Snub square tiling

Snub square tiling
Snub square tiling
Type	Semiregular tiling
Vertex configuration	120px 3.3.4.3.4
Schläfli symbol	s{4,4} sr{4,4}
Wythoff symbol	| 4 4 2
Coxeter diagram
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. It has Schläfli symbol of 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	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).

320px

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 will produce a snub tiling with perfect equilateral triangle faces.

Example:

Regular octagons alternately truncated	→ (Alternate truncation)	150px Isosceles triangles (Nonuniform tiling)
150px Nonregular octagons alternately truncated	→ (Alternate truncation)	Equilateral triangles

Related 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.

The snub square tiling can be seen related to this 3-colored square tiling, with the yellow and red squares being twisted rigidly and the blue tiles being distorted into rhombi and then bisected into two triangles.

Related polyhedra and tilings

The snub square tiling is similar to the elongated triangular tiling with vertex configuration 3.3.3.4.4, and two 2-uniform dual tilings and 2 3-uniform duals which mix the two types of pentagons:

80px 3.3.3.4.4

80px 3.3.4.3.4

snub square tiling	2-uniform
p4g, (4*2)	p2, (2222)	cmm, (2*22)
150px 3.3.4.3.4	150px (3.3.3.4.4; 3.3.4.3.4)	150px (3.3.3.4.4; 3.3.4.3.4)
Elongated triangular tiling	3-uniform
cmm, (2*22)	p2, (2222)
3.3.3.4.4	150px (3.3.3.4.4; 3.3.4.3.4)	150px (3.3.3.4.4; 3.3.4.3.4)

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
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
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
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								60px
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)
{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
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

• List of uniform planar tilings
• Snub square prismatic honeycomb
• Tilings of regular polygons
• Elongated triangular tiling

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
• Richard Klitzing, 2D Euclidean tilings, s4s4s - snasquat - O10
• (Chapter 2.1: Regular and uniform tilings, p. 58-65)
• p38
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 115

External links

• Weisstein, Eric W., "Semiregular tessellation", MathWorld.