geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the snub square tiling is a

semiregular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Eucl ...

of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

. There are three triangles and two squares on each vertex. Its

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

is ''s''.

Conway Conway may refer to: Places United States * Conway, Arkansas * Conway County, Arkansas * Lake Conway, Arkansas * Conway, Florida * Conway, Iowa * Conway, Kansas * Conway, Louisiana * Conway, Massachusetts * Conway, Michigan * Conway Townshi ...

calls it a snub quadrille, constructed by a 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 consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

(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.)

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).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 snub 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 () 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, which alternates two types of edges. A truncated octagon, t is a ...

s and 1

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

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 polygon, 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 ...

, 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 general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

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-Strauss, ''The Symmetries of Things'' 2008,

* * (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * p38 * Dale Seymour and Jill Britton, ''Introduction to Tessellations'', 1989, , pp. 50–56, dual p. 115

External links

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