Justifying the means
Ian Stewart

WHO was the leading mathematician of all time? To many mathematicians, the top candidate is Carl Friedrich Gauss, a bricklayer's son born in 1777 in Brunswick. So wide-ranging were Gauss's ideas, and so high were his standards of publication, that later mathematicians repeatedly found that Gauss had anticipated their life's work but kept it to himself. Examples are Bolyai and Lobachevskii's discovery of non-euclidean geometry, Cauchy's work in complex analysis, and Jacobi's theory of elliptic integrals. It would not be a great surprise if an obscure manuscript by Gauss on the Mandelbrot set were to turn up.

Although no such work is known, Gauss had some closely related ideas, which are now being revived as the Mandelbrot set focuses attention on what is now called complex dynamics. They are interpreted from the modern viewpoint, and generalized , in a recent paper by Shaun Bullett (Topology 30, 171-190; 1991).

The arithmetic mean of two numbers is half their sum; the geometric mean is the square root of their product. Gauss invented a mixture of the two, which he called the arithmetico geometric mean (a.g.m. for short). Start with a pair of numbers (a, b). Calculate their arithmetic and geometric means, to get a new pair (a, b) = (½(a+b), Öab); repeat this process indefinitely. The two numbers converge to a common limit, the a.g.m. of the original numbers. This is a deceptively simple procedure: it conceals some remarkably deep mathematics.

Gauss kept a mathematical diary, and the entry for 30 May 1799 records that the a.g.m. of 1 and Ö2 is 2p divided by the total length of a figure-of-eight shaped curve known as a lemniscate, whose formula given in polar coordinates is r2=cos 2q. He discovered this by evaluating the length to 11 decimal places. He correctly predicted that "the demonstration of this fact will open up an entirely new field of analysis". That field occupied the attention of many nineteenth-century mathematicians: it is the theory of elliptic functions, and it too is experiencing something of a revival, for various reasons.

Gauss had a great interest in complex numbers x+iy where i=Ö-1. The above discussion of the a.g.m. assumes the initial numbers are real and positive, because square roots are involved in the geometric mean, and negative real numbers do not have a real square root. All complex numbers possess square roots, so the concept of a.g.m. extends naturally to arbitrary pairs of complex numbers. However, any complex number other than 0 has two distinct square roots, so the calculation of the a.g.m. involves an arbitary choice at each step. Gauss wondered what the set of all possible values of the a.g.m., obtained by making all possible sequences of choices of square root, would look like.

The modern approach, adopted by Bullett, is to view his procedure as a 'quadratic correspondence', a two-valued map z®w. Suppose a and b are multiplied by some real constant. Then so are the arithmetic and geometric means. In other words, we can always scale the value of b to equal 1. Then the pair (z,1) leads to a new pair (w,1), given by the quadratic equation 4w2z-z-(z+1)2=0. Call this the a.g.m. correspondence. It is an example of a more general concept, that of a quadratic correspondence. Conventional discrete dynamics studies the effect of iterating a single-valued mapping; here a two-valued mapping must be iterated, but the situation is closely analogous. The famous Mandlebrot set, and its relatives, the Julia sets, arise from conventional dynamics in the complex plane. Gauss was studying a two-valued analogue.

Dynamicists study orbits - the sets traced out by a given initial point under iteration of a mapping. The simplest orbits are fixed points, which map to themselves; the next simplest are periodic points, which repeat a cycle of values; and at the other extreme are chaotic orbits that display no obvious pattern whatsoever. The fixed points for the a.g.m. correspondence are 1 and ¥. The point 1 is attractive: nearby points move towards it when the mapping is applied. The ¥ point is repulsive.

Orbits of the a.g.m. correspondence reveal an intricate packing of Siegel disks (white regions).

But the dynamics is far more intricate than just a steady flow away from ¥  towards 1, as Bullett's computer plot (see figure) of a restricted set of orbits illustrates. The blank regions in this picture are called Siegel disks. At the centre of each Siegel disk lives a periodic point; and these points are surrounded by closed curves, along which points move much as if the curves were rotating through various angles at each application of the mapping. The fractal nature of the picture is evident, and similar Siegel disks can be seen in The Beauty of Fractals by H.-O. Peitgen and P. Richter (Springer, New York, 1988), a book that is largely about Mandelbrot and Julia sets.

The connection with the lemniscate led Gauss in a rather different direction (though one might speculate on what would have happened if there had been a workstation on his desk). While trying to calculate the length of the curve, he invented what are now called theta functions, subsequently developed extensively by Jacobi. They are defined by certain infinite series, for example

p(t) =Sexp(n2pit)


summed over all positive and negative integers n. They satisfy some remarkable identities; in particular the arithmetic mean of p(t)2 and q(t)2 is p(2t)2, and their geometric mean is q(2t)2 . In short, if we start the procedure for computing the a.g.m. not with (a,b) but with p(t)2,q(t)2), then the next step just re places t by 2t. The map t®2t thus 'induces' the a.g.m. correspondence, but is far simpler to think about.

Using his version of these ideas, Gauss came very close indeed to solving his question about the set of values produced by making arbitrary choices of square roots in the a.g.m. calculation. Suppose the starting values are (a, b). Let l and m be the 'simplest' values of a.g.m. (a,b) and a.g.m. (a+b,a-b). Then the other possible values m' of a.g.m. (a,b) are given by 1/m'=d/m+ic/l where c and d are integers with no common factor, c is a multiple of 4, and d-1 is a multiple of 4. This may seem a complicated answer, but it is direct and explicit once the theta functions have been introduced. In a sense, their role is to 'linearize' the a.g.m. correspondence by a cunning change of variable. The conditions on c and d also reveal deep connections with number theory.
In a recent preprint (Critically finite correspondences and subgroups of the modular group",Queen Mary and Westfield College,1991),Bullet obtains a remarkable classification of a more general class of correspondences with strongly constrained dynamics.Among them is a finite set of quadratic correspondences having curious connections with regular solids: the a.g.m. correspondence is one of the simplest,and its solid is an octahedron.Many problems about such correspondences remain unsolved,but the subject now has a starting point.Gauss's intellectual legacy still contains unrevealed niches.

Ian Stewart is in the Mathematics Institute,Warwick University,Coventry CV4 7AL,UK.






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