Snub square tiling

Snub square tiling

Type Semiregular tiling
Vertex configuration
3.3.4.3.4
Schläfli symbol s{4,4}
sr{4,4} or ${\displaystyle s{\begin{Bmatrix}4\\4\end{Bmatrix}}}$
Wythoff symbol | 4 4 2
Coxeter diagram
or
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 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).[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 will produce a snub tiling with perfect equilateral triangle faces.

Example:

 Regular octagons alternately truncated →(Alternatetruncation) Isosceles triangles (Nonuniform tiling) Nonregular octagons alternately truncated →(Alternatetruncation) 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.

Fractalizing and Dissection

Every uniform planar tiling can be reduced to a tiling in equilateral triangles and squares by dissecting regular hexagons into six triangles and dissecting regular dodecagons into hexakis rhombitrihexagonal rotundas:

Dissecting a Hexagon Dissecting a Dodecagon
Insetting a Degree 6 Vertex Completely Insetting a Degree 12 Vertex

Nonetheless, the duals of the resulting tilings may also contain regular hexagons in addition to Cairo and prismatic pentagons.

For instance, starting from a truncated-hexagonal-fractalizing of the elongated square tiling, the dodecagons are progressively dissected in either orientation. The corresponding fractalizing dual is progressively inset. Below, we see that the kites (ties), skew quadrilaterals, and isosceles triangles are reduced to Cairo pentagons, prismatic pentagons, and hexagons; with most of the tiles Cairo pentagons:

Fractalizing the Elongated Triangular Tiling Dissecting some dodecagons in 1 orientation Dissecting all dodecagons in both orientations
Dual Fractalizing the Elongated Triangular Tiling Insetting some degree 12 vertices in 1 orientation Insetting all degree 12 vertices in both orientations

Related polyhedra and tilings

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.
The approximate dual tiling of the double snub square tiling. It contains four types of polygons, which are all pentagons. These pentagons are a mixture of Cairo-Floret (snub square = Cairo, snub hexagon = Floret). It is a non-uniform-4 tiling with 3-1 vertex configuration (turquoise and pink pentagons are topology-congruent).

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 (dual planigons to scale):[2][3]

 Unit of the Double Gyro Square Tiling 3.3.3.4.4 3.3.4.3.4 V3.3.3.4.4 V3.3.4.3.4

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

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

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)