THIS week a puzzle that has confounded mathematicians for over three
centuries is officially laid to rest. In the 17th century, the French
mathematician Pierre de Fermat casually noted in the margin of a book that he
had a proof of an apparently simple theorem. Unfortunately, the mischievous
Frenchman also wrote that there was not enough space in the margin to contain
the proof, and in fact he never got round to committing it to paper.
After Fermat鈥檚 death, the frustrating margin note was discovered and, ever
since, mathematicians have tried to prove Fermat鈥檚 Last Theorem. Fermat鈥檚
brainteaser tickled some of the greatest minds on the planet. It attracted
tremendous prizes and generated intense rivalries. This week, on 27 June,
Fermat鈥檚 challenge to the world officially comes to an end when Andrew Wiles, an
Englishman working at Princeton University, goes to G枚ttingen to collect
the prestigious Wolfskehl Prize
(see 鈥淟ast will and testament鈥�), for his proof
of the Last Theorem.
But this has left mathematicians twiddling their thumbs. Now that Fermat鈥檚
riddle has been solved, what next? 鈥淭here is a sense of melancholy,鈥� says Wiles.
鈥淲e鈥檝e lost something that鈥檚 been with us for so long, and something that drew a
lot of us into mathematics.鈥� Fortunately for despondent mathematicians, there
are still plenty of conundrums left to be solved (see 鈥淢athematical mystery
迟辞耻谤鈥�). However, a worthy successor for Fermat鈥檚 Last Theorem must match its
charm and allure. Kepler鈥檚 sphere-packing conjecture is just such a
problem鈥攊t looks simple at first sight, but reveals its subtle horrors to
those who try to solve it.
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Sorting spheres
Kepler鈥檚 conundrum is even older than Fermat鈥檚鈥攊t first reared its head
when Sir Walter Raleigh, explorer and Spanish Armada veteran, asked his
mathematical friend Thomas Harriot if there was a formula for identifying the
number of cannonballs in a stack. This led Harriot to ponder the different ways
that spheres of all kinds can be arranged. In 1606, he wrote to Johannes Kepler,
famed for his astronomical discoveries, outlining different possibilities.
Intrigued, Kepler wrote a paper in 1611 exploring how the way spheres arrange
themselves could explain the formation of natural shapes and structures. As part
of this research he began to investigate the most efficient way to stack spheres
so they occupy the least possible space. He quickly realised that however they
are arranged there will always be gaps between them. The challenge was to
minimise the gaps.
After much trial and error, Kepler concluded that the most efficient
arrangement is the face-centred cubic lattice (see Diagram). It鈥檚 the way
greengrocers stack oranges. First you make a bottom layer with each sphere
surrounded by six others, then you make the second layer by putting spheres in
the dimples of the lower layer. The second layer looks just like the first but
it has been shifted sideways so that it fits snugly into position. Other layers
are formed in a similar way.
Just because the face-centred cubic lattice was the best Kepler could come up
with, that doesn鈥檛 mean there isn鈥檛 a better way. The sphere-packing problem has
turned out to be even more intractable than Fermat鈥檚 Last Theorem鈥攊s the
face-centred cubic lattice really the most efficient method of packing
spheres?
It鈥檚 a real poser鈥攜ou have to check so many arrangements before you can
be sure. And you can鈥檛 limit yourself to neat regular arrangements鈥攖here
is an infinite number of random ways to pack spheres, too. Which is why, in
nearly 400 years nobody has been able to prove Kepler鈥檚 conjecture. On the other
hand, since nobody has discovered a more efficient way to pack spheres, most
people assume that the conjecture is true. As the sphere-packing expert
Claude Ambrose Rogers from University College London puts it, Kepler鈥檚 claim is
one that 鈥渕ost mathematicians believe, and all physicists know鈥�.
Mathematicians are never really satisfied without a rigorous proof, and so
they have been sweating over their notepads for centuries. Progress was
painfully slow until the summer of 1990, when the mathematical world was rocked
by the news that the Kepler conjecture had been 鈥減roved鈥�. Wu-Yi Hsiang of the
University of California at Berkeley published a paper that seemed to succeed
where generations of others had failed. All that remained was for a team of
referees to go over his calculation with a fine-tooth comb.
