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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 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
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
The 3-dimensional bitruncated cubic honeycomb 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.
:
The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed 
* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * * Dale Seymour and
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 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 '' Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619).
Notation of Eu ...
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 ...
with one 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 ...
and two 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 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 ...
. This is the only edge-to-edge tiling by 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 ...
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 a ...
s which contains an octagon. It has 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 ...
of ''t''.
Conway 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. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of th ...
(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 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 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, 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
:
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 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 longitudin ...
pattern also has the same topology, with 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 flattened rectangles.
Related polyhedra and tilings
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
The 3-dimensional bitruncated cubic honeycomb 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.
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: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 ...
, truncated square tiling, snub square tiling.
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 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 ...
with each square divided into four isosceles right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right ...
s from the center point, forming an infinite arrangement of lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestra ...
. 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 .
Conway calls it a kisquadrille, represented by a kis operation that adds a center point and triangles to replace the faces of 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). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices..
See also
*Tilings of 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 '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619).
Notation of ...
* List of uniform tilings
* Percolation threshold
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a ...
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''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 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
External links
* * {{Tessellation Euclidean tilings Isogonal tilings Semiregular tilings Square tilings Truncated tilings