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A Pythagorean tiling or two squares tessellation is a tessellation 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. Because of numerous proofs of the Pythagorean theorem based on such a tiling,[2] it is called a Pythagorean tiling.[1] It also is commonly used as a pattern for floor tiles; in this context it is also known as a hopscotch pattern.[3]

Topology and symmetry

A Pythagorean tiling (*)

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).[4]

Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons.[5] 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 symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic set of symmetries around the corresponding points, giving it p4 symmetry.[6] 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 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.[7]

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 by the ninth-century Arabic mathematicians Al-Nayrizi and Thābit ibn Qurra, and by the 19th-century British amateur mathematician Henry Perigal.[1][8][9][10] 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 of a right triangle 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 (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.[8]
Aperiodic cross sections
An aperiodic sequence generated from tilings by two squares whose side lengths form the golden ratio

Although the Pythagorean tiling is itself periodic (it has a square lattice of translational symmetries) its sections can be used to generate one-dimensional aperiodic sequences.[11]

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 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.[11]

If x is chosen as the golden ratio, the sequence of 0s and 1s generated in this way has the same recursive structure as the Fibonacci word: 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.[11]
Related results

According to Keller's conjecture, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge.[12] None of the squares in the Pythagorean tiling meet edge-to-edge,[4] 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 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 hypercubes of two different sizes.[13]

Burns and Rigby found several prototiles, including the Koch snowflake, that may be used to tile the plane only by using copies of the prototile in two or more different sizes.[14] An earlier paper by Danzer, Grünbaum, and Shephard provides another example, a convex pentagon that tiles the plane only when combined in two sizes.[15] 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.

Nelsen, Roger B. (November 2003), "Paintings, plane tilings, and proofs" (PDF), Math Horizons: 5–8. Reprinted in Haunsperger, Deanna; Kennedy, Stephen (2007), The Edge of the Universe: Celebrating Ten Years of Math Horizons, Spectrum Series, Mathematical Association of America, pp. 295–298, ISBN 978-0-88385-555-3. See also Alsina, Claudi; Nelsen, Roger B. (2010), Charming proofs: a journey into elegant mathematics, Dolciani mathematical expositions 42, Mathematical Association of America, pp. 168–169, ISBN 978-0-88385-348-1.
Wells, David (1991), "two squares tessellation", The Penguin Dictionary of Curious and Interesting Geometry, New York: Penguin Books, pp. 260–261, ISBN 0-14-011813-6.
Hopscotch: It's more than a kid's game (PDF), Tile Inc., August 2008.
Martini, Horst; Makai, Endre; Soltan, Valeriu (1998), "Unilateral tilings of the plane with squares of three sizes", Beiträge zur Algebra und Geometrie 39 (2): 481–495, MR 1642720.
Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, W. H. Freeman, p. 171.
Grünbaum & Shephard (1987), p. 42.
Grünbaum & Shephard (1987), pp. 73–74.
Frederickson, Greg N. (1997), Dissections: Plane & Fancy, Cambridge University Press, pp. 30–31.
Aguiló, Francesc; Fiol, Miquel Angel; Fiol, Maria Lluïsa (2000), "Periodic tilings as a dissection method", American Mathematical Monthly 107 (4): 341–352, doi:10.2307/2589179, MR 1763064.
Grünbaum & Shephard (1987), p. 94.
Steurer, Walter; Deloudi, Sofia (2009), " The Klotz construction", Crystallography of Quasicrystals: Concepts, Methods and Structures, Springer Series in Materials Science 126, Springer, pp. 91–92, doi:10.1007/978-3-642-01899-2, ISBN 978-3-642-01898-5.
The truth of his conjecture for two-dimensional tilings was known already to Keller, but it was since proven false for dimensions eight and above. For a recent survey on results related to this conjecture, see Zong, Chuanming (2005), "What is known about unit cubes", Bulletin of the American Mathematical Society, New Series 42 (2): 181–211, doi:10.1090/S0273-0979-05-01050-5, MR 2133310.
Bölcskei, Attila (2001), "Filling space with cubes of two sizes", Publicationes Mathematicae Debrecen 59 (3-4): 317–326, MR 1874434. See also Dawson (1984), which includes an illustration of the three-dimensional tiling, credited to "Rogers" but cited to a 1960 paper by Richard K. Guy: Dawson, R. J. M. (1984), "On filling space with different integer cubes", Journal of Combinatorial Theory. Series A 36 (2): 221–229, doi:10.1016/0097-3165(84)90007-4, MR 734979.
Burns, Aidan (1994), "78.13 Fractal tilings", Mathematical Gazette 78 (482): 193–196, JSTOR 3618577. Rigby, John (1995), "79.51 Tiling the plane with similar polygons of two sizes", Mathematical Gazette 79 (486): 560–561, JSTOR 3618091.
Figure 3 of Danzer, Ludwig; Grünbaum, Branko; Shephard, G. C. (1982), "Unsolved Problems: Can All Tiles of a Tiling Have Five-Fold Symmetry?", The American Mathematical Monthly 89 (8): 568–570+583–585, doi:10.2307/2320829, MR 1540019.

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