Ian Stewart Read online

  Two positions of a concertina.

  Imagine the concertina partially closed, like the left-hand picture, and then opened, like the right-hand one. We’re viewing it from the side here. If the faces don’t bend or otherwise distort, the line AB can’t change length. Now, the sides AC and BD actually slope away from us, and we’re seeing them sideways on, but, even so, because those lengths don’t change in three dimensions, the points C and D in the right-hand picture have to be further apart than they are in the left-hand one. But this contradicts lengths being unchanged. Therefore the faces must change shape. In practice, the material that hinges them together can stretch a bit, which is why a concertina works.

  The Bellows Conjecture

  Whenever mathematicians make a discovery, they try to push their luck by asking further questions. So when flexible polyhedra were discovered, mathematicians soon realised that there might be another reason why concertinas don’t satisfy the mathematical definition. So they did some experiments, making a small hole in a cardboard flexible polyhedron, filling it with smoke, flexing it, and seeing if the smoke puffed out.

  It didn’t. If you’d done that with a concertina, or bellows, and compressed it, you’d have seen a puff.

  Then they did some careful calculations to confirm the experiment, turning it into genuine mathematics. These showed that when you flex one of the known flexible polyhedra, its volume doesn’t change. Dennis Sullivan conjectured that the same goes for all flexible polyhedra, and in 1997 Robert Connelly, Idzhad Sabitov and Anke Walz proved he was right.

  It doesn’t work for polygons.

  Before sketching what they did, let me put the ideas into context. The corresponding theorem in two dimensions is false. If you take a rectangle and flex it to form a parallelogram, the area gets smaller. So there must be some special feature of three-dimensional space that makes a mathematical bellows impossible. Connelly’s group suspected it might relate to a formula for the area of a triangle, credited to Heron of Alexandria (see note on page 282).6 This formula involves a square root, but it can be rearranged to give a polynomial equation relating the area of the triangle to its three sides. That is, the terms in the equation are powers of the variables, multiplied by numbers.

  Sabitov wondered whether there might be a similar equation for any polyhedron, relating its volume to the lengths of its sides. This seemed highly unlikely: if there was one, how come the great mathematicians of the past had missed it?

  Nevertheless, suppose this unlikely formula does exist. Then the bellows conjecture follows immediately. As the polyhedron flexes, the lengths of its sides don’t change - so the formula stays exactly the same. Now, a polynomial equation may have many solutions, but the volume clearly changes continuously as the polyhedron flexes. The only way to change from one solution of the equation to a different one is to make a jump, and that’s not continuous. Therefore the volume cannot change.

  All very well, but does such a formula exist? There is one case where it definitely does: a classical formula for the volume of a tetrahedron in terms of its sides. Now, any polyhedron can be built up from tetrahedra, so the volume of the polyhedron is the sum of the volumes of its tetrahedral pieces.

  However, that’s not good enough. The resulting formula involves all the edges of all the pieces, many of which are ‘diagonal’ lines that cut across from one corner of the polyhedron to another. These are not edges of the polyhedron, and, for all we know, their lengths may change as the polyhedron flexes. Somehow the formula has to be tinkered with to get rid of these unwanted edges.

  A heroic calculation led to the amazing conclusion that such a formula does exist for an octahedron - a solid with eight triangular faces. It involves the 16th power of the volume, not the square. By 1996, Sabitov had found a way to do the same for any polyhedron, but it was very complicated, which may have been why the great mathematicians of earlier times had missed it. In 1997, however, Connelly, Sabitov and Walz found a far simpler approach, and the bellows conjecture became a theorem.

  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.

  Digital Cubes

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

  13 + 53 + 33 = 1 + 125 + 27 = 153

  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

  Nothing Which Appeals Much to a Mathematician

  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:

  • When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

  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:

  122 + 332 = 1, 233

  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

  a2 + b2 = 100a + b

  which is what you get by stringing the digits of a and b together. Then some algebra shows that

  (100 - 2a)2 + (2b - 1)2 = 10,001

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

  10,001 = 1002 + 12

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

  10,001 = 762 + 652

  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

  882 + 332 = 8,833

  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

  5882 + 2,3532 = 5,882,353

  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

  163 + 503 + 333 = 165,033

  1663 + 5003 + 3333 = 166,500,333

  1,6663 + 5,0003 + 3,3333 = 166,650,003,333

  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.

  What Is the Area of an Ostrich Egg?

  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

  A = the area

  a = half the long axis

  c = half the short axis

  e = the eccentricity, which equals

  How to rotate the ellipse.

