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SEVENTY years ago, the Polish mathematicians Stefan Banach and Alfred Tarski proved that you can cut a sphere into six pieces and reassemble them to make two spheres, each the same size as the original. This certainly sounds not so much strange as completely barmy. Surely, it is obvious that the bits you get by cutting up a sphere can鈥檛 be put together again to make two spheres the same size as the original, because the volume would have to double.

Or would it? For a mathematician, it all depends on what you mean by 鈥渂its鈥�, 鈥渃ut up鈥� and 鈥減ut together鈥�. The bits needed to perform this mathemagical marvel are very complicated, with infinitely fine structure. The change in volume is a red herring because bits that complicated don鈥檛 have volumes. You can鈥檛 do it with a sphere of gold and become a trillionaire, because 鈥� among other things 鈥� gold is made up of atoms. The maths works only because a sphere is, technically speaking, made up of an infinite number of points, and it鈥檚 impossible to cut atoms up into infinitely small pieces.

Nonetheless, Banach and Tarski鈥檚 theorem (commonly known as the Banach-Tarski paradox, though it is not a true paradox, being counterintuitive rather than self-contradictory) is important for the way it forces us to reconsider what we mean by concepts such as volume and infinity. The following is an attempt to explain how the theorem can be proved, what its limitations are and why volumes are irrelevant.

We will be talking about sets 鈥� collections of points in space. Such sets of points can make up the traditional objects of geometry, such as spheres, but they can also be used to make up more complicated objects.

Inevitably we will need some set jargon, otherwise the explanation will become too unwieldy to handle. A set is said to be a subset of another set if it can be obtained by removing some points. For example, the High Court judges of England are a subset of the legal profession. A set is said to be split into disjoint subsets if every point of the set lies in one, and only one, of the subsets. In Britain, the House of Commons is split into disjoint subsets known as parties comprising Labour, Conservative, Liberal Democrat, Ulster Unionist and a few others, and every member of the Commons is in one or other of the subsets (assuming, of course, that any rebellious Tory MPs who have lost the Whip still count as members of the Conservative Party). Finally, two sets of points are said to be 鈥渄issection-equivalent鈥� if you can split one set into a finite number of disjoint subsets that can be moved around and fitted together to form the other set, as long as this can be done without changing the relative positions of the component points of each subset 鈥� that is, the subsets of points stay rigid. For example the jigsaw pieces in a box and the assembled puzzle are dissection-equivalent.

In this language, the Banach-Tarski paradox says that a solid sphere of unit radius is dissection-equivalent to a set composed of two disjoint solid spheres of unit radius. What it does not say is that the dissection is simple enough to be performed with an orange and a knife 鈥� because it isn鈥檛.

While spheres seem ordinary, the fact that they contain infinitely many points leaves room for strange behaviour. For example, an infinite set of points can be made to look identical to a subset of itself. To see why, place points along a line at whole number distances 0, 1, 2, 3, 鈥� going on forever (see Diagram). If you slide this set one unit to the right you get another set consisting of 1, 2, 3, 4, 鈥�, also going on forever. Point 0 is missing from the second set, but because the two sets are infinite, losing one point doesn鈥檛 matter. If you pick up set number one and lay it over set number two, the two sets coincide, point for point 鈥� in other words they can be made to look identical.

This idea leads into a result that turns out to be important for the Banach-Tarski paradox: a circle is dissection-equivalent to itself minus one point. To prove it, you need to understand what I鈥檒l call the dishwasher principle. First decide which point on the circle you want to remove. Choose an angle that is not an exact fraction of a whole turn, such as 2掳, and mark off successive points at spacings of that angle, starting from your chosen point (see Diagram). The choice of angle ensures that no two of these points ever coincide, but as you keep marking them off round and round the circle they begin to bunch up. Now split the circle into two parts: the first one containing all the points in this set, and the second all the other points on the circle.

Next, think what happens when the first set you鈥檝e just constructed is 鈥渃licked on one step鈥� like the knob on a dishwasher. You get a new set that begins with point number two and contains all the points that the previous set did apart from the first one (the one that you chose). The trick is the same as for the dots on the line, but 鈥渨rapped round the circle鈥�. In other words, splitting the circle into these two parts and then reassembling it with one part clicked one place on means that you get rid of the chosen point. By repeated application of the dishwasher principle, you can remove any finite set of points from almost any set you wish.

