Professor Stewart's Hoard of Mathematical Treasures (8 page)

Same edges, different volumes.

I’d better point out that the existence of this formula does not imply that the volume of a polyhedron is uniquely determined by the lengths of its edges. A house with a roof has a smaller volume if you turn the roof upside down. These are two
different solutions of the same polynomial equation, and that causes no problems in the proof of the bellows conjecture - you can’t flex the roof into the downward position without bending something.

The number 153 is equal to the sum of the cubes of its digits:

There are three other 3-digit numbers with the same property, excluding numbers like 001 with a leading zero. Can you find them?

 

Answer on page 283

In his celebrated book A Mathematician’s Apology of 1940, the English mathematician Godfrey Harold Hardy had this to say about the digital cubes puzzle:

‘This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician ... One reason ... is the extreme speciality of both the enunciation and the proof, which is not capable of any significant generalisation.’

In his 1962 book Profiles of the Future, Arthur C. Clarke stated three laws about prediction. The first is:

This is called Clarke’s first law, or often just Clarke’s law, and there are good reasons to claim that it applies to Hardy’s statement. To be fair, the point Hardy was trying to make is a good one, but you can pretty much guarantee that, whenever anyone cites a specific example to drive such an argument home, it will turn out to be a bad choice. In 2007, a trio of mathematicians - Alf van der Poorten, Kurth Thomsen and Mark Weibe - took an imaginative look at Hardy’s assertion. Here’s what they found.

It was all triggered by a ‘cute observation’ made by the number-theorist Hendrik Lenstra:

This is about squares, not cubes, but it hints that maybe there is more to this sort of question than first meets the eye. Suppose that a and b are 2-digit numbers, and that
which is what you get by stringing the digits of a and b together. Then some algebra shows that

So we can find a and b by splitting 10,001 into a sum of two squares. There is an easy way:

But 100 has three digits, not two. However, there is also a less obvious way:

So 100 - 2a = 76 and 2b - 1 = 65. Therefore a = 12 and b = 33, which leads to Lenstra’s observation.

A second solution is hidden here, because we could take 2a - 100 = 76 instead. Now
a
= 88, and we discover that

Similar examples can be found by splitting numbers like 1,000,001 or 100,000,001 into a sum of squares. Number theorists know a general technique for this, based on the prime factors of those numbers. After a lot of detail that I won’t go into here, this leads to things like

This is all very well, but what about cubes? Most mathematicians would probably guess that 153 is a special accident. However, it turns out that
and a bit of algebra proves that this pattern continues indefinitely.

These facts depend on our base-10 notation, of course, but that opens up further opportunities: what happens in other number bases?

Hardy was trying to explain a valid point, about what constitutes interesting mathematics, and he plucked the 3-digit puzzle from thin air as an example. If he had given it more thought, he would have realised that although that particular puzzle is special and trivial, it motivates a more general class of puzzles, whose solutions lead to serious and intriguing mathematics.

Who cares, you may ask, and the answer is ‘archaeologists’. To be precise, the archaeological team led by Renée Friedman, investigating the ancient Egyptian site of Nekhen, better known by its Greek name Hierakonpolis.

Hierakonpolis was the main centre of Predynastic Egypt, about 5,000 years ago, and it was the cult centre for the falcon-god
Horus. It was probably first settled several thousand years earlier. Until recently the site was dismissed as a featureless, barren waste, but beneath the desert sands lie the remains of an ancient town, the earliest known Egyptian temple, a brewery, a potter’s house that burnt down when his nearby kiln set it on fire, and the only known burial of an elephant in ancient Egypt.

My wife and I visited this extraordinary site in 2009, under the auspices of the ‘Friends of Nekhen’. And there we saw the ostrich eggs whose broken shells were excavated from the area known as HK6. They had been deposited there, intact, as foundation deposits - artefacts deliberately placed in the foundations of a new building. Over the millennia, the eggs had broken into numerous fragments, so the first question was ‘how many eggs were there?’ The Humpty-Dumpty project - to reassemble the eggs - turned out to be too time-consuming. So the archaeologists settled for an estimate: work out the total area of the shell fragments and divide by the area of a typical ostrich egg.

Typical ostrich egg fragments from Hierakonpolis.

It is here that the mathematics comes in. What is the (surface) area of an ostrich egg? Or, for that matter, what is the area of an egg? Our textbooks list formulas for the areas of spheres, cylinders, cones, and lots of other shapes - but no eggs. Fair enough, since eggs come in many different shapes, but the typical chicken’s-egg shape fits ostrich eggs pretty well too, and is one of the commonest shapes found in eggs.

One helpful aspect of eggs is that (to a good approximation, a phrase that you should attach to every statement I make from now on) they are surfaces of revolution. That is, they can be formed by rotating some specific curve around an axis. The curve is a slice through the egg along its longest axis, and has the expected ‘oval’ shape. The best-known mathematical oval is the ellipse - a circle that has been stretched uniformly in one direction. But eggs aren’t ellipses, because one end is more rounded than the other. There are fancier egg-shaped mathematical curves, such as Cartesian ovals, but those don’t seem to help.

If you rotate an ellipse about its axis, you get an ellipsoid of revolution. More general ellipsoids do not have circular cross-sections, and are essentially spheres that have been stretched or squashed in three mutually perpendicular directions. Arthur Muir, in charge of the Hierakonpolis eggs, realised that an egg is shaped like two half-ellipsoids joined together. If you can find the surface area of an ellipsoid, you can divide by 2 and then add the areas of the two pieces.

Forming an egg from two ellipsoids.

There is a formula for the area of an ellipsoid, but it involves esoteric quantities called elliptic functions. By a stroke of good fortune, the ostrich’s propensity to lay surfaces of revolution,
which is a consequence of the tubular geometry of its egg-laying apparatus, comes to the aid of both archaeologist and mathematician. There is a relatively simple formula for the area of an ellipsoid of revolution:
where