In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the Pythagorean tiling or two squares tessellation is a tessellation

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

 of the plane by square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s of two different sizes, in which each square touches four squares of the other size on its four sides. A tiling of this type may be formed by squares of any two different sizes. It also is commonly used as a pattern for floor tiles; in this context it is also known as a hopscotch pattern.

Topology and symmetry

The Pythagorean tiling is the unique tiling by squares of two different sizes that is both unilateral (no two squares have a common side) and equitransitive (each two squares of the same size can be mapped into each other by a symmetry of the tiling).

Topologically, the Pythagorean tiling has the same structure as the truncated square tiling

Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon...

 by squares and regular octagons. The smaller squares in the Pythagorean tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling, however, the two tilings have different sets of symmetries: the truncated square tiling has dihedral

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

 symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 set of symmetries around the corresponding points, giving it p4 symmetry. It is a chiral pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations.

A uniform tiling

Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane...

 is a tiling in which each tile is a regular polygon and in which there is a symmetry, mapping every vertex to every other vertex. Usually, uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed then there are eight additional uniform tilings: four formed from infinite strips of squares or equilateral triangles, three formed from equilateral triangles and regular hexagons, and one more, the Pythagorean tiling.

Pythagorean theorem and dissections

This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

 by the ninth-century Arabic mathematicians Al-Nayrizi

Al-Nayrizi

Abū’l-‘Abbās al-Faḍl ibn Ḥātim al-Nairīzī was a 9th-10th century Persian mathematician and astronomer from Nayriz, Fars, Iran.He flourished under al-Mu'tadid, Caliph from 892 to 902, and compiled astronomical tables, writing a book for al-Mu'tadid on atmospheric phenomena.Nayrizi wrote...

 and Thābit ibn Qurra

Thabit ibn Qurra

' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...

, and by the 19th-century British amateur mathematician Henry Perigal

Henry Perigal

Henry Perigal, Jr. FRAS MRI was a British stockbroker and amateur mathematician, known for his dissection-based proof of the Pythagorean theorem and for his unorthodox belief that the moon does not rotate.-Biography:...

. If the sides of the two squares forming the tiling are the numbers a and b, then the closest distance between corresponding points on congruent squares is c, where c is the length of the hypotenuse

Hypotenuse

In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...

 of a right triangle

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

 having sides a and b. For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...

 (5,12,13).

By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides a and b into a single square of side c, showing that the two smaller squares have the same area as the larger one. Similarly, overlaying two Pythagorean tilings may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares.

Aperiodic tilings

Although the Pythagorean tiling is itself periodic (it has a square lattice

Square lattice

In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...

 of translational symmetries) its sections can be used to generate one-dimensional aperiodic

Aperiodic tiling

An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. Properly speaking, aperiodicity is a property of particular sets of tiles; any given finite tiling is either periodic or non-periodic...

 sequences.

In the "Klotz construction" for aperiodic sequences (Klotz is a German word for a block), one forms a Pythagorean tiling with two squares for which the ratio between the two side lengths is an irrational number

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 x. Then, one chooses a line parallel to the sides of the squares, and forms a sequence of binary values from the sizes of the squares crossed by the line: a 0 corresponds to a crossing of a large square and a 1 corresponds to a crossing of a small square. In this sequence, the relative proportion of 0s and 1s will be in the ratio x:1. This proportion cannot be achieved by a periodic sequence of 0s and 1s, because it is irrational, so the sequence is aperiodic.

If x is chosen as the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

, the sequence of 0s and 1s generated in this way has the same recursive structure as the Fibonacci word

Fibonacci word

thumb|350px|Characterization by a [[cutting sequence]] with a line of slope 1/\varphi or \varphi-1, with \varphi the [[golden ratio]].A Fibonacci word is a specific sequence of binary digits...

: it can be split into substrings of the form "01" and "0" (that is, there are no two consecutive ones) and if these two substrings are consistently replaced by the shorter strings "0" and "1" then another string with the same structure results.

Related results

According to Keller's conjecture

Keller's conjecture

In geometry, Keller's conjecture is the conjecture introduced by that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to...

, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge. None of the squares in the Pythagorean tiling meet edge-to-edge, but this fact does not violate Keller's conjecture because the tiles are not all congruent to each other.

The Pythagorean tiling may be generalized to a three-dimensional tiling of Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 by cubes of two different sizes, which also is unilateral and equitransitive. Attila Bölcskei calls this three-dimensional tiling the Rogers filling. He conjectures that, in any dimension greater than three, there is again a unique unilateral and equitransitive way of tiling space by hypercube

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

s of two different sizes.

Burns and Rigby found several prototile

Prototile

In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others...

s, including the Koch snowflake

Koch snowflake

The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...

, that may be used to tile the plane only by using copies of the prototile in two or more different sizes. An earlier paper by Danzer, Grünbaum, and Shephard provides another example, a convex pentagon that tiles the plane only when combined in two sizes. Although the Pythagorean tiling uses two different sizes of squares, the square does not have the same property as these prototiles of only tiling by similarity, because it also is possible to tile the plane using only squares of a single size.