Alien alphabet

This principle will be very useful for part of the proof, but for the rest of it, we need a more powerful idea 鈥� the dictionary principle. I鈥檒l sneak up on it by way of a story. The alien inhabitants of the planet Xodarap have a very short alphabet containing only the letters A and B. To make up for it, their language contains very long words indeed, words like BBBAAABBBABBABABABABBB (which means 鈥淪orry, I seem to have stepped on your pet bog-lizard鈥�). In fact, the language contains all possible finite sequences of letters.

The planet鈥檚 lexicographers decided to compile a dictionary listing all of the words in alphabetical order. Even though it did not attach definitions to any of the words, it was rather a large dictionary because it contained infinitely many words, so they decided to make it more portable by splitting it into volume A, containing all the words beginning with A, and volume B for all those beginning with B. Then some genius noticed that since all words in volume A began with A, the first letter could be omitted to save space; and similarly for volume B. All was well until the chief lexicographer examined the proof copies of volumes A and B. First, he observed that they were identical except for their covers: they both began A, AA, AAA, and got round to the bog-lizard apology word at exactly the same place. But what was worse, each was exactly the same as the original one-volume dictionary. The lexicographers of Xodarap had discovered, quite by accident, that their dictionary could be cut up and reassembled to make two copies of itself. They had hit on a version of the dictionary principle 鈥� a way of dividing up an infinite number of points to generate more than one copy of the original set.

Armed with these two ideas 鈥� the dishwasher principle, which allows us to remove troublesome points from a set, and the dictionary principle 鈥� we can now turn to Banach and Tarski鈥檚 proof of their peculiar theorem.

Banach and Tarski relied very heavily on a discovery made by the German topologist Felix Hausdorff in the early 1920s. He managed to prove that, like the dictionary of the Xodarapians, the surface of a sphere is dissection-equivalent to two copies of itself. Here鈥檚 how.

First, Hausdorff thought about ways to move points from one place to another on a sphere鈥檚 surface. One way is to rotate the sphere round different axes, like twisting a Rubik鈥檚 cube. Hausdorff came up with three possible rotations around two different axes of the sphere inclined at 45掳 (see Diagram). Let鈥檚 use the symbol r to denote a rotation through 180掳 about one of these axes, and let p be rotation through 120掳 about the other. If you do p twice the effect is a rotation of 240掳, or 120掳 in the opposite direction, and I鈥檒l call this q because it looks like p backwards.

Like the inhabitants of Xodarap, we can make up a 鈥渄ictionary鈥� consisting of 鈥渨ords鈥� containing the 鈥渓etters鈥� p, q and r. Remember that there is an infinite number of such words but that they are all of finite length. Each word corresponds to a series of rotations. For example, pqrq means 鈥減erform p then q then r then q again鈥�. If you take any point on the sphere鈥檚 surface and apply these four rotations to generate a new point somewhere else, the net result is exactly the same as performing just one simple rotation about a completely different axis. In fact, every individual 鈥渨ord鈥� in Hausdorff鈥檚 dictionary corresponds to a simple rotation about a single axis.

Next, and this is the clever bit, Hausdorff discovered that it was possible to split his dictionary into three disjoint books called A, B and C, with the properties qB = A, pC = A, rA = B + C (see 鈥淢aking two out of one鈥�). To understand what this means, imagine applying all of the rotations in book A to a point on the sphere鈥檚 surface. Each rotation generates a new point until you end up with a whole set of points. Similarly, applying all the rotations in B and in C to the same initial point generates two different sets of points, which are different again from those using A.

Now, supposing you first generate all the points corresponding to the A rotations, and then apply rotation r to them all. Because rA = B + C, this process generates not just all the points corresponding to B, but the ones corresponding to C as well. Next, separate out the 鈥淏鈥� points and apply q to them all. This generates A back again. Now take all of the 鈥淐鈥� points and apply p to them. Once again, you regenerate A. As a result of this set of rotations, you have started with one copy of A and ended up with two.

Dissected dictionary

So now, we have a way of dissecting one book of the dictionary to make two identical copies of the original. In fact, it is also possible to dissect B into two copies of itself using a slightly different set of rotations, and the same goes for C. In other words, this same principle can be used to dissect the whole dictionary. If Hausdorff鈥檚 dictionary of rotations could be used to generate every point on the surface of a sphere, we would now have more or less proved a two-dimensional version of the Banach-Tarski paradox.

