Snub square tiling Information

From Wikipedia
https://en.wikipedia.org/wiki/Snub_square_tiling
Snub square tiling
Snub square tiling
Type Semiregular tiling
Vertex configuration Tiling snub 4-4 left vertfig.svg
3.3.4.3.4
Schläfli symbol s{4,4}
sr{4,4} or
Wythoff symbol | 4 4 2
Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png or CDel node h.pngCDel split1-44.pngCDel nodes hh.png
Symmetry p4g, [4+,4], (4*2)
Rotation symmetry p4, [4,4]+, (442)
Bowers acronym Snasquat
Dual Cairo pentagonal tiling
Properties Vertex-transitive

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 Uniform tiling 44-h01.png
11212
Uniform tiling 44-snub.png
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 CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png

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:

Uniform tiling 44-t012.png
Regular octagons alternately truncated
(Alternate
truncation)
Nonuniform tiling 44-snub.png
Isosceles triangles (Nonuniform tiling)
Nonuniform tiling 44-t012-snub.png
Nonregular octagons alternately truncated
(Alternate
truncation)
Uniform tiling 44-snub.png
Equilateral triangles

Related tilings

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)
1-uniform n9.svg
[32434]
1-uniform n8.svg
[3342]
2-uniform n17.svg
[3342; 32434]
2-uniform n16.svg
[3342; 32434]
3-uniform 53.svg
[2: 3342; 32434]
3-uniform 55.svg
[3342; 2: 32434]
Vertex type 3-3-4-3-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-4-3-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-4-3-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-4-3-4.svg Vertex type 3-3-3-4-4.svg Vertex type 3-3-4-3-4.svg 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.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Spherical square antiprism.png Spherical snub cube.png Uniform tiling 44-snub.png H2-5-4-snub.svg Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
Spherical tetragonal trapezohedron.png Spherical pentagonal icositetrahedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg H2-5-4-floret.svg
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Digonal antiprism.png Pseudoicosahedron-3.png Uniform tiling 44-snub.png Uniform tiling 552-snub.png Uniform tiling 66-snub.png Uniform tiling 77-snub.png Uniform tiling 88-snub.png Uniform tiling ii2-snub.png
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Digonal trapezohedron.png Pyritohedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg Infinitely-infinite-order floret pentagonal tiling.png
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.3.∞
Uniform tilings based on square tiling symmetry
Symmetry: [4,4], (*442) [4,4]+, (442) [4,4+], (4*2)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 44-t0.svg Uniform tiling 44-t01.png Uniform tiling 44-t1.png Uniform tiling 44-t12.svg Uniform tiling 44-t2.png Uniform tiling 44-t02.png Uniform tiling 44-t012.png Uniform tiling 44-snub.png Uniform tiling 44-h01.png
{4,4} t{4,4} r{4,4} t{4,4} {4,4} rr{4,4} tr{4,4} sr{4,4} s{4,4}
Uniform duals
CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Uniform tiling 44-t0.png Tetrakis square tiling.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
V4.4.4.4 V4.8.8 V4.4.4.4 V4.8.8 V4.4.4.4 V4.4.4.4 V4.8.8 V3.3.4.3.4

See also

References

  1. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C
  2. ^ 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.
  3. ^ "Archived copy". Archived from the original on 2006-09-09. Retrieved 2006-09-09.{{ cite web}}: CS1 maint: archived copy as title ( link)

External links