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One of the more annoying things that mathematicians do is cast doubt upon things that we imagine we understand perfectly well. For example, we all know what areas and volumes are, don’t we? I am going to try to convince you that there is far more to areas and volumes than we normally imagine. The tale begins somewhere in the prehistory of the human race, and culminates in the dramatic solution in 1988 of a problem posed in 1925 by the mathematical logician Alfred Tarski. He asked whether it is possible to dissect a circular disc into a finite number of pieces, which can be reassembled to form a square. Tarski could prove, with some difficulty, that if such a dissection exists, then the resulting square must have the same area as the circle; but he was not sure whether such a dissection was possible.

Hang on-isn’t it obvious that the resulting square must have the same area? The total area won’t be affected by cutting the circle into pieces, will it? And isn’t it obvious that you can’t cut a circle up to get a square, because the curved edges won’t fit together properly? No, it’s not. Indeed, one of those two ‘obvious’ statements is false. The other is true, but not at all obvious.

Tarski’s problem is often compared to the famous problem of ‘squaring the circle’, attributed to the ancient Greeks. Actually it is very different. In the classical formulation, you must construct a square of area equal to that of a circle using ruler and compasses. This was proved impossible by Ferdinand Lindemann in 1882. But Tarski asks for a dissection, not a construction, and you can use any methods you like.

There is a fascinating mathematical toy, which can be made from card but looks best in polished hardwood and brass. It is made by hinging together four oddly-shaped pieces, and attaching handles. If the hinges are closed in one direction, the result is a square; if they are closed the other way, you get an equilateral triangle (see Figure 1). This is an ingenious example of a type of puzzle that was very popular towards the end of the 19th century, when a lot of people thought that Euclidean geometry was a fun thing. Such dissection puzzles ask for ways to cut up a given shape or shapes, so that the pieces can be reassembled to form some other shape or shapes. For example, can you dissect a square into a regular pentagon? A regular octagon? Yes on both counts (see Figure 2). What else can you dissect a square into?

For simplicity, suppose we restrict ourselves to straight line cuts. Then we always end up with a polygon-a shape with straight edges. It also seems clear that when you dissect a square, its area can’t change. Later, we’ll have to re-examine that belief more critically, but for polygonal dissections it is correct. So, if a square can be dissected to form a polygon, then the polygon must have the same area as the square. Are all such polygons possible? The answer is yes. This could have been proved by the ancient Greeks, but as far as we know, it wasn’t. It is usually called the Bolyai-Gerwien theorem, because Wolfgang Bolyai raised the question, and P. Gerwien answered it in 1833. However, William Wallace gave a proof in 1807 (see Figure 3).FIG-mg17645403.jpg

Area, along with volume, is actually quite a subtle idea, as primary school teachers know to their cost. It takes a lengthy process of experiment with scissors and paper, and vessels of various sizes, and water all over the floor, before children acquire the concept. In particular, the idea of conservation-that the amount of fluid you have does not depend upon the shape of the container that holds it-doesn’t come naturally. Most adults can be fooled, by clever packaging, into thinking that they are buying more shampoo than they really are. One difficulty is that areas and volumes are most easily defined for rectangles and boxes: it is not so clear how they work for other shapes, especially ones with curves. We have all been taught that the area of a circle is pi r2, but have you ever wondered why? What does pi have to do with ‘how much stuff’ there is inside a circle? How does that relate to how much stuff there is inside a square? You can’t do it with scissors.

Adding up the infinite

Area and volume are such useful concepts that they really do deserve to be put on a sound logical basis. The Bolyai-Gerwien theorem is not just a recreational curiosity: it is important because it justifies one possible method for defining area. Start by defining the area of a square to be the square of the length of its side, so that for instance a square with sides 3 centimetres long has an area of 9 square centimetres. Then the area of any other polygon is defined to be the area of the square into which it can be dissected. This approach brings out two key properties of area. The first is rigidity-the area of a shape stays the same during a rigid motion. The second is ‘additivity’-if a number of shapes are joined without overlapping, the resulting area can be found by adding their individual areas.

