Squaring the square
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A square with sides equal to a unit length multiplied by an integer is called an integral square. Squaring the square is the problem of tiling one integral square using only other integral squares.
Squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square, where all contained squares are of different size (see below).
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[edit] Perfect squared squares
A "perfect" squared square is a square such that each of the smaller squares has a different size. The name was coined in humorous analogy with squaring the circle.
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. They transformed the square tiling into an equivalent electrical circuit — they called it "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 square was found by Roland Sprague in 1939.
If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.
It is still an unsolved problem, however, whether the plane can be tiled with a set of integral tiles such that each natural number is used exactly once as size of a square tile.
Martin Gardner has published an extensive article written by W. T. Tutte about the early history of squaring the square.
[edit] Simple squared squares
A "simple" squared square is one where no subset of the squares forms a rectangle or square, otherwise it is "compound". The smallest simple perfect squared square was discovered by A. J. W. Duijvestijn using a computer search. His tiling uses 21 squares, and has been proved to be minimal. The smallest perfect compound squared square was discovered by T.H. Willcocks and has 24 squares.
[edit] 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.
[edit] External links
[edit] 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 but solvable problem, cubing the cube is impossible. This can be shown by a relatively simple argument. Consider a hypothetical cubed cube. The bottom face of this cube is a squared square. If you take this squared square, and consider the smallest square S in it, and stand cubes of the same side on each of the squares, they 'tower over' the cube standing at S, and surround it since the smallest square of a squared square cannot be on its edge. Since it is not allowed to place another cube of the same size atop this one, and the cubes on the sides obstruct the placement of a larger cube, only smaller cubes may stand upon S. This means that the top face of S must be a squared square, and the argument continues by infinite descent. Thus it is not possible to dissect a cube into finitely many smaller cubes of different sizes.
[edit] References
- Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. The Dissection of Rectangles into Squares, Duke Math. J. 7, 312-340, 1940
- Martin Gardner, "Squaring the square," in The 2nd Scientific American Book of Mathematical Puzzles and Diversions.
- C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25, Eindhoven Univ. Technology, Dept. of Math., Report 92-WSK-03, Nov. 1992.
- C.J.Bouwkamp and A.J.W.Duijvestijn, Album of Simple Perfect Squared Squares of order 26, Eindhoven University of Technology, Faculty of Mathematics and Computing Science, EUT Report 94-WSK-02, December 1994.
[edit] External links
- Perfect squared squares:
- Nowhere-neat squared squares:
