One might have guessed that, from the nature of the equation. But no one could possibly have intuited its real appearance. If the question had been put to me in virginal pre-Mandelbrot days, I would probably have hazarded: ‘Something like an ellipse, squashed along the Y-axis.’ I might even (though I doubt it) have correctly guessed that it would be shifted toward the left, or minus, direction.
At this point, I would like to try a thought experiment on you. The M-Set being literally indescribable, here’s my attempt to describe it:
Imagine you’re looking straight down on a rather plump turtle, swimming westward. It’s been crossed with a swordfish, so has a narrow spike pointing ahead of it. Its entire perimeter is festooned with bizarre marine growths — and with baby turtles of assorted sizes, which have smaller weeds growing on them. . . .
I defy you to find a description like that in any math textbook. And if you think you can do better when you’ve seen the real beast, you’re welcome to try. (I suspect that the insect world might provide better analogies; there may even be a Mandelbeetle lurking in the Brazilian rain forests. Too bad we’ll never know.)
Here is the first crude approximation, shorn of details — much like Conroy Castle’s ‘Lake Mandelbrot’ (Chapter 18). If you like to fill its blank spaces with the medieval cartographers’ favorite ‘Here be dragons’ you will hardly be exaggerating.
First of all, note that — as I’ve already remarked — it’s shifted to the left (or west, if you prefer of the S-set, which of course extends from +1 to –1 along the X-axis. The M-Set only gets to 0.25 on the right along the axis, though above and below the axis it bulges out to just beyond 0.4.
On the left-hand side, the map stretches to about –1.4, and then it sprouts a peculiar spike — or antenna — which reaches out to exactly –2.0. As far as the M-Set is concerned, there is nothing beyond this point; it is the edge of the Universe. Mandelbrot fans call it the ‘Utter West,’ and you might like to see what happens when you make c equal to –2. Z doesn’t converge to zero — but it doesn’t escape to infinity either, so the point belongs to the set — just. But if you make c the slightest bit larger, say –2.00000 . . . 000001, before you know it you’re passing Pluto and heading for Quasar West.
Now we come to the most important distinction between the two sets. The S-set has a nice, clean line for its boundary. The frontier of the M-Set is, to say the least, fuzzy. Just how fuzzy you will begin to understand when we start to ‘zoom’ into it; only then will we see the incredible flora and fauna which flourish in that disputed territory.
The boundary — if one can call it that — of the M-Set is not a simple line; it is something which Euclid never imagined, and for which there is no word in ordinary language. Mandelbrot, whose command of English (and American) is awesome, has ransacked the dictionary for suggestive nouns. A few examples: foams, sponges, dusts, webs, nets, curds. He himself coined the technical name fractal, and is now putting up a spirited rearguard action to stop anyone from defining it too precisely.
Computers can easily make ‘snapshots’ of the M-Set at any magnification, and even in black and white they are fascinating. However, by a simple trick they can be colored, and transformed into objects of amazing, even surreal, beauty.
The original equation, of course, is no more concerned with color than is Euclid’s Elements of Geometry. But if we instruct the computer to color any given region in accordance with the number of times z goes around the loop before it decides whether or not it belongs to the M-Set, the results are gorgeous.
Thus the colors, though arbitrary, are not meaningless. An exact analogy is found in cartography. Think of the contour lines on a relief map, which show elevations above sea level. The spaces between them are often colored so that the eye can more easily grasp the information conveyed. Ditto with bathymetric charts; the deeper the ocean, the darker the blue. The mapmaker can make the colors anything he likes, and is guided by aesthetics as much as geography.
It’s just the same here — except that these contour lines are set automatically by the speed of the calculation — I won’t go into details. I have not discovered what genius first had this idea — perhaps Monsieur M. himself — but it turns them into fantastic works of art. And you should see them when they’re animated. . . .
One of the many strange thoughts that the M-Set generates is this. In principle, it could have been discovered as soon as the human race learned to count. In practice, since even a ‘low magnification’ image may involve billions of calculations, there was no way in which it could even be glimpsed before computers were invented! And such movies as Art Matrix’ Nothing But Zooms would have required the entire present world population to calculate night and day for years — without making a single mistake in multiplying together trillions of hundred-digit numbers. . . .
I began by saying that the Mandelbrot Set is the most extraordinary discovery in the history of mathematics. For who could have possibly imagined that so absurdly simple an equation could have generated such — literally — infinite complexity, and such unearthly beauty?
The Mandelbrot Set is, as I have tried to explain, essentially a map. We’ve all read those stories about maps which reveal the location of hidden treasure.
Well, in this case — the map is the treasure!
Colombo, Sri Lanka
1990 February 28