Previously, mathematicians had projected the infinite box of spheres onto a
conventional Cartesian system of coordinates with three perpendicular
axes鈥攛, y and z. Hsiang鈥檚 apparent breakthrough involved analysing the
problem using a different system of coordinates.
He employed spherical coordinates in which each point in space is defined by
the length of a radius and its direction measured from the origin.
This seemed to be a far more natural approach for a problem involving
spheres. Hsiang then broke down the infinite space into manageable chunks. He
began by looking at a small cluster of spheres, rather than the infinite
problem. Having tackled the local problem, he then developed a semi-global
argument. Then, after a hundred pages of geometrical gymnastics, the argument
matured into a truly infinite solution. This step-by-step extension from a local
to a global solution was the key.
Although the proof was still being examined by other mathematicians, there
was so much confidence in Hsiang鈥檚 method that he was invited to give a lecture
entitled 鈥淭he proof of Kepler鈥檚 conjecture on the sphere-packing problem鈥� at the
joint meeting of the American Mathematical Society and the Mathematical
Association of America.
However, as time passed, experts in the field began to discover flaws in the
proof, which forced Hsiang to issue a revised proof in the summer of 1991.
Although the obvious errors were now fixed, the new proof was viewed with
greater scepticism and scrutinised intently. Some mathematicians identified
steps in the revised proof which they believed were unsubstantiated (New
快猫短视频, Science, 2 May 1992, p 16). For instance, in a letter to Hsiang,
Thomas Hales, now at the University of Michigan, wrote: 鈥淵ou state that `the
best way (that is, volume minimising) of adding a second layer of packing is to
cap as many holes as possible鈥� 鈥� Your argument seems to rely heavily and
essentially on this assumption. Nowhere is there even a hint of a proof.鈥�
The argument has rumbled on. While Hsiang claims that he has cleared up all
the glitches and that his proof is rigorous, his critics maintain that it is
still incomplete. After attending a major conference in 1996 to discuss
breakthroughs in geometry in 1996, Doug Muder, of the Mitre Corporation based in
Massachusetts, who had been working on the sphere-packing problem since the
1980s, declared: 鈥淭he community has reached a consensus on [Hsiang鈥檚 proof]: no
one buys it鈥� At best it is a sketch (a 100-page sketch!) of how such a
proof might go.鈥�
Hsiang鈥檚 proof is still shrouded in controversy鈥攕ome say it has been
discredited. Either way, the door is still open for anybody who wants to prove
Kepler鈥檚 conjecture beyond doubt. If you want to claim your place in the
mathematicians鈥� hall of fame, you could try patching up Hsiang鈥檚 proof to
everyone鈥檚 satisfaction. Alternatively, you could follow the strategy being
developed by his rival Hales.
Hales鈥檚 research is based on the work of L谩szl贸
Fejes-T贸th, a Hungarian mathematician who spearheaded research into
Kepler鈥檚 conjecture in the 1950s. Fejes-T贸th suggested that the most
efficient stacking arrangement could be identified by examining a finite cluster
of spheres, rather than an infinite number of spheres. Unlike Hsiang鈥檚 approach,
Fejes-T贸th鈥檚 method did not even require the gradual extension to a
global cluster. Having analysed 50 spheres, the proof would be complete. This
seemed to bring a proof within reach, but the calculation required to analyse
this finite cluster in enough detail was beyond Fejes-T贸th.
The calculation involves looking at all the positions of the 50 spheres
within the cluster. For each set of positions, you can then calculate a packing
efficiency. Because the position of each sphere is defined by three coordinates,
the equation you need to calculate the packing efficiency contains 150
variables. The task is to find the maximum value for this equation. If the
maximum efficiency turns out to be 74.04 per cent, the efficiency for the
face-centred cubic, Kepler鈥檚 conjecture would be proved.
But how do you find the maximum value? For an equation this complex, standard
methods would be pretty hopeless. Hales鈥檚 approach is to plot the efficiency on
a 鈥済raph鈥� of 150 variables, building a hilly landscape in 150-dimensional
hyperspace. All he needs then is a way to find the height of the tallest
hill.
His approach to this involves constructing hyperplanes鈥攔ather like
mathematical beams鈥攚hich form a roof over the landscape. He checks that
every hyperplane is clear above the landscape, so that he can be confident that
the tallest hill sits below the tallest point of the hyperplane roof. Next, he
calculates the highest point of the hyperplane roof. The skill is in lowering
the roof until it just touches the tallest hill鈥攚hich would give the value
for the maximum possibly packing efficiency.