Unfortunately it can鈥檛. There simply aren鈥檛 enough words in his dictionary to cover all of the points. It may seem surprising that an infinite number of words is still not enough, but in fact infinity isn鈥檛 everything. For example, imagine a set made up of exact fractions, known as rational numbers. There is an infinite number of these but they don鈥檛 cover every possible number. Lots of numbers like 蟺 and the 鈭�2 are missing.

But all is not lost. Hausdorff managed to show that you can choose a set of points, called H, and then apply all the rotations in the dictionary to each of the points in that set, rather than to one individual point. If you have chosen the right points for H, doing this generates virtually all the points on the sphere鈥檚 surface. (The missing ones are the 鈥減oles鈥� about which the rotations are performed, but some clever jiggery-pokery using a reversal of the dishwasher principle means that they can be inserted as necessary.)

So now, we have a proof that the surface of a sphere can be dissected and reconstructed to make two identical copies of itself. What Banach and Tarski realised (and Hausdorff didn鈥檛) was that virtually the same proof also works for a solid sphere. To see how, imagine the sphere as a nested onion of surfaces. For each surface, you can dissect it and reconstruct two identical copies. So, you can dissect the whole sphere and reconstruct two identical copies. (There is a slight problem with the central point, but it can be removed using the dishwasher principle for long enough to prevent it from causing trouble.) Hausdorff must have been a bit peeved when Banach and Tarski published what he鈥檇 just missed.

What use is the Banach-Tarski paradox? Sadly, it鈥檚 no good for creating matter such as gold or diamonds out of nothing. The dissection is infinitely fine, and it鈥檚 impossible to cut atoms up into infinitely small points (even if we had infinitely fine knives and an infinite amount of time to do it in). But it does illuminate the limitations of concepts such as length, area and volume. Counterintuitive examples, such as that of Banach and Tarski, made the mathematicians of the early 20th century realise that such concepts cannot be applied willy-nilly to any old set. They make sense only for sets with a relatively special structure.

The intuitive objection to the paradox is that it seems to involve doubling the volume of something without adding anything to it 鈥� contravening conservation of matter. The way round this is the idea that the bits we divide the sphere into are so complicated that they don鈥檛 have well-defined volumes. If they did have volume, then the paradoxical dissection couldn鈥檛 exist. But it does, so they don鈥檛. Quod erat dissectandum, as Euclid never said.

Making two out of one

IN the early 1920s, German topologist Felix Hausdorff came up with an idea that turns out to be very important for proving the Banach-Tarski paradox. He imagined a dictionary made up of an infinite number of finite-length words containing some or all of the letters p, q and r. He then split the dictionary up into three volumes: A, B and C. But he couldn鈥檛 just do it any old way. The volumes had to have the property that qB = A, pC = A and rA = B + C. In other words, adding q to the beginning of every word in volume B produces volume A, and so on.

Not surprisingly, working out how to divide up the dictionary in this way wasn鈥檛 easy, but in 1988, Robert French of the University of Michigan devised a wonderful way of demonstrating how Hausdorff did it. French saw the procedure as a kind of sausage machine. It has three units, A, B and C, each consisting of a breeder, a sorter, a copier and a bag in which to store words.

The process begins by dropping an 鈥渆mpty鈥� word (one with no letters at all) into the top of breeder A. Breeders produce all the possible words containing the letters p, q and r that are one letter longer than those that are dropped in. For example, the empty word 鈥渂reeds鈥� the three one-letter words r, p and q. In general a word breeds new words by adding r, p or q to the beginning of the original word 鈥� duplicates of existing words are discarded.

The old words and the newly bred ones drop through the the sorter, which directs them along three tubes according to their initial letter p, q or r. They fall into the copier of another unit, which sends one copy of the new words back to the top of the unit and drops both new and old words into its bag.

After machine A has done its stuff, machine B repeats the operation, using any words that happen to have fallen into its copier. Then C does the same. This completes one full cycle of the process, which now starts all over again, and so on forever. After an infinite number of cycles the words in the three bags A, B and C give Hausdorff鈥檚 strange decomposition of his dictionary (see Diagram)