Exploring the Fractal Universe

Abstract

In November 1989, when receiving the Association of Space Explorers’ Special Achievement Award in Riyadh, Saudi Arabia, I had the privilege of addressing the largest gathering of astriiionauts and cosmonauts ever assembled at one place (more than fifty, including Apollo 11’s Buzz Aldrin and Mike Collins, and the first ‘space walker’ Alexei Leonov … I decided to expand their horizons by introducing them to something really large, and, with astronaut Prince Sultan bin Salman bin Abdul Azïz in the chair, delivered a lavishly illustrated lecture ‘The Colours of Infinity: Exploring the Fractal Universe’

2010 © The Estate of Arthur C Clarke.

Mathematical Appendix

One way of appreciating where the curiously-shaped country of the M-set is located on the map of all possible (complex) numbers is to pin down its Eastern and Western frontiers, ignoring everything to the North and South.

The Western, or negative, limit is easily identified; for once, the calculation can be done mentally, without the aid of a computer! If we take the basic equation: Z = z² + c and set the initial value of c equal to −2, the first time round the loop gives Z = −2. The second value is Z = (−2)² − 2 = 2. The third value is Z = 2² − 2 = 2.

And so on for ever: Z is stuck at 2! It does not shrink to zero, but neither does it go racing off to infinity. Thus the point at −2 on the X-axis, or 2 units to the left of the origin, definitely belongs to the M-set. It masks the Utter West – the very tip of the strangely ornamented spike that extends in that direction.

It’s interesting to see what happens for values of c on either side of −2, and for that we certainly do need a computer. Take c = −1.99999. Table 1 shows what happens to Z as it goes round and round the loop:

Table 1 c = −1.99999 (reading the numbers left to right)

The value then goes on oscillating, presumably forever (my computer has been round the loop only about ten thousand times) between the limits of plus and minus 2. Perhaps after ten million iterations Z might change its mind and suddenly shoot off to infinity, but it seems reasonable to assume that this value of c is definitely inside the M-set.

The fate of the point only 0.00002 units further ‘west’, on the other hand, is very quickly decided, as we see in Table 2:

Table 2 c = −2.00001

As far as an Apple Mac is concerned, the numbers in that last line are infinite, and I doubt if even a Super Cray would disagree. So −2.00001 is definitely outside the M-set.

On the Eastern, or positive side of the Set, the limit is not so easily defined.

Obviously, it is closer to the origin (0,0) than the point + 1, which gives a value shooting off to infinity after only a few times round the loop. A few minutes’ work with pencil and paper shows that it is even closer than 0.5, for putting c = ¹/₂ also gives a rapidly soaring Z. It is, in fact, at 0.25 – though this is by no means easy to prove.

When I set c = 0.25 in the program I have painfully written, the screen is flooded with a torrent of numbers, which after hundreds of iterations finally settle down to the odd value 0.4998505. I assume that this should be exactly 0.5, with the difference due to rounding-off errors. In any event, Z doesn’t shoot off to infinity, so the Eastern limit of the M-set is definitely at 0.25. (On the centre line, that is; above and below, it bulges considerably further eastwards.)

It’s interesting to check what happens when bracketing this value and setting c equal to 0.24999 and 0.25001. Table 3 gives the result of the first:

Table 3 c=24999

Although all these calculations involve only the X-coordinate, and ignore complex numbers by setting Y = 0, they can be very time-consuming. Tables 3 and 4 demonstrate how impossible it would have been to discover – let alone map in detail! – the Mandelbrot Set before the advent of modern computers.

Table 4 c=25001