When the servants (servants! — that was another novelty) had cleared the remnants of an excellent Chinese dinner flown in especially from Dublin, Bradley and his hosts retreated to a set of comfortable armchairs in the adjoining room.
‘We won’t let you get away,’ said Donald, ‘without giving you our Child’s Guide to the M-Set. Edith can spot a Mandel-virgin at a hundred meters.’
Bradley was not sure if he qualified for this description. He had finally recognized the odd shape of the lake, though he had forgotten its technical name until reminded of it. In the last decade of the century, it had been impossible to escape from manifestations of the Mandelbrot Set — they were appearing all the time on video displays, wallpaper, fabrics, and virtually every type of design. Bradley recalled that someone had coined the word ‘Mandelmania’ to describe the more acute symptoms; he had begun to suspect that it might be applicable to this odd household. But he was quite prepared to sit with polite interest through whatever lecture or demonstration his hosts had in store for him.
He realized that they too were being polite, in their own way. They were anxious to have his decision, and he was equally anxious to give it.
He only hoped that the call he was expecting would come through before he left the castle. . . .
Bradley had never met the traditional stage mother, but he had seen her in movies like — what was that old one called? — ah, Fame. Here was the same passionate determination on the part of a parent for a child to become a star, even if there was no discernible talent. In this case, he did not doubt that the faith was fully justified.
‘Before Ada begins,’ said Edith, ‘I’d like to make a few points. The M-Set is the most complex entity in the whole of mathematics — yet it doesn’t involve anything more advanced than addition and multiplication — not even subtraction or division! That’s why many people with a good knowledge of math have difficulty in grasping it. They simply can’t believe that something with too much detail to be explored before the end of the Universe can be generated without using logs or trig functions or higher transcendentals. It doesn’t seem reasonable that it’s all done merely by adding numbers together.’
‘Doesn’t seem reasonable to me, either. If it’s so simple, why didn’t anyone discover it centuries ago?’
‘Very good question! Because so much adding and multiplying is involved, with such huge numbers, that we had to wait for high-speed computers. If you’d given abacuses to Adam and Eve and all their descendants right up to now, they couldn’t have found some of the pictures Ada can show you by pressing a few keys. Go ahead, dear. . . .’
The holoprojector was cunningly concealed; Bradley could not even guess where it was hiding. Very easy to make this old castle a haunted one, he thought, and scare away any intruders. It would beat a burglar alarm.
The two crossed lines of an ordinary x-y diagram appeared in the air, with the sequence of integers 0, 1, 2, 3, 4 . . . marching off in all four directions.
Ada gave Bradley that disconcertingly direct look, as if she were once again trying to estimate his I.Q. so that her presentation could be appropriately calibrated.
‘Any point on this plane,’ she said, ‘can be identified by two numbers — its x- and y-coordinates. Okay?’
‘Okay,’ Bradley answered solemnly.
‘Well, the M-Set lies in a very small region near the origin — it doesn’t extend beyond plus or minus two in either direction, so we can ignore all the larger numbers.’
The integers skittered off along the four axes, leaving only the numbers one and two marking distances away from the central zero.
‘Now suppose we take any point inside this grid, and join it to the center. Measure the length of this radius — let’s call it r.’
This, thought Bradley, is putting no great strain on my mental resources. When do we get to the tricky part?
‘Obviously, in this case r can have any value from zero to just under three — about two point eight, to be exact. Okay?’
‘Okay.’
‘Right. Now Exercise One. Take any point’s r value, and square it. Keep on squaring it. What happens?’
‘Don’t let me spoil your fun, Ada.’
‘Well, if r is exactly one, it stays at that value — no matter how many times you square it. One times one times one times one is always one.’
‘Okay,’ said Bradley, just beating Ada to the draw.
‘If it’s even a smidgin more than one, however, and you go on squaring it, sooner or later it will shoot off to infinity. Even if it’s 1.0000 . . . 0001, and there are a million zeros to the right of the decimal point. It will just take a bit longer.
‘But if the number is less than one — say .99999999 . . . with a million nines — you get just the opposite. It may stay close to one for ages, but as you keep on squaring it, suddenly it will collapse and dwindle away down to zero — okay?’
This time Ada got there first, and Bradley merely nodded. As yet, he could not see the point in this elementary arithmetic, but it was obviously leading somewhere.
‘Lady — stop bothering Mr. Bradley! So you see, simply squaring numbers — and going on squaring them, over and over — divides them into two distinct sets. . . .’
A circle had appeared on the two crossed axes, centered on the origin and with radius unity.
‘Inside that circle are all the numbers that disappear when you keep on squaring them. Outside are all those that shoot off to infinity. You could say that the circle of radius one is a fence — a boundary — a frontier — dividing the two sets of numbers. I like to call it the S-set.’
‘S for squaring?’
‘Of cour — Yes. Now, here’s the important point. The numbers on either side are totally separated; yet though nothing can pass through it, the boundary hasn’t any thickness. It’s simply a line — you could go on magnifying it forever and it would stay a line, though it would soon appear to be a straight one because you wouldn’t be able to see its curvature.’
‘This may not seem very exciting,’ interjected Donald, ‘but it’s absolutely fundamental — you’ll soon see why — sorry, Ada.’
‘Now, to get the M-Set we make one teeny-weeny change. We don’t just square the numbers. We square and add . . . square and add. You wouldn’t think it would make all that difference — but it opens up a whole new universe. . . .
‘Suppose we start with one again. We square it and get one. Then we add them to get two.
‘Two squared is four. Add the original one again — answer five.
‘Five squared is twenty-five — add one — twenty-six.
‘Twenty-six squared is six hundred seventy-six — you see what’s happening! The numbers are shooting up at a fantastic rate. A few more times around the loop, and they’re too big for any computer to handle. Yet we started with — one! So that’s the first big difference between the M-Set and the S-set, which has its boundary at one.
‘But if we started with a much smaller number than one — say zero point one — you’ll probably guess what happens.’
‘It collapses to nothing after a few cycles of squaring and adding.’
Ada gave her rare but dazzling smile.
‘Usually. Sometimes it dithers around a small, fixed value — anyway, it’s trapped inside the set. So once again we have a map that divides all the numbers on the plane into two classes. Only this time, the boundary isn’t something as elementary as a circle.’
‘You can say that again,’ murmured Donald. He collected a frown from Edith, but pressed on. ‘I’ve asked quite a few people what shape they thought would be produced; most suggested some kind of oval. No one came near the truth; no one ever could. All right, Lady! I won’t interrupt Ada again!’
‘Here’s the first approximation,’ continued Ada, scooping up her boisterous puppy with one hand while tapping the keyboard with the other. ‘You’ve already seen it today.’
The now-familiar outline of Lake Mandelbrot had appeared superimposed on the grid of unit squares, but in far more detail than Bradley had seen it in the garden. On the right was the largest, roughly heart-shaped figure, then a smaller circle touching it, a much smaller one touching that — and the narrow spike running off to the extreme left and ending at — 2 on the x-axis.
Now, however, Bradley could see that the main figures were barnacled — that was the metaphor that came instantly to mind — with a myriad of smaller subsidiary circles, many of which had short jagged lines extending from them. It was a much more complex shape than the pattern of lakes in the garden — strange and intriguing, but certainly not at all beautiful. Edith and Ada, however, were looking at it with a kind of reverential awe, which Donald did not seem to entirely share.