|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
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
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
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
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
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
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
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
NATURE VOL 354 21