Snub square tiling

Snub square tiling

This coloring has xx (pg) symmetry
Type Semiregular tiling
Vertex configuration 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)
p4, [4,4]+, (442)
Rotation symmetry p4, [4,4]+, (442)
Bowers acronym Snasquat
Dual Cairo pentagonal tiling
Properties Vertex-transitive

Vertex figure: 3.3.4.3.4

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 Symmetry 11212 11213 4*2, [4+,4], (p4g) 442, [4,4]+, (p4) s{4,4} sr{4,4} | 4 4 2

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).

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) Isosceles triangles (Nonuniform tiling) 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.

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

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
242
[2,4]+
342
[3,4]+
442
[4,4]+
542
[5,4]+
642
[6,4]+
742
[7,4]+
842
[8,4]+...
∞42
[∞,4]+
Snub
figure

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.∞
Coxeter
Schläfli

sr{2,4}

sr{3,4}

sr{4,4}

sr{5,4}

sr{6,4}

sr{7,4}

sr{8,4}

sr{∞,4}
Snub
dual
figure

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.∞
Coxeter

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

Dimensional family of snub polyhedra and tilings: 3.3.n.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
222
[2,2]+
322
[3,3]+
442
[4,4]+
552
[5,4]+
662
[6,6]+
772
[7,7]+
882
[8,8]+...
∞∞2
[∞,∞]+
Snub
figure

3.3.2.3.2

3.3.4.3.3

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.∞
Coxeter
Schläfli

sr{2,2}

sr{3,3}

sr{4,4}

sr{5,5}

sr{6,6}

sr{7,7}

sr{8,8}

sr{∞,∞}
Snub
dual
figure

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.∞
Coxeter
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

Cairo pentagonal tiling

The dual of the snub square tiling is the Cairo pentagonal tiling, given its name because several streets in Cairo are paved in this design.[1][2] It is one of 14 known isohedral pentagon tilings.

It is also called MacMahon's net[3] after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes.[4]

Conway calls it a 4-fold pentille.[5]