Actually, there are two versions of additivity. The weaker one, ‘finite additivity’, applies to a finite number of pieces; the stronger one applies to an infinite number of pieces. The technique of integration, basic to the calculus, extends the property of additivity to shapes formed by joining together infinitely many pieces. The total area can then be defined as the sum of the infinite series formed by the areas of the pieces-this is ‘infinite additivity’. Because infinite series are tricky, this approach requires extreme caution. It is, however, unavoidable if we wish to assign areas to curved regions, such as a circle. A circle cannot be cut into a finite number of triangles, but it can be cut into an infinite number. Pursuing a similar line of argument, Archimedes proved that the area of a circle is the same as that of a rectangle whose sides are its radius and half its circumference. Because pi is defined as the ratio of circumference to diameter, you can now see how it gets in on the act.

David Hilbert, a German whose interests ranged from number theory to physics, and was one of the greatest mathematicians of all time, wondered whether an equally ‘clean’ approach to volume might be possible. Mathematicians already knew what volume was, of course; in particular they could calculate the volume of any pyramid (one-third of the height times the area of the base). But again they wanted sound logical foundations. Hilbert asked whether any polyhedron (a solid bounded by flat surfaces) can always be dissected into a finite number of pieces, and reassembled to form a cube of the same volume. In 1900, at the International Congress of Mathematicians in Paris, he listed the question among 23 major unsolved problems.

Unlike the other 22, it did not survive very long. In the following year Max Dehn, one of the founders of topology, found the answer. Surprisingly, it is no. Dehn’s proof uses the angles of the solid to define what is now called the Dehn invariant, a number that, like volume, remains unchanged under dissection and reassembly. However, solids of the same volume can have different Dehn invariants. For example, this happens for a cube and a regular tetrahedron. So you can not dissect a tetrahedron into a cube of equal volume. The Dehn invariant is the only new obstacle to dissectability in three dimensions, and the correct analogue of the Bolyai-Gerwien theorem is that polyhedra can be dissected into each other if and only if they have the same volume and the same Dehn invariant.

Though differing in this respect, area and volume share the basic features of rigidity and infinite additivity. A general theory of such concepts was developed by Henri Lebesgue, who called them ‘measures’, and devised the great-granddaddy of all such concepts, the Lebesgue measure. Along the way, it transpired that sufficiently messy and complicated sets may not possess well-defined areas or volumes at all. Only the ‘measurable’ sets do. These include all of the familiar shapes from geometry, and a wide range of much more bizarre sets as well, but not everything. Measure theory is important for a variety of reasons: in particular, it is the foundation for probability theory.

Tarski belonged to a group of Polish mathematicians who frequented the ‘Scottish Cafe’ in Lvov. Another member was Stefan Banach. All sorts of curious ideas arose through talking shop in the Scottish Cafe. Among them is a theorem so ridiculous that it is almost unbelievable, known as the Banach-Tarski paradox. It dates from 1924, and states that it is possible to dissect a solid sphere into six pieces, which can be reassembled, by rigid motions, to form two solid spheres the same size as the original one.

But what about the volume? It increases eightfold. Surely that is impossible? The trick is that the pieces are so complicated that they do not have volumes. The total volume can change. Because the pieces are so complicated, with arbitrarily fine detail, you cannot actually carry out this dissection on a lump of physical matter. A good job too, it would ruin the gold market.

Once we start thinking about very complicated pieces, we have to be careful. For instance, the ‘pieces’ need not actually come as connected lumps. Each piece might consist of many disconnected components-possibly an infinite number. When we apply a rigid motion to such a set, we must not only keep each component the same shape, we must also preserve the mutual relationship of the parts. And it is not just the notion of rigid motion that needs to be made precise. When we dissected a square into an equilateral triangle, for example, we did not worry about the edges of the pieces. Do they abut or overlap? If a point is used in one edge, does that exclude it from the adjacent edge? The question was not raised before because it would have been distracting; and also because the edges of polygons have zero area and, therefore, do not really matter. But with highly complicated sets, we need to be more careful: every single point must occur in precisely one of the pieces. Banach proved that the Bolyai-Gerwien theorem is still true for this more careful definition of ‘dissection’. He did so by inventing a method for ‘losing’ unwanted edges.

