A History of Mathematics From Mesopotamia to Modernity

248 A History ofMathematics

the screen. The most popular of all was thequadratic map(justax^2 +bx+c), in two forms: real

(Feigenbaum) and complex (Mandelbrot).

The underlying mathematics, after these simplifications, is not particularly difficult, although dif-

ficult and interesting results have been proved about such systems. Also, naturally, not everything

which was studied was chaotic; but quite simple maps could have both chaotic regions and other

more stable ones. The theory was simple, and the results were often surprising. The presentation,

particularly in Robert Devaney’s classic textbook (1992), could be clear and (relatively) access-

ible, and one would have the benefit of the striking pictures, absent from the average differential

equations course. What really seized the imagination of teachers, students, and popularizers alike

was the possibility that the computer—necessarily a finite system—could provide an image of

infinite complexity, the ‘Mandelbrot set’ being the universal icon which symbolized this. In one

well-known example, ‘Douady’s rabbit’ (Fig. 4), one is considering the behaviour of thecomplex

functionf(z)=z^2 +cwherec=−0.12+0.75iunder iteration. Within the grey area,fn(z)→∞

asn→∞; while within the black area there is a periodic orbit of period 3:

z 0 ,z 1 =f(z 0 ), z 2 =f^2 (z 0 ), f^3 (z 0 )=z 0

which isattracting;ifzis in this area,fn(z)tends to cycle round the periodic orbit asn→∞. And

the diminishing ‘rabbit-ears’ of the picture invite the viewer to visualize an infinite process, present

in the idealized mathematics if only suggested on screen.

The inside and the outside are regions wherefshows stable, non-chaotic behaviour, but the

boundary which separates the two—the ‘Julia set’—is, as might be expected, chaotic. This is true

both in the obvious sense that an arbitarily small deviation from the boundary will land you in

one or other of the stable sets, and also in the sense that nearby points on the boundary behave

completely differently under iteration. Considering such images one can ask, is chaos theory going

to be used more as a guide for analysing systems or as a means of producing art?

Fig. 4‘Douady’s rabbit’. Letf(z)=z^2 +c(a quadratic function), andc=0.12+0.75i. Then the black area, a region in the

complex plane, represents allzsuch that iterations off(z)do not tend to infinity.