Dissecting a square into smaller squares all of different sizes

(Notes for the exhibition Very Important Knitting at the High Street Project Gallery in Christchurch)

Can you chop up a square into smaller squares all of different sizes? It's easy enough to chop a square into smaller squares all the same size, but all of different sizes? That was the question addressed by the Important Persons of the Trinity Mathematical Society. (TMS) in 1936-8. (The Important Persons were the committee of the TMS, and the TMS was—and still is—a society for mathematics students at the University of Cambridge who belong to Trinity College.)

One of the Important Persons was William Tutte , who went on to do Important Things in cryptography in WWII. (I could tell you about them but I would then have to kill you.) He was a friend of Martin Gardner, the world's first (and probably only) mathematical journalist, who wrote up their story in More Mathematical Puzzles and Diversions (London G. Bell and Sons 1963 (pp 136--153)).

En route to finding a square dissected into squares the Important Persons tried various strategies, one of which was to draw rectangles dissected into squares, and then—by looking at how the squares fitted together—try to deduce what the lengths of the sides must be.

The knitted square that you see below and on the left is a realisation of one of these experimental rectangles, with the working still showing. You will see inside each square a formula in x and y. Two of the squares—one might think of them as the seeds— have the simplest possible formulae inside them: x and y. The formula inside each square gives you the linear dimensions of that square in terms of the linear dimensions of the two seed squares. If you examine them carefully you can see how the lengths of many of the sides can be deduced from the assumption that the length-of-side of the two seed squares are x and y. (The whole rectangle is not actually an exact square, being 176 by 177 rather than 176 by 176, but the difference is not apparent to the naked eye.) The long side of the rectangle on the top right must be both 14y - 3x and (3x - 3y) + (3x + y), from which we deduce that x = 16 and y = 9. The reason why you do not see the intended companion piece—in which the squares are supplied in colour and without the workings—is that the companion piece was stolen! In fact over the years I have knitted no fewer three versions of this rectangle, and the first two were stolen. (The third panel was presented to the committee of the TMS and is now at a secret location in Trinity College Cambridge, and that is the one you see below). The first one was stolen by my sister from my parents' house, and the second one was stolen from the Department of Philosophy and Religious Studies at Canterbury University in Christchurch (the Christchurch constabulary have not closed the file on this theft, and this makes me the only Mathematical Logic Ph.D. in the world one of whose works is on the register of stolen art). Perhaps it was stolen by a Religious Studies student who was under the impression that it has magical powers, like the magic squares or the Cirencester word square;

S  A  T  O  R
A  R  E  P  O
T  E  N  E  T
O  P  E  R  A
R  O  T  A  S

...perhaps the thief has made it into an altarpiece somewhere and is—even as we speak—sacrificing babies in front of it to make the sun come up. I am not sure that I want to find out.

Fortunately these squares brought to you by The Important Persons are not from the Age of Magic. In the Ancient world and well into the Middle Ages people believed that numbers had magical properties, and could be used in spells and the like. When St Augustine warned his readers against mathematicians, he was really talking about numerologists and astrologers. The question of whether or not we could dissect a square in this way is one that the mathematicians of Ancient Greece would have understood had they encountered it, but there is no reason to suppose they ever did. It's tempting to speculate about what the Ancient World would have made of the Important Persons' squares had they found any. Given the intellectual climate of the time it would surely have been incorporated into religion or popular superstition—somehow. The Greeks wrote about politics and they thought about how to design a perfect society. Perhaps this square—dissected into little squares which are all of different sizes and are all in precisely the right place—would have come to represent how perfect individuals can be fitted together to form a perfect society.

(Annoyingly the right-hand knitted square needs to be reflected in a horizontal axis if the two squares are to match...)