In recent decades, others have tried to lower the roof, and in 1958 Rogers
calculated an upper limit of 77.97 per cent. This meant that there was a 4 per
cent space in which a rogue hill (or arrangement) could exist and prove Kepler
wrong. Reducing the upper limit further has turned out to be a slow and
difficult process, and in 1993 Muder had only managed to reduce it to 77.31 per
cent.
It鈥檚 a painstaking process, because lowering the ceiling involves introducing
new, lower hyperplanes. For each new one you introduce, you need to ensure that
it is completely clear of every hill, which takes a colossal amount of
calculation. However, Hales believes he has discovered short cuts which speed up
the checking procedure. Instead of checking the entire hyperplane, he just tests
certain points in each region of the landscape. Even with these short cuts,
Hales has to resort to hefty computer power to perform his calculations.
This computational approach may eventually succeed where paper-and-pencil
logic has failed, but many feel that such a proof is less illuminating than
traditional proofs. A mathematical proof, the argument goes, should not just
answer a question, it should also provide some insight. When a computer was used
to solve the Four-Colour Problem (which poses the question, are four colours
enough to colour any conceivable map, such that no two bordering regions are
coloured the same?) some mathematicians were dismayed. Philip Davis from Brown
University in Providence, Rhode Island, described his response to the news that
a proof had been found: 鈥淢y first reaction was. `Wonderful! How did they do it?鈥�
I expected some brilliant new insight, a proof which had in its kernel an idea
whose beauty would transform my day. But when I received the answer: `they did
it by breaking it down into thousands of cases, and then running them all on the
computer, one after the other鈥�, I felt disheartened.鈥�
So while the smart money is backing Hales鈥檚 approach, many experts in the
field are hoping that a more traditional proof will emerge. But time is running
out. Hales confidently predicts that a complete proof is on the horizon: 鈥淚f I
haven鈥檛 done it by the year 2000, I鈥檒l be disappointed.鈥�
* * *
Last will and testament
IN THE 19th century, a German industrialist called Paul Wolfskehl became
obsessed by Fermat鈥檚 Last Theorem, and eventually bequeathed 100 000 marks
(equivalent to 拢1 million in today鈥檚 money) to whoever could find a
proof.
Wolfskehl鈥檚 legacy inspired thousands of amateur mathematicians to take up
Fermat鈥檚 challenge. The Wolfskehl Committee was inundated with flawed proofs,
and monitoring the entries soon became a nuisance. Edmund Landau, a member of
the committee and head of the department of mathematics at the University of
G枚ttingen, dealt with the matter by handing entries to his students along
with a card which read: 鈥淭hank you for your entry. The first error occurs on
page . . . line . . .鈥� It was the student鈥檚 task to fill in the gaps.
Hyperinflation after the First World War devalued the prize considerably, but
Wiles will still collect 拢30 000 for penning what has been called the
proof of the century.
* * *
Mathematical mystery tour
SOME of the most profound problems in mathematics involve some of the
simplest concepts. For example, there are still mysteries surrounding 鈥減erfect
numbers鈥�. The factors of perfect numbers add up to the number itself. For
example, 28 is a perfect number because 1, 2, 4, 7, and 14 divide into 28 and 28
= 1 + 2 + 4 + 7 + 14. Ren茅 Descartes said that 鈥減erfect numbers, like
perfect men, are very rare鈥�. Over the last few thousand years only 30 have been
discovered. But is there an inexhaustible supply of perfect numbers for
mathematicians to find? Whoever succeeds in answering this question will earn a
place in history.
Another ancient area of mathematics rich in unsolved problems is the theory
of prime numbers. A prime number can only be divided by 1 and itself. For
example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are all primes. The sequence of
primes forms no obvious pattern, and they have been described as weeds growing
randomly among the other numbers. For centuries mathematicians have searched for
an underlying pattern, but it could be that no pattern exists, in which case
mathematicians would be advised to tackle another less ambitious prime
problem鈥搕he Twin Prime Conjecture.
Twin primes are pairs of primes which differ by only two, for example 17 and
19, and 1000 000 000 061 and 1000 000 000 063. Two thousand years ago Euclid
proved that there is an infinite quantity of prime numbers, but no-one has yet
managed to prove that there is also an infinite number of twin primes.