Snub square tiling
Template:Uniform tiling stat table In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.
Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)
| Coloring | File:Uniform tiling 44-h01.svg 11212 |
File:Uniform tiling 44-snub.svg 11213 |
|---|---|---|
| Symmetry | 4*2, [4+,4], (p4g) | 442, [4,4]+, (p4) |
| Schläfli symbol | s{4,4} | sr{4,4} |
| Wythoff symbol | 4 4 2 | |
| Coxeter diagram | Template:CDD | Template:CDD |
Circle packing
The snub square 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).[1]
Wythoff construction
The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.
An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.
If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.
Example:
| File:Uniform tiling 44-t012.png Regular octagons alternately truncated |
→ (Alternatetruncation) |
File:Nonuniform tiling 44-snub.png Isosceles triangles (Nonuniform tiling) |
| File:Nonuniform tiling 44-t012-snub.svg Nonregular octagons alternately truncated |
→ (Alternatetruncation) |
File:Uniform tiling 44-snub.svg Equilateral triangles |
Related tilings
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A snub operator applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.
-
A related isogonal tiling that combines pairs of triangles into rhombi
-
A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.
-
The Cairo pentagonal tiling is dual to the snub square tiling.
Related k-uniform tilings
This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many k-uniform tilings.[2][3]
| Related tilings of triangles and squares | ||||||
|---|---|---|---|---|---|---|
| snub square | elongated triangular | 2-uniform | 3-uniform | |||
| p4g, (4*2) | p2, (2222) | p2, (2222) | cmm, (2*22) | p2, (2222) | ||
| File:1-uniform n9.svg [32434] |
File:1-uniform n8.svg [3342] |
File:2-uniform n17.svg [3342; 32434] |
File:2-uniform n16.svg [3342; 32434] |
File:3-uniform 53.svg [2: 3342; 32434] |
File:3-uniform 55.svg [3342; 2: 32434] | |
| File:Vertex type 3-3-4-3-4.svg | File:Vertex type 3-3-3-4-4.svg | File:Vertex type 3-3-3-4-4.svg File:Vertex type 3-3-4-3-4.svg | File:Vertex type 3-3-3-4-4.svg File:Vertex type 3-3-4-3-4.svg | File:Vertex type 3-3-3-4-4.svg File:Vertex type 3-3-3-4-4.svg File:Vertex type 3-3-4-3-4.svg | File:Vertex type 3-3-3-4-4.svg File:Vertex type 3-3-4-3-4.svg File:Vertex type 3-3-4-3-4.svg | |
Related topological series of polyhedra and tiling
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. Template:Snub4 table The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n. Template:Snub5 table
See also
- List of uniform planar tilings
- Snub (geometry)
- Snub square prismatic honeycomb
- Tilings of regular polygons
- Elongated triangular tiling
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:Isbn [1]
- Template:KlitzingPolytopes
- Script error: No such module "citation/CS1". (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- Template:The Geometrical Foundation of Natural Structure (book) p38
- Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, Template:Isbn, pp. 50–56, dual p. 115
External links
- Script error: No such module "Template wrapper".