Of course squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square (see below). Other possible conditions that lead to interesting results are nowhere neat squared squares and no-touch squared squares (see tiling).
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2 Simple squared squares 3 Mrs. Perkins' Quilt 4 External links |
A "perfect" squared square is such 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, and then applied Kirchhoff's 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 written an extensive article about the early history of squaring the square.
Perfect Squared Squares
Lowest order perfect square |
A "simple" squared square is one where no subset of the squares forms a rectangle. The smallest simple perfect squared square was discovered by A. J. W. Duijvestin using a computer search. His tiling uses 21 squares, and has been proved to be minimal.
When the constraint of all the squared being different sizes is relaxed, the resulting squared square problem is often called the "Mrs. Perkins' Quilt" problem.
References:
Perfect Squares:
Simple squared squares
Mrs. Perkins' Quilt
External links