“We’re taking the long-term view,” said Parkinson. “When our grandchildren come to Florida to dive on the Titanic, they won’t care whether we raised her up in 2012 or 2020—though of course we hope to make the centennial date.” He turned to the archaeologist. “I almost wish we could use Portsmouth, and arrange for a simultaneous opening. It would be nice to have Nelson’s Victory, Henry Eight’s Mary Rose, and Titanic side by side. Four hundred years of British shipbuilding. Quite a thought.”
“I’d be there,” said Kilford. “But now I’d like to raise a couple of more serious matters. First of all, there’s still much talk of… well, ‘desecration’ seems too strong a word, but what do you say to the people who regard Titanic as a tomb, and say she should be left in peace?”
“I respect their views, but it’s a little late now. Hundreds of dives have been made on her—and on countless other ships that have gone down with great loss of life. People only seem to raise objections to Titanic! How many people died in Mary Rose, Dr. Thornley? And has anyone protested about your work?”
“About six hundred—almost half as many casualties as Titanic—and for a ship a fraction of the size! No—we’ve never had any serious complaints; in fact the whole country approved of the operation. After all, it was mostly supported by private funds.”
“Another point which isn’t widely realized,” added Parkinson: “Very few people could have actually died in the Titanic; most of them got off, and were drowned or frozen.”
“No chance of bodies?”
“None whatsoever. There are lots of very hungry creatures down there.”
“Well, I’m glad we’ve disposed of that depressing subject. But there’s something perhaps more important….” Kilford picked up one of the little glass spheres, and rolled it between thumb and forefinger. “You’re putting billions of these in the sea. Inevitably, lots of them will be lost. What about the ecological impact?”
“I see you’ve been reading the Bluepeace literature. Well, there won’t be any.”
“Not even when they wash to shore—and our beaches are littered with broken glass?”
“I’d like to shoot the copywriter who coined that phrase—or hire him. First of all, it will take centuries—maybe millennia—for these spheres to disintegrate. And please remember what they’re made of—silica! So when they do eventually crumble, do you realize what they’ll turn into? That well-known beach pollutant—sand!”
“Good point. But what about the other objection? Suppose fish or marine animals eat them?”
Parkinson picked up one of the microspheres, and twirled it between his fingers just as Kilford had done.
“Glass is totally nonpoisonous—chemically inert. Anything big enough to swallow one of these won’t be hurt by it.”
And he popped the sphere into his mouth.
Behind the control panel, the producer turned to Roy Emerson.
“That was terrific—but I’m still sorry you wouldn’t go on.”
“Parky did very well without me. Do you think I’d have gotten in any more words than poor Dr. Thornley?”
“Probably not. And that was a neat trick, swallowing the microsphere—don’t think I could manage it. And I’ll make a bet that from now on, everyone’s going to call them Parky’s Pills.”
Emerson laughed. “I wouldn’t be surprised. And he’ll be asked to repeat the act, every time he goes on TV.”
He thought it unnecessary to add that, besides his many other talents, Parkinson was quite a good amateur conjurer. Even with freeze-frame, no one would be able to spot what had really happened to that pill.
And there was another reason why he preferred not to join the panel—he was an outsider, and this was a family affair.
Though they lay centuries apart, Mary Rose and Titanic had much in common. Both were spectacular triumphs of British shipbuilding genius—sunk by equally spectacular examples of British incompetence.
20.
INTO THE M-SET
It was hard to believe, Jason Bradley told himself, that people actually lived like this, only a few generations ago. Though Conroy Castle was a very modest example of its species, its scale was still impressive to anyone who had spent most of his life in cluttered offices, motel rooms, ships’ cabins—not to mention deep-diving minisubs, so cramped that the personal hygiene of your companions was a matter of crucial importance.
The dining room, with its ornately carved ceiling and enormous wall mirrors, could comfortably seat at least fifty people. Donald Craig felt it necessary to explain the little four-place table that looked lost and lonely at its center.
“We’ve not had time to buy proper furniture. The castle’s own stuff was in terrible shape—most of it had to be burned. And we’ve been too busy to do much entertaining. But one day, when we’ve finally established ourselves as the local nobility…”
Edith did not seem to approve of her husband’s flippancy, and once again Bradley had the impression that she was the leader in this enterprise, with Donald a reluctant—or at best passive—accomplice. He could guess the scenario: people with enough money to squander on expensive toys often discovered that they would have been happier without them. And Conroy Castle—with all its surrounding acres and maintenance staff—must be a very expensive toy indeed.
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 Mandelvirgin 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.
“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.”
The Ghost from the Grand Banks (Arthur C. Clarke Collection) Page 9