‘This is the complete set with no magnification,’ said Ada, in a voice that was now a little less self-assured — in fact, almost hushed.
‘Even on this scale, though, you can see how different it is from the plain, zero-thickness circle bounding the S-set. You could zoom that up forever and ever, and it would remain a line — nothing more. But the boundary of the M-Set is fuzzy — it contains infinite detail: you can go in anywhere you like, and magnify as much as you please — and you’ll always discover something new and unexpected — look!’
The image expanded; they were diving into the cleft between the main cardioid and its tangent circle. It was, Bradley told himself, very much like watching a zip-fastener being pulled open — except that the teeth of the zipper had the most extraordinary shapes.
First they looked like baby elephants, waving tiny trunks. Then the trunks became tentacles. Then the tentacles sprouted eyes. Then, as the image continued to expand, the eyes opened up into black whirlpools of infinite depth. . . .
‘The magnification’s up in the millions now,’ Edith whispered. ‘The picture we started with is already bigger than Europe.’
They swept past the whirlpools, skirting mysterious islands guarded by reefs of coral. Flotillas of seahorses sailed by in stately procession. At the screen’s exact center, a tiny black dot appeared, expanded, began to show a haunting familiarity — and seconds later revealed itself as an exact replica of the original set.
This, Bradley thought, is where we came in. Or is it? He could not be quite sure; there seemed to be minor differences, but the family resemblance was unmistakable.
‘Now,’ continued Ada, ‘our original picture is as wide as the orbit of Mars — so this mini-set’s really far smaller than an atom. But there’s just as much detail all around it. And so on forever.’
The zooming stopped; for a moment it seemed that a sample of lacework, full of intricate loops and whorls that teased the eye, hung frozen in space. Then, as if a paintbox had been spilled over it, the monochrome image burst into colors so unexpected, and so dazzlingly beautiful, that Bradley gave a gasp of astonishment.
The zooming restarted, but in the reverse direction, and in a micro-universe now transformed by color. No one said a word until they were back at the original complete M-Set, now an ominous black fringed with a narrow border of golden fire, and shooting off jagged lightnings of blues and purples.
‘And where,’ asked Bradley when he had recovered his breath, ‘did all those colors come from? We didn’t see them on the way in.’
Ada laughed. ‘No — they’re not really part of the set — but aren’t they gorgeous? I can tell the computer to make them anything I like.’
‘Even though the actual colors are quite arbitrary,’ Edith explained, ‘they’re full of meaning. You know the way map makers put shades of blue and green between contour lines, to emphasize differences in level?’
‘Of course; we do just the same thing in oceanography. The deeper the blue, the deeper the water.’
‘Right. In this case, the colors tell us how many times the computer’s had to go around the loop before it decides whether a number definitely belongs to the M-Set — or not. In borderline cases, it may have to do the squaring and adding routine thousands of times.’
‘And often for hundred-digit numbers,’ said Donald. ‘Now you understand why the set wasn’t discovered earlier.’
‘Mighty good reason.’
‘Now watch this,’ said Ada.
The image came to life as waves of color flowed outward. It seemed that the borders of the set itself were continually expanding — yet staying in the same place. Then Bradley realized that nothing was really moving; only the colors were cycling around the spectrum, to produce this completely convincing illusion of movement.
I begin to understand, Bradley thought, how someone could get lost in this thing — even make it a way of life.
‘I’m almost certain,’ he said, ‘that I’ve seen this program listed in my computer’s software library — with a couple of thousand others. How lucky I’ve never run it. I can see how addictive it could get.’
He noticed that Donald Craig glanced sharply at Edith, and realized that he had made a somewhat tactless remark. However, she still seemed engrossed by the flow of colors, even though she must have seen this particular display countless times.
‘Ada,’ she said dreamily, ‘give Mr. Jason our favorite quotation from Einstein.’
That’s asking a lot from a ten-year-old, thought Bradley — even one like this; but the girl never hesitated, and there was no trace of mechanical repetition in her voice. She understood the words, and spoke from the heart:
‘ ‘The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand wrapt in awe, is as good as dead.’ ‘
I’ll go along with that, thought Bradley. He remembered calm nights in the Pacific, with a sky full of stars and a glimmering trail of bioluminescence behind the ship; he recalled his first glimpse of the teeming life-forms — as alien as any from another planet — gathered around the scalding cornucopia of a Galápagos mid-ocean vent, where the continents were slowly tearing apart; and he hoped that before long he would feel awe and wonder again, when the tremendous knife edge of Titanic’s prow came looming up out of the abyss.
The dance of colors ceased: the M-Set faded out. Although nothing had ever been really there, he could somehow sense that the virtual screen of the holograph projector had switched off.
‘So now,’ said Donald, ‘you know more about the Mandelbrot Set than you want to.’ He glanced momentarily at Edith, and once again Bradley felt that twinge of sympathy toward him.
It was not at all the feeling he had expected, when he came to Conroy Castle; ‘envy’ would have been a better word. Here was a man with great wealth, a beautiful home, and a talented and attractive family — all the ingredients which were supposed to guarantee happiness. Yet something had obviously gone wrong. I wonder, Bradley thought, how long it is since they went to bed together. It could be as simple as that — though that, of course, was seldom simple. . . .
Once again he glanced at his watch; they must think he was deliberately avoiding the issue — and they were perfectly right. Hurry up, Mr. Director-General! he pleaded silently.
As if on cue, he felt the familiar tingling in his wrist.
‘Excuse me,’ he said to his hosts. ‘I’ve a very important call coming through. It will only take a minute.’
‘Of course. We’ll leave you to it.’
How many million times a day this ritual was now carried out! Strict etiquette dictated that everyone else offer to leave the room when a personal call was coming through; politeness demanded that only the recipient leave, with apologies to all. There were countless variations according to circumstances and nationalities. In Japan, so Kato was fond of complaining, the formalities often lasted so long that the caller hung up in disgust.
‘Sorry for the interruption,’ Bradley said as he came back in through the French windows. ‘That was about our business — I couldn’t give you a decision until I’d received it.’
‘I hope it’s a favorable one,’ said Donald. ‘We need you.’
‘And I would like to work with you — but — ‘
‘Parky’s made you a better offer,’ said Edith, with scarcely veiled contempt.
Bradley looked at her calmly, and answered without rancor.
‘No, Mrs. Craig. Please keep these figures confidential. The Parkinson group’s offer was generous — but it was only half of yours. And the offer which I’ve just received is much less than one tenth of that. Nevertheless, I’m considering it very seriously.’
There was a resounding silence, broken at last by an uncharacteristic giggle from Ada.
‘You must be crazy,’ said Edith. Donald merely grinned.
‘You may be right. But I’ve reached the stage when I don’t need the money, though it’s always good to have some around.’ He paused, and chuckled softly.
‘Enough is enough. I don’t know if you ever heard the wisecrack that Titanic’s most famous casualty, J. J. Astor, once made: ‘A man who has a million dollars is as well off as one who is rich.’ Well, I’ve made a few million during my career, and some of it’s still in the bank. I don’t really need any more; and if I do, I can always go down and tickle another octopus.
‘I didn’t plan this — it was a bolt from the blue — two days ago I’d already decided to accept your offer.’
Edith now seemed more perplexed than hostile.