Snub square tiling

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}.

Snub square tiling

TypeSemiregular tiling
Vertex configuration
3.3.4.3.4
Schläfli symbols{4,4}
sr{4,4} or
Wythoff symbol| 4 4 2
Coxeter diagram
or
Symmetryp4g, [4+,4], (4*2)
Rotation symmetryp4, [4,4]+, (442)
Bowers acronymSnasquat
DualCatalan_Cairo pentagonal tiling
PropertiesVertex-transitive

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
11212
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

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:


Regular octagons alternately truncated
→
(Alternate
truncation)
Isosceles triangles (Nonuniform tiling)

Nonregular octagons alternately truncated
→
(Alternate
truncation)
Equilateral triangles

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.

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)

[32434]
[3342]
[3342; 32434]
[3342; 32434]
[2: 3342; 32434]
[3342; 2: 32434]

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

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

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.CS1 maint: archived copy as title (link)
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