So think of a computer’s VDU as a radar screen, with a single blip on it, whose movements are going to trace out the M-Set. However, before we switch on our radar, I want to make the equation even simpler, to:
Z = z2
I’ve thrown c away, for the moment, and left only the z’s. Now let me define them more precisely.
Small z is the initial range of the blip — the distance at which it starts. Big Z is its final distance from the origin. Thus if it was initially 2 units away, by obeying this equation it would promptly hop to a distance of 4.
Nothing to get very excited about, but now comes the modification that makes all the difference:
That double arrow is a two-way traffic sign, indicating that the numbers flow in both directions. This time, we don’t stop at Z = 4; we make that equal to a new z — which promptly gives us a second Z of 16, and so on. In no time we’ve generated the series
256, 65536, 4294967296 . . .
and the spot that started only 2 units from the center is heading toward infinity in giant steps of ever-increasing magnitude.
This process of going around and around a loop is called ‘iteration.’ It’s like a dog chasing its own tail, except that a dog doesn’t get anywhere. But mathematical iteration can take us to some very strange places indeed — as we shall soon discover.
Now we’re ready to turn on our radar. Most displays have range circles at 10, 20 . . . 100 kilometers from the center. We will require only a single circle, at a range of 1. There’s no need to specify any units, as we’re dealing with pure numbers. Make them centimeters or light-years, as you please.
Let’s suppose that the initial position of our blip is anywhere on this circle — the bearing doesn’t matter. So z is 1.
And because 1 squared is still 1, so is Z. And it remains at that value, because no matter how many times you square 1, it always remains exactly 1. The blip may hop around and around the circle, but it always stays on it.
Now consider the case where the initial z is greater than 1. We’ve already seen how rapidly the blip shoots off to infinity if z equals 2 — but the same thing will happen sooner or later, even if it’s only a microscopic shade more than 1 — say 1.000000000000000000001. Watch:
At the first squaring, Z becomes
1.000000000000000000002
then
1.000000000000000000004
1.000000000000000000008
1.000000000000000000016
1.000000000000000000032
and so on for pages of printout. For all practical purposes, the value is still exactly 1. The blip hasn’t moved visibly outward or inward; it’s still on the circle at range 1.
But those zeros are slowly being whittled away, as the digits march inexorably across from the right. Quite suddenly, something appears in the third, second, first decimal place — and the numbers explode after a very few additional terms, as this example shows:
1.001 1.002 1.004 1.008 1.016 1.032
1.066 1.136 1.292 1.668 2.783 7.745
59.987 3598.467 12948970
167675700000000
28115140000000000000000000000
(Overflow)
There could be a million — a billion — zeros on the right-hand side, and the result would still be the same. Eventually the digits would creep up to the decimal point — and then Z would take off to infinity.
Now let’s look at the other case. Suppose z is a microscopic amount less than 1 — say something like
.99999999999999999999
As before, nothing much happens for a long time as we go around the loop, except that the numbers on the far right get steadily smaller. But after a few thousand or million iterations — catastrophe! Z suddenly shrinks to nothing, dissolving in an endless string of zeros. . . .
Check it out on your computer. It can only handle twelve digits? Well, no matter how many you had to play with, you’d still get the same answer. Trust me. . . .
The results of this ‘program’ can be summarized in three laws that may seem too trivial to be worth formulating. But no mathematical truth is trivial, and in a few more steps these laws will take us into a universe of mind-boggling wonder and beauty.
Here are the three laws of the ‘Squaring’ Program:
1. If the input z is exactly equal to 1, the output Z always remains 1.
2. If the input is more than 1, the output eventually becomes infinite.
3. If the input is less than 1, the output eventually becomes zero.
That circle of radius 1 is therefore a kind of map — or, if you like, fence — dividing the plane into two distinct territories. Outside it, numbers which obey the squaring law have the freedom of infinity; numbers inside it are prisoners, trapped and doomed to ultimate extinction.
At this point, someone may say: ‘You’ve only talked about ranges — distances from the origin. To fix the blip’s position, you have to give its bearing as well. What about that?’
Very true. Fortunately, in this selection process — this division of the z’s into two distinct classes — bearings are irrelevant; the same thing happens whichever direction r is pointing. For this simple example — let’s call it the S-set — we can ignore them. When we come on to the more complicated case of the M-Set, where the bearing is important, there’s a very neat mathematical trick which takes care of it, by using complex or imaginary numbers (which really aren’t at all complex, still less imaginary). But we don’t need them here, and I promise not to mention them again.
The S-set lies inside a map, and its frontier is the circle enclosing it. That circle is simply a continuous line with no thickness. If you could examine it with a microscope of infinite power, it would always look exactly the same. You could expand the S-set to the size of the Universe; its boundary would still be a line of zero thickness. Yet there are no holes in it; it’s an absolutely impenetrable barrier, forever separating the z’s less than one from those greater than one.
Now, at last, we’re ready to tackle the M-Set, where these commonsense ideas are turned upside down. Fasten your seat belts.
During the 1970s, the French mathematician Benoit Mandelbrot, working at Harvard and IBM, started to investigate the equation which has made him famous, and which I will now write in the dynamic form:
The only difference between this and the equation we have used to describe the S-set is the term c. This — not z — is now the starting point of our mapping operation. The first time around the loop, z is put equal to zero.
It seems a trifling change, and no one could have imagined the universe it would reveal. Mandelbrot himself did not obtain the first crude glimpses until the spring of 1980, when vague patterns started to emerge on computer printouts. He had begun to peer through Keats’
Charm’d magic casements, opening on the foam
Of perilous seas, in faery lands forlorn . . .
As we shall learn later, that word ‘foam’ is surprisingly appropriate.
The new equation asks and answers the same question as the earlier one: What shape is the ‘territory’ mapped out when we put numbers into it? For the S-set it was a circle with radius 1. Let’s see what happens when we start with this value in the M-equation. You should be able to do it in your head — for the first few steps. After a few dozen, even a supercomputer may blow a gasket.
For starters, z = 0, c = 1. So Z = 1
First loop: Z = 12 + 1 = 2
Second loop: Z = 22 + 1 = 5
Third loop: Z = 52 + 1 = 26
Fourth loop: Z = 262 + 1 . . . and so on.
I once set my computer to work out the higher terms (about the limit of my programming ability) and it produced only two more values before it had to start approximating:
1, 2, 5, 26, 677, 458330,
21006640000
4412789000000000000000
At that point it gave up, because it doesn’t believe there are any numbers with more than 38 digits.
However, even the first two or three terms are quite enough to show that the M-Set must have a very different shape from the perfectly circular S-set. A point at distance 1 is in the S-set; indeed, it defines its boundary. A point at that same distance may be outside the boundary of the M-Set.
Note that I say ‘may,’ not ‘must.’ It all depends on the initial direction, or bearing, of the starting point, which we have been able to ignore hitherto because it did not affect our discussion of the (perfectly symmetrical) S-set. As it turns out, the M-Set is only symmetrical about the X, or horizontal, axis.