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 truncated square tiling is a semiregular

tiling by regular polygons 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 ...

with one

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 ...

and two

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 on each vertex. This is the only edge-to-edge tiling by

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

s which contains an octagon. It has

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 ...

of ''t''.

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 truncated quadrille, constructed as a

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

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). Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges. There are 3

and 8 semiregular tilings in the plane.

Uniform colorings

There are two distinct

uniform coloring In geometry, a uniform coloring is a property of a uniform figure ( uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following diff ...

s of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)

Circle packing

The truncated square tiling can be used as a

circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...

, placing equal diameter circles at the center of every point. Every circle is in contact with 3 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 H : 1-uniform-2-circlepack

Variations

One variations on this pattern, often called a ''Mediterranean pattern'', is shown in stone tiles with smaller squares and diagonally aligned with the borders. Other variations stretch the squares or octagons. The

Pythagorean tiling A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are ...

alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices. A

weaving Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, crocheting, felting, and braiding or plaiting. The longitudinal ...

pattern also has the same topology, with

s flattened

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

Related polyhedra and tilings

Optical-illusion-checkerboard-twisted-cord2

The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with

vertex figures 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines acr ...

4.2n.2n, extending into the hyperbolic plane: The 3-dimensional

bitruncated cubic honeycomb The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of truncated octahedron, truncated octahedra (or, equivalently, Bitruncation (geometry), bitruncated cubes). It has 4 ...

projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling, or combined can be seen as a

chamfered square tiling In geometry, the chamfered square tiling or semitruncated square tiling is a tiling of the Euclidean plane. It is a square tiling with each edge Chamfer (geometry), chamfered into new hexagonal faces. It can also be seen as the intersection of tw ...

Wythoff constructions from square tiling

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms:

, truncated square tiling,

snub square tiling 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''. Conway calls it a snub quadrille, constructed by a snub operation appli ...

Related tilings in other symmetries

Tetrakis square tiling

: The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed

with each square divided into four

isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides ...

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

s from the center point, forming an infinite

arrangement of lines In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons, the ''cells'' of the arrangement, line segments and rays, the ''edg ...

. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of .

calls it a kisquadrille, represented by a kis operation that adds a center point and triangles to replace the faces of a

(quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.. P1 dual

References

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

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

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

, ''Introduction to Tessellations'', 1989, , pp. 50–56

External links

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