Squaring the square

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The first perfect squared square discovered, a compound one of side 4205 and order 55. Each number denotes the side length of its square. Sprague squared square.svg
The first perfect squared square discovered, a compound one of side 4205 and order 55. Each number denotes the side length of its square.

Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

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Perfect squared squares

Smith diagram of a rectangle Smith diagram.png
Smith diagram of a rectangle

A “perfect” squared square is a square such that each of the smaller squares has a different size.

It is first recorded as being studied by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte at Cambridge University between 1936 and 1938. They transformed the square tiling into an equivalent electrical circuit — they called it a "Smith diagram" — by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit. The first perfect squared squares they found were of order 69.

The first perfect squared square to be published, a compound one of side 4205 and order 55, was found by Roland Sprague in 1939. [2]

Martin Gardner published an extensive article written by W. T. Tutte about the early history of squaring the square in his mathematical games column in November 1958. [3]

Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2-4) - all are simple squared squares Smallest perfect squared squares.svg
Lowest-order perfect squared square (1) and the three smallest perfect squared squares (24) all are simple squared squares

Simple squared squares

A "simple" squared square is one where no subset of more than one of the squares forms a rectangle or square, otherwise it is "compound".

In 1978, A. J. W. Duijvestijn  [ de ] discovered a simple perfect squared square of side 112 with the smallest number of squares using a computer search. His tiling uses 21 squares, and has been proved to be minimal. [4] This squared square forms the logo of the Trinity Mathematical Society. It also appears on the cover of the Journal of Combinatorial Theory.

Duijvestijn also found two simple perfect squared squares of sides 110 but each comprising 22 squares. Theophilus Harding Willcocks, an amateur mathematician and fairy chess composer, found another. In 1999, I. Gambini proved that these three are the smallest perfect squared squares in terms of side length. [5]

The perfect compound squared square with the fewest squares was discovered by T.H. Willcocks in 1946 and has 24 squares; however, it was not until 1982 that Duijvestijn, Pasquale Joseph Federico and P. Leeuw mathematically proved it to be the lowest-order example. [6]

Mrs. Perkins's quilt

When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the greatest common divisor of all the smaller side lengths should be 1.

The Mrs. Perkins's quilt problem is to find a Mrs. Perkins's quilt with the fewest pieces for a given n × n square.

No more than two different sizes

A square cut into 10 pieces (an HTML table)
    
    
  

A cute number means a positive integer n such that some square admits a dissection into n squares of no more than two different sizes, without other restrictions. It can be shown that aside from 2, 3, and 5, every positive integer is cute. [7]

Squaring the plane

Tiling the plane with different integral squares using the Fibonacci series
1. Tiling with squares with Fibonacci-number sides is almost perfect except for 2 squares of side 1.
2. Duijvestijn found a 110-square tiled with 22 different integer squares.
3. Scaling the Fibonacci tiling by 110 times and replacing one of the 110-squares with Duijvestijn's perfects the tiling. Squaring the plane.svg
Tiling the plane with different integral squares using the Fibonacci series
1. Tiling with squares with Fibonacci-number sides is almost perfect except for 2 squares of side 1.
2. Duijvestijn found a 110-square tiled with 22 different integer squares.
3. Scaling the Fibonacci tiling by 110 times and replacing one of the 110-squares with Duijvestijn's perfects the tiling.

In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares, one of each integer edge-length, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his Scientific American column and appeared in several books, but it defied solution for over 30 years.

In Tilings and Patterns, published in 1987, Branko Grünbaum and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares grew exponentially. For example, the plane can be tiled with different integral squares, but not for every integer, by recursively taking any perfect squared square and enlarging it so that the formerly smallest tile now has the size of the original squared square, then replacing this tile with a copy of the original squared square.

In 2008 James Henle and Frederick Henle proved that this, in fact, can be done. [8] Their proof is constructive and proceeds by "puffing up" an L-shaped region formed by two side-by-side and horizontally flush squares of different sizes to a perfect tiling of a larger rectangular region, then adjoining the square of the smallest size not yet used to get another, larger L-shaped region. The squares added during the puffing up procedure have sizes that have not yet appeared in the construction and the procedure is set up so that the resulting rectangular regions are expanding in all four directions, which leads to a tiling of the whole plane.

