Perfect Squared Squares
It is easy to tile a square area using smaller square tiles that are all the same size as each other, but tiling a square area using smaller square tiles that are all of different sizes
is much harder.
This picture shows one possible way to tile a 1376 by 1376 square using smaller square tiles that are all of different sizes. The numbers in each smaller square denote the length of each of its sides, as opposed to its area. The solution in the picture has another remarkable property: it is not possible to form a rectangle from the union of a subset of the tiles, except in trivial ways.
A rectangle that is partitioned into smaller squares is known as a squared rectangle
. If the rectangle happens to be square, then it is known as a squared square
. If all of the smaller squares have different sizes, the squared square is called perfect
, and if no proper nontrivial union of the smaller squares forms a rectangle, the squared square is called simple
. The configuration in the picture is therefore an example of a simple perfect squared square
, or SPSS for short.
The history of perfect squared squares is surprisingly long. It starts in 1902 when H.E. Dudeney
published a puzzle called Lady Isabel's Casket
. The puzzle involves partitioning a square into different sized smaller squares together with a rectangle. In 1903, Max Dehn
proved the key result that a rectangle can be tiled by squares if and only if its length is a rational multiple of its width. This result can be used to show that if a tiling by squares is possible, then the rectangle can be scaled in such a way that all of its side lengths, as well as all the side lengths of its constituent squares, are integers. Since 1903, a large number of examples of squared rectangles have been produced by many mathematicians. The history of the subject is surprisingly rich, and is recounted in detail in the paper http://arxiv.org/abs/1303.0599
by Stuart E. Anderson
The simple perfect squared square in the picture was discovered in 2013 by James B. Williams
using a computer search. It has the distinctive feature that none of the seven largest constituent squares (shaded in purple) appears as a corner piece (shaded in green).
Now I know what you're thinking. What happens if you try to construct a similar type of square tiling on a cylinder? Or a Möbius strip? Or a Klein bottle? Or a projective plane? And what happens if we use triangles instead of squares? Well, you'll be pleased to know that the site http://www.squaring.net
treats all these cases in detail. (You should have a look at the site, because it probably contains more detail than you are expecting.) The illustration here comes from that website, which is maintained by Stuart E. Anderson.
On the other hand, if you're wondering why people would want to study this, I don't have a good answer, except that it is challenging and aesthetically appealing.#mathematics #sciencesunday #spnetwork