  Putting this all together, using measurements from modern ostrich eggs and intact ancient ones, led to an average figure of 570 square centimetres for one egg. This seemed quite large, but experiments with a modern egg confirmed it. The sums then indicated that at least six eggs had been deposited in Structure 07, the largest concentration of ostrich eggs in any single Predynastic deposit.

  You never know when mathematics will be useful.

  For the archaeological details, see:

  www.archaeology.org/interactive/hierakonpolis/field07/6.html

  Order into Chaos

  Many puzzles, indeed most of them, lead to more serious mathematical ideas as soon as you start to ask more general questions. There is a class of word puzzles in which you have to start with one word and turn it into a different one in such a way that only one letter is changed at each step and that every step is a valid word.7 Both words must have the same number of letters, of course. To avoid confusion, you are not allowed to rearrange the letters. So CATS can legitimately become BATS, but you can’t go from CATS to CAST in one step. You can using more steps, though: CATS-CARS-CART-CAST.

  Here are two for you to try:

  • Turn SHIP into DOCK.

  • Turn ORDER into CHAOS.

  Even though these puzzles involve words, with all the accidents and irregularities of linguistic history, they lead to some important and intriguing mathematics. But I’ll postpone that until the Answers section, so that I can discuss these two examples there without giving anything away here.

  Answers on page 283

  Big Numbers

  Big numbers have a definite fascination. The Ancient Egyptian hieroglyph for ‘million’ is a man with arms outstretched - often likened to a fisherman indicating the size of ‘the one that got away’, although it is often found as part of a symbolic representation of eternity, with the two hands holding staffs that represent time. In ancient times, a million was pretty big. The Hindu arithmeticians recognised much bigger numbers, and so did Archimedes in The Sand Reckoner, in which he estimates how many grains of sand there are on the Earth and demonstrates that the number is finite.

  The million that got away . . .

  In mathematics and science the usual way to represent big numbers is to use powers of 10:

  102 = 100 (hundred)

  103 = 1,000 (thousand)

  106 = 1,000,000 (million)

  109 = 1,000,000,000 (billion)

  1012 = 1,000,000,000,000 (trillion)

  There was a time when an English billion was 1012, but the American usage now prevails almost universally - if only because a billion is now common in financial transactions and we need a snappy name for it. The obsolete ‘milliard’ doesn’t have the right ring. In this age of collapsing banks, trillions of pounds or dollars are starting to be headline material. Billions are passé.

  In mathematics, far bigger numbers arise. Not just for the sake of it, but because they are needed to express significant discoveries. Two relatively well-known examples are:

  10100 = 10,000, . . . ,000 (googol)

  with one hundred zeros, and

  10googol = 1,000, . . . ,000 (googolplex)

  which is 1 followed by a googol of zeros. Don’t try to write it down that way: the universe won’t last long enough and you won’t be able to get a big enough piece of paper. These two names were invented in 1938 by Milton Sirotta, the American mathematician Edward Kasner’s nine-year-old nephew, during an informal discussion of big numbers (Cabinet, page 213). The official name for googol is ten duotrigintillion in the American system and ten thousand sexdecillion in the obsolescent English system. The name of the internet search engine Google™ is derived from googol.

  Kasner introduced the googol to the world in his book Mathematics and the Imagination, written with James Newman, and they tell us that a group of children in a kindergarten worked out that the number of raindrops falling on New York in a century is much less than a googol. They contrast this with the claim (in a ‘very distinguished scientific publication’) that the number of snowflakes needed to form an ice age is a billion to the billionth power. This is 109000000000, and you could just about write it down if you covered every page of every book in a very large library with fine print - all but one symbol being the digit 0.
A more reasonable estimate is 1030. This makes the point that it is easy to get confused about big numbers, even when a systematic notation is available.

  All of this pales into insignificance when compared with Skewes’ number, which is the magnificent

  When considering these repeated exponentials, the rule is to start at the top and work backwards. Form the 34th power of 10, then raise 10 to that power, and finally raise 10 to the resulting power. A South African mathematician, Stanley Skewes, came across this number in his work on prime numbers. Specifically, there is a well-known estimate for the number of primes π(x) less than or equal to any given number x, given by the logarithmic integral

  In all cases where π(x) can be computed exactly, it is less than Li(x), and mathematicians wondered whether this might always be true. Skewes proved that it is not, giving an indirect argument that it must be false for some x less than his gigantic number, provided that the so-called Riemann hypothesis is true (Cabinet, page 215).

  To avoid complicated typesetting, and in computer programs, exponentials ab are often written as a^b. Now Skewes’ number becomes

  10^10^10^34

  In 1955 Skewes introduced a second number, the corresponding one without assuming the Riemann hypothesis, and it is

  10^10^10^963