geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the snub square tiling is a

semiregular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...

of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

. There are three triangles and two squares on each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

. Its

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

is ''s''. Conway calls it a snub quadrille, constructed by a

snub A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub. In Awards and Lists For awards, the term "snub" ...

operation applied to a

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

(quadrille). There are 3

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

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.)

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 In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

).Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C : 1-uniform-9-circlepack

Wythoff construction

The snub square tiling can be constructed as a

operation from the

, 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

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A ''regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, wh ...

s and 1

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

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 In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symm ...

, will produce a snub tiling with perfect equilateral triangle faces. Example:

Related tilings

File:Snub snub square tiling.svg, 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. File:Isogonal snub square tiling-8x8.svg, A related isogonal tiling that combines pairs of triangles into rhombi File:Triangular heptagonal tiling.svg, A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons. File:P2_dual.png, The Cairo pentagonal tiling is dual to the snub square tiling.

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.

Related topological series of polyhedra and tiling

The ''snub square tiling'' is third in a series of snub polyhedra and tilings with

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

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

3.3.''n''.3.''n''.

References

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008,

* * (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * p38 * Dale Seymour and

Jill Britton Jill E. Britton (6 November 1944 – 29 February 2016) was a Canadian mathematics educator known for her educational books about mathematics. Career Britton was born on 6 November 1944. She taught for many years, at Dawson College in Westmount ...

, ''Introduction to Tessellations'', 1989, , pp. 50–56, dual p. 115

External links

* {{Tessellation Euclidean tilings Isogonal tilings Semiregular tilings Square tilings Snub tilings