Professor Stewart's Hoard of Mathematical Treasures (27 page)

There is a generalisation of the problem, to so-called Gaussian integers, which are complex numbers
p
+
q
i, where
p
and
q
are ordinary integers and i =
. Here, there exist two non-trivial powers that differ not by 1, but by i:

As far as I know, the corresponding conjecture - that this or minor variations are the only new cases where two Gaussian integer powers differ by 1, -1, i, or -i - remains open.

An extensive history of the problem can be found at:
www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf

The square root symbol
has a wonderfully arcane look, like something out of an ancient manuscript on alchemy. It’s the sort of symbol wizards would write, and formulas that contain it always look impressive and mysterious. But where did it come from?

Before 1400, European writers on mathematics generally used the word ‘radix’ for ‘root’ when referring to square roots. By the late medieval period, they abbreviated the word to its initial letter, a capital R with a short stroke through it:

The Renaissance Italian algebraists Girolamo Cardano, Luca
Pacioli, Rafael Bombelli and Tartaglia (Niccolò Fontana) all used this symbol.

The symbol √ is in fact a distorted letter r. How mundane! It first appeared in print in the first German algebra text, Christoff Rudolff’s Coss of 1525, but it took several centuries to become standard.

The site

www.roma.unisa.edu.au/07305/symbols.htm
discusses the history of many other mathematical symbols.

Q: What’s a polar bear?

A: A Cartesian bear after a change of coordinates.

I’m not making this up: that’s what it is called. It says that if you make a ham sandwich from two slices of bread and a slice of ham, then it is possible to cut the sandwich along some plane so that each of these three components is divided exactly in half, by volume.

Start with this . . .

. . . to get this - easy!

This is fairly obvious if the bread and the ham form nice square slabs, neatly arranged. It is less obvious if you appreciate that mathematicians are referring to generalised bread and ham,
which may take any shape whatsoever. (One immediate consequence is the cheese sandwich theorem, which might otherwise need a separate proof. Generality and power go hand in hand.)

A mathematician’s ham sandwich.

There are some technical conditions: in particular, the three pieces must not be so terribly complicated that they don’t have well-defined volumes (see Cabinet, page 163). In compensation, there is no requirement for a ‘piece’ to be connected - all in one lump, so to speak - but if it’s not you only have to divide the overall lump in half, not each separate part of it. Otherwise you’re trying to prove the ham and cheese sandwich theorem, which is false - see below.

The ham sandwich theorem is actually quite tricky to prove, and it is mostly an exercise in topology. To give you a flavour of the proof, I’ll show you how to deal with the simpler case of two shapes in two dimensions - the Flatland cheese-on-toast theorem.

Here’s the problem:
Here’s how to prove it can be solved. Pick a direction and find
a line pointing that way, which splits the cheese in half. It is not hard to prove that precisely one such line exists.

Start with a line in some direction (shown by the arrow) that splits the cheese in half.

Of course, unless you’ve got lucky, this line won’t split the toast in half too, but there will be two parts A and B on opposite sides of the line, with A on the left and B on the right if you look along the arrow. (Here B includes both chunks of the toast on that side. In general, either A or B might be empty - that doesn’t change the proof.) Suppose that, as shown, A has a bigger area than B.

Now gradually rotate the direction you’re thinking of, and do the same thing for each new direction.

Gradually rotate the line, always splitting the cheese in half.

Eventually you will have rotated the direction by 180°. Since only one line splits the cheese in half, this line must coincide with the original one, except that the arrow now points the other way:

Because the arrow points the other way, parts A and B of the toast have changed places. At the start, A was bigger than B, so now B must be bigger than A. (The pieces are the same as they were at the start, only the labels A and B have swapped.) However, the areas of A and B vary continuously as the angle of the line rotates. (This is where the topology comes in.) Since initially area(A) > area(B) and finally area(A) < area(B), there must be some angle in between for which area(A) = area(B). (Why? The difference area(A) - area(B) also varies continuously, starts positive, and ends negative. Somewhere in between it must be zero.) This proves the Flatland cheese-on-toast theorem.