Cubing the cube

Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a cube C, the problem of dividing it into finitely many smaller cubes, no two congruent.

Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a rectangular cuboid C into a finite number of unequal cubes.

To prove this, we start with the following claim: for any perfect dissection of a rectangle in squares, the smallest square in this dissection does not lie on an edge of the rectangle. Indeed, each corner square has a smaller adjacent edge square, and the smallest edge square is adjacent to smaller squares not on the edge.

Now suppose that there is a perfect dissection of a rectangular cuboid in cubes. Make a face of C its horizontal base. The base is divided into a perfect squared rectangle R by the cubes which rest on it. The smallest square s1 in R is surrounded by larger, and therefore higher, cubes. Hence the upper face of the cube on s1 is divided into a perfect squared square by the cubes which rest on it. Let s2 be the smallest square in this dissection. By the claim above, this is surrounded on all 4 sides by squares which are larger than s2 and therefore higher.

The sequence of squares s1, s2, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition. [9]

If a 4-dimensional hypercube could be perfectly hypercubed then its 'faces' would be perfect cubed cubes; this is impossible. Similarly, there is no solution for all cubes of higher dimensions.

See also

Related Research Articles

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Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid.

Cube A geometric 4-dimensional object with 6 square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Pentomino Geometric shape formed from five squares

Derived from the Greek word for '5', and "domino", a pentomino is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.

Rhombicuboctahedron Archimedean solid with 26 faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

Rectangle Quadrilateral with four right angles

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

Polyomino Geometric shapes formed from squares

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.

In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cube. In math language a cuboid is convex polyhedron, whose polyhedral graph is the same as that of a cube.

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Prism (geometry) Solid with parallel bases connected by parallelograms

In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Rhombicosidodecahedron Archimedean solid

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Tessellation Tiling of a plane in mathematics

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Hexomino Geometric shape formed from six squares

A hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes. When reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes.

Polyabolo Shape formed from isosceles right triangles

In recreational mathematics, a polyabolo is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gardner in his June 1967 "Mathematical Games column" in Scientific American.

Tromino Geometric shape formed from three squares

A tromino is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.

Nonomino Geometric shape formed from nine squares

A nonomino is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different free nonominoes. When reflections are considered distinct, there are 2,500 one-sided nonominoes. When rotations are also considered distinct, there are 9,910 fixed nonominoes.

Plastic number Algebraic number, approximately 1.325

In mathematics, the plastic numberρ is a mathematical constant which is the unique real solution of the cubic equation

Rhombille tiling Tiling of the plane with 60° rhombi

In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.

Domino tiling Geometric construct

In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.

Pythagorean tiling Tiling by squares of two sizes

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 based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.

Rep-tile Shape subdivided into copies of itself

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

References

  1. "o55-4205-sprague.pdf" (PDF). Retrieved 25 August 2015.
  2. "5. Towards a theory for combinatorial games". American Mathematical Society . Retrieved 2017-06-30.
  3. "Brooks, Smith, Stone and Tutte, II". www.squaring.net. Retrieved 19 April 2018.
  4. W., Weisstein, Eric. "Perfect Square Dissection". mathworld.wolfram.com. Retrieved 19 April 2018.
  5. Gambini, Ian (1999). "A method for cutting squares into distinct squares". Discrete Applied Mathematics . 98 (1–2): 65–80. doi: 10.1016/S0166-218X(99)00158-4 . MR   1723687.
  6. Duijvestijn, A. J. W.; Federico, P. J.; Leeuw, P. (1982). "Compound perfect squares". American Mathematical Monthly . 89 (1): 15–32. doi:10.2307/2320990. JSTOR   2320990. MR   0639770.
  7. Henry, JB; Taylor, PJ (2009). Challenge! 1999 - 2006 Book 2. Australian Mathematics Trust. p. 84. ISBN   978-1-876420-23-9.
  8. Henle, Frederick V.; Henle, James M. (2008). "Squaring the plane". American Mathematical Monthly. 115 (1): 3–12. doi:10.1080/00029890.2008.11920491. JSTOR   27642387. S2CID   26663945.
  9. Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; Tutte, W. T. (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:10.1215/S0012-7094-40-00718-9. MR   0003040.

Further reading