Professor Stewart's Hoard of Mathematical Treasures (7 page)

Curiously, there is some truth to this adage. Robert Matthews has analysed the dynamics of falling toast, which does in fact have a propensity to land in a way that gets butter (or in my case marmalade) all over the carpet and ruins the toast. This lends support to Murphy’s law: Anything that can go wrong, will go wrong.

Matthews applied some basic mechanics to explain why toast tends to land buttered-side down. It turns out that tables are just the right height for the toast to make one half turn before it hits the floor. This may not be an accident, because the height of tables is related to the height of humans, and if we were much taller then the force of gravity would smash our skulls if we tripped. Matthews thus traces the trajectory of toast to a universal feature of the fundamental constants of the universe in relation to intelligent life forms. To my mind, this is probably the most convincing example of ‘cosmological fine-tuning’.

Suppose we put the previous two pieces of folklore together:

Therefore
. . .
what?
The buttered cat paradox takes these statements as given, and asks what would happen to a cat, dropped from a considerable height, to whose back is firmly attached a slice of buttered toast - buttered-side outwards from the cat, of course.
⁴

At the time of writing, the favoured answer is that, as the cat nears the ground, some kind of antigravity effect kicks in, and the cat hovers just off the ground while spinning madly over and over.

However, this argument has some logical loopholes, and it ignores basic mechanics. We’ve just seen that the mathematics of falling cats, and falling toast, lends scientific support to both adages. So what does the same mathematics say about a buttered cat?

What happens depends on how massive the toast is compared with the cat. If the toast is an ordinary slice, the cat has no difficulty in coping with the small amount of extra angular momentum that the toast contributes, and still lands on its feet. The toast doesn’t land at all.

However, if the toast is made of some kind of incredibly dense bread,
⁵
so that its mass is much larger than that of the cat, then Matthews’s analysis applies and the toast lands buttered-side down with the cat upside down waving its paws frantically in the air.

What happens for intermediate masses? The simplest possibility is that there is a critical cat-to-toast mass ratio [C : T]
_crit
, below which the toast wins and above which the cat wins. But it wouldn’t surprise me to find a range of mass ratios for which the cat lands on its side or, indeed, exhibits more complex transitional behaviour. Chaos cannot be ruled out, as any cat owner knows.

Abraham Lincoln once asked: ‘How many legs will a dog have if you call its tail a leg?’

OK, how many?

 

Discussion on page 281

Grumpelina, the Great Whodunni’s beautiful assistant, placed a blindfold over the eyes of the famous stage magician. A member of the audience then rolled three dice.

‘Multiply the number on the first dice by 2 and add 5,’ said Whodunni. ‘Then multiply the result by 5 and add the number on the second dice. Finally, multiply the result by 10 and add the number on the third dice.’

As he spoke, Grumpelina chalked up the sums on a blackboard which was turned to face the audience so that Whodunni could not have seen it, even if the blindfold had been transparent.

‘What do you get?’ Whodunni asked.

‘Seven hundred and sixty-three,’ said Grumpelina.

Whodunni made strange passes in the air. ‘Then the dice were—’

What? And how did he do it?

 

Answer on page 282

A polyhedron is a solid whose faces are polygons. It has been known since 1813 that a convex polyhedron (one with no indentations) is rigid: it cannot flex without changing the shapes of its faces. This was proved by Augustin-Louis Cauchy. For a long time, no one could decide whether a non-convex polyhedron must also be rigid, but in 1977 Robert Connelly discovered a flexible polyhedron with 18 faces. His construction was gradually simplified by various mathematicians, and Klaus Steffen improved it to a flexible polyhedron with 14 triangular faces. This is known to be the smallest possible number of triangular faces in a flexible polyhedron. You can watch it flex on:

You can make one by cutting the diagram from thin card, folding it, and joining the edges marked with the same letters. You can add flaps to do this, or use sticky tape. The dark lines show ‘hill’ folds, the grey ones ‘valley’ folds.

Cut out and fold: dark lines are ‘hill’ folds, grey lines are ‘valley’ folds.

Join the edges as marked to get Steffen’s flexible polyhedron.

Hang on a mo - isn’t there an obvious way to make a flexible polyhedron? What about the bellows used by blacksmiths to blow air into a fire? Or for that matter, what about a concertina? That has a flexible series of zigzag flaps. If you replace the two big pieces on the ends by flat-sided boxes, which they almost are anyway, then it’s a polyhedron. And it’s flexible. So what’s the big deal?

Although a concertina is a polyhedron, and flexible, it is not a flexible polyhedron. Remember, the shapes of its faces are not permitted to change. They start out flat, so they have to stay flat, which means they can’t bend. Not even the tiniest bit. But when you play a concertina, and the flexible bit opens up, the faces do bend. Very slightly.

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.

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.