The Kepler Problem // 231

That ought to give some interesting new kinds of curves. So why don't the mathematics books mention them?

...........................................

Answer on page 304

Mathematical Jokes 3 Two mathematicians in a cocktail bar are arguing about how much maths the ordinary person knows. One thinks they're hopelessly ignorant; the other says that quite a few people actually know a lot about the subject.

`Bet you twenty pounds I'm right,' says the first, as he heads for the gents. While he is gone, his colleague calls the waitress over.

`Listen, there's ten pounds in it for you if you come over when my friend gets back and answer a question. The answer is ``one-third x cubed.'' Got that?'

`Ten pounds for saying ``One thirdex cue?'' '

`No, one-third x cubed.'

`One thir dex cubed?'

`Yeah, that'll do.'

The other mathematician comes back, and the waitress comes over.

`Hey ­ what's the integral of x squared?'

`One third x cubed,' says the waitress. As she walks away, she

........................................... adds, over her shoulder, `Plus a constant.'

The Kepler Problem Mathematicians have learned that apparently simple questions are often hard to answer, and apparently obvious facts may be false, or may be true but extremely hard to prove. The Kepler problem is a case in point: it took nearly three hundred years to 232 // The Kepler Problem

solve it, even though everyone knew the correct answer from the start.

It all began in 1611 when Johannes Kepler, a mathematician and astrologer (yes, he cast horoscopes; lots of mathematicians did at that time ­ it was a quick way to make money) wanted to give his sponsor a New Year's gift. The sponsor rejoiced in the

¨ name Johannes Mathaus Wacker of Wackenfels, and Kepler wanted to say `thanks for all the cash' without actually spending any of it. So he wrote a book, and presented it to his sponsor. Its title (in Latin) was The Six-Cornered Snowflake. Kepler started with the curious shapes of snowflakes, which often form beautiful sixfold symmetric crystals, and asked why this happened.

A typical `dendritic'

snowflake.

It is often said that `no two snowflakes are alike'. The logician in me objects `How can you tell?' but a back-of-the-envelope calculation suggests that there are so many features in a `dendritic' snowflake, of the kind illustrated, that the chance of two being identical is pretty much zero.

No matter. What matters here is that Kepler's analysis of the snowflake led him to the idea that its sixfold symmetry arises because that's the most efficient way to pack circles in a plane.

Take a lot of coins, of the same denomination ­ pennies, say. If you lay them on a table and push them together tightly, you The Kepler Problem // 233

quickly discover that they fit perfectly into a honeycomb pattern, or `hexagonal lattice':

(Left) The closest way to

pack circles, and (right)

a less efficient lattice

packing.

And this is the closest packing ­ the one that fills space most efficiently, in the ideal case of infinitely many circles arranged on a plane. Alternatives, such as the square lattice on the right, are less efficient.

Mind you, this innocent assertion wasn't proved until 1940,

´ ´ ´ when Laszlo Fejes Toth managed it. (Axel Thue sketched out a proof in 1892, and gave more details in 1910, but he left some

´ gaps.) Toth's proof was quite hard. Why the difficulty? We don't know, to begin with, that the most efficient packing forms a regular lattice. Maybe something more random could work better. (For finite packings, say inside a square, this can actually happen ­ see the next puzzle, about a milk crate.)

Along the way, Kepler came very close indeed to the idea that all matter is made from tiny indivisible components, which we now call `atoms'. This is impressive, given that he did no experiments in the course of writing his book. Atomic theory, introduced by the Greek Democritus, was not established experimentally until about 1900.

Kepler had his eye on something a bit more complicated, though: the closest way to pack identical spheres in space. He was aware of three regular `lattice' packings, which we now call the hexagonal, cubic and face-centred cubic lattices. The first of these is formed by stacking lots of honeycomb layers of spheres on top of one another, with the centres of corresponding spheres forming a vertical line. The second is made from square-lattice layers, also stacked vertically. For the third, we stack hexagonal layers, but fit the spheres in any given layer into the hollows in the one below. 234 // The Kepler Problem

You can get the same result, though tilted, by similarly stacking square-lattice layers so that the spheres in any given layer fit into the hollows in the one below ­ this isn't entirely obvious, and ­ like the milk crate puzzle ­ it shows that intuition may not be a good guide in this area. The picture shows how this happens: the horizontal layers are square, but the slanting layers are hexagonal.

Part of a face-centred

cubic lattice.

Now, every greengrocer knows that the way to stack oranges is to use the face-centred cubic lattice.* By thinking about pomegranate seeds, Kepler was led to the casual remark that with the face-centred cubic lattice, `the packing will be the tightest possible'.

That was in 1611. The proof that Kepler was right had to wait until 1998, when Thomas Hales announced that he had achieved this with massive computer assistance. Basically, Hales consid- ered all possible ways to surround a sphere with other spheres, and showed that if the arrangement wasn't the one found in the face-centred cubic lattice, then the spheres could be shoved

´ closer together. Toth's proof in the plane used the same ideas, but he only had to check about forty cases.

Hales had to check thousands, so he rephrased the problem in terms that could be verified by a computer. This led to a huge computation ­ but each step in it is essentially trivial. Almost all of the proof has been checked independently, but a very tiny level of doubt still remains. So Hales has started a new computer-

* They don't say it that way, but they stack it that way.

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