The spirit (but not the details) of how the Banach-Tarski paradox works can be understood by thinking about a dictionary rather than a sphere. This is an idealised mathematician’s dictionary, the Hyperwebster, which contains all possible words-sense or nonsense-that can be formed from the 26 letters of the English alphabet. They are arranged in alphabetical order. It begins with the words A, AA, AAA, AAAA . . . and only infinitely many words later does it get round to AB. Nevertheless, every word, including AARDVARK, BANACH, TARSKI, or ZYMOLOGY, finds its place. I’ll show you how to dissect a Hyperwebster into 26 copies of itself, each maintaining the correct alphabetical order-with a spare alphabet thrown in.

The first of the 26 copies, ‘volume A’, consists of all words that begin with A, apart from A itself. The second, ‘volume B’, consists of all words that begin with B, apart from B itself; and so on. Let’s think about volume B. Like gentlemen songsters out on a spree, it begins BA, BAA, BAAA, and so on. Indeed, it contains every word in the entire Hyperwebster exactly once, except that a B has been stuck on the front of every one. BAARDVARK, BBANACH, BTARSKI, and BZYMOLOGY are in volume B. Moreover, they are in the same order as AARDVARK, BANACH, TARSKI and ZYMOLOGY.

The same goes for volume A and volumes C to Z. Each is a perfect copy of the entire Hyperwebster, with an extra letter stuck on the front of every word. Conversely, every word in the Hyperwebster, apart from those containing a single letter, appears in precisely one of these 26 volumes. AARDVARK, for instance, is in volume A, in the position reserved for ARDVARK in the Hyperwebster itself. BANACH is in volume B, and its position corresponds to that of ANACH in the original Hyperwebster. The cartoonist’s snooze symbol ZZZZZ is in volume Z, in the position corresponding to the slightly shorter snooze ZZZZ.

In short, one Hyperwebster can be cut up, and rearranged, without altering the orders of the words, to form 26 identical Hyperwebsters plus a spare alphabet. Order-preserving dissections of Hyperwebsters are not ‘volume’-preserving. For ‘Hyperwebster’ now read ‘sphere’, for ‘word’ read ‘point’, and for ‘without altering the order of’ read ‘without altering the distances between’, and you’ve got the Banach-Tarski paradox. In fact, the analogy is closer than it might appear, because the Banach-Tarski paradox is proved by using sequences of rigid motions in much the same way as I have used sequences of letters. The possibility of doing this was discovered by Felix Hausdorff, who showed that there exist ‘independent’ rotations, for which every different sequence of combinations leads to a distinct result-just as every different sequence of letters leads to a distinct word in the Hyperwebster. In spirit, the Banach-Tarski paradox is just the Hyperwebster paradox wrapped round a sphere.

Banach and Tarski actually proved something much stronger. Take any two sets in space, subject to two conditions: they don’t extend to infinity, and they each contain a solid sphere, which can be as small or as large as you wish. Then you can dissect one into the other. Squaring the circle may be tough, but Banach and Tarski could cube the sphere-getting any size of cube, to boot. They could dissect a football into a statue of Lady Godiva, a scale model of a rabbit 3 light years high, or a pinhead in the shape of the Taj Mahal.

