One third elongated |
|---|---|---|---|
| Файл:2-uniform n4.svg | Файл:2-uniform n3.svg | Файл:2-uniform n14.svg | Файл:2-uniform n15.svg |
Prismatic pentagonal tiling
Шаблон:Infobox face-uniform tiling 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 an Шаблон:Not a typo.[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:
| Файл:P5-type6.png | Файл:Prototile p5-type6.png 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 (the prismatic pentagon is half-square half-hexagon).
| Dual: Double Elongated | Dual: Triple Elongated | Dual: Half Elongated | Dual: One-Third Elongated |
|---|---|---|---|
| Файл:2-uniform 4 dual.svg | Файл:2-uniform 3 dual.svg | Файл:2-uniform 14 dual.svg | Файл:2-uniform 15 dual.svg |
| Dual: [44; 33.42]1 (t=2,e=4) | Dual: [44; 33.42]2 (t=3,e=5) | Dual: [36; 33.42]1 (t=3,e=4) | Dual: [36; 33.42]2 (t=4,e=5) |
See also
- Tilings of regular polygons
- Elongated triangular prismatic honeycomb
- Gyroelongated triangular prismatic honeycomb
Notes
References
Шаблон:Commons category Шаблон:Commons category
- Шаблон:Cite book (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- Шаблон:The Geometrical Foundation of Natural Structure (book) p37
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Шаблон:ISBN [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, Шаблон:ISBN, pp. 50–56
External links
- ↑ 1,0 1,1 Conway, 2008, p.288 table
- ↑ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern F
- ↑ aperiodic tilings by towers Andrew Osborne 2018
- ↑ Two Dimensional symmetry Mutations by Daniel Huson
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