The plane truth

The Banach-Tarski paradox does not arise in two dimensions. You cannot change areas in the plane by dissection. (You can change areas on the surface of a sphere.) The basic reason is that in the plane, the result of performing two rotations in succession is independent of the order. Analogously, in the corresponding Dyslexicon, the order of the letters does not matter. SPOTTER means the same as POTTERS. Now the dissection trick goes haywire: if we put SPOTTER in volume S, corresponding to POTTER in the original, then volume P fails to contain anything corresponding to OTTERS. So the dictionary approach certainly will not work. Tarski proved that nothing else would either. Banach had proved the existence, in the plane, of what is now called a Banach measure. This is a concept of area, corresponding to the usual one for those sets that have areas in the usual sense, and defined for every set whatsoever, even those that do not have an area, which is rigid and finitely additive. It follows easily that no two-dimensional Banach-Tarski paradox is possible.

That is the point Tarski had reached when he posed his circle-squaring problem. If a circular disc can be cut into a finite number of pieces, which can be reassembled to form a square, then the square must have the ‘correct’ area. However, it was not clear to him whether such a process is possible.

The problem proved difficult, so mathematicians concentrated on special cases: restricting the nature of the pieces, or the type of rigid motion allowed. In 1963, Lester Dubins, Morris Hirsch, and J. Karush proved that you cannot cut up a circle along continuous curves and reassemble the pieces to form a square. You cannot market a jigsaw puzzle whose pieces can be put together one way to form a circle and another way to form a square. If the circle can be squared by dissection, then the pieces have to be far more complex than anything that can be cut by the finest jigsaw.

In 1988, at a conference in Capri, an American mathematician named Richard Gardner gave a lecture. In it, he proved that, in any number of dimensions and not just two, there can be no solution to a version of Tarski’s circle-squaring problem in which a limited system of rigid motions is permitted (selected from any ‘discrete group’). He conjectured that the same is true if a rather broader system of rigid motions is allowed (‘amenable group’). In the audience was Miklos Laczkovich, a Hungarian mathematician from Lorand Eotvos University in Budapest. Laczkovich had been making a careful study of dissection problems in one dimension, and during Gardner’s talk he realised that his conjecture is false in the one-dimensional setting.

The discovery triggered a whole series of new ideas on Tarski’s circle-squaring problem itself, and within two months Laczkovich had a complete solution. You can square the circle by making a finite number of dissections. Even more surprisingly, the pieces do not even have to be rotated. The same goes for any shape whose boundary is composed of smoothly curving pieces, such as an ellipse, or a crescent, or the outline of an amoeba. Laczkovich’s method requires about 1050 pieces, so it is not something you can draw. The proof occupies about 40 pages, and involves a whole series of novel ideas: naturally, it gets rather technical.

One trick worth mentioning is a version of the Marriage theorem. A (rather old-fashioned) dating agency has a rule that clients may date only clients of the opposite sex to whom they have been properly introduced. Under what conditions can all clients secure a date simultaneously? Obviously, the numbers of men and women have to be equal, but that is not enough-for instance, some men may not have been introduced to any women at all. Even if each man has been introduced to at least one woman, there can still be problems: 200 men trying to date one woman, who has been introduced to them all. The more you think about it, the trickier it gets; but there is a very elegant solution. According to the Marriage theorem, a complete set of dates is possible if and only if every subset of men collectively has been introduced to at least as many women, and conversely. In Laczkovich’s proof the ‘men’ are possible candidates for pieces of the square, the ‘women’ are candidates for pieces of the circle, and ‘being introduced’ is the same as ‘related by a rigid motion’. The Marriage theorem lets Laczkovich ‘marry’ a suitable collection of pieces from the square to a corresponding collection from the circle. Much of the remainder of the proof tackles the construction of those candidate sets.

Despite Laczkovich’s brilliant breakthrough, the last word has not been said. Here is a (deceptively) simple example of the questions that remain unanswered. Can you cut a circular disc into three pieces which, apart from rigid motions, are identical? It is easy to do this if overlaps at edges are allowed (see Figure 3a), or for a disc with its central point missing (see Figure 3b). Fame and fortune-or at least a mention in the newspapers-await anyone who can work out how to deal with that one tricky point.

Ian Stewart is a professor of mathematics at the University of Warwick, and the author of Does God Play Dice? and Game, Set, and Math, both published by Blackwell.