Kilford turned to his other guest.
“You could have done with these on the Mary Rose, back in 1982—couldn’t you, Dr. Thornley?”
The marine archaeologist shook her head. “Not really. That was a totally different problem. Mary Rose was in shallow water, and our divers were able to place a cradle under her. Then the biggest floating crane in the world pulled her up.”
“It was touch and go, wasn’t it?”
“Yes. A lot of people nearly had heart attacks when that metal strap gave way.”
“I can believe it. Now, that hull has been sitting in Southampton Dock for a quarter century—and it still isn’t ready for public display. Will you do a quicker job on Titanic, Mr. Parkinson—assuming that you do get her up?”
“Certainly; it’s the difference between wood and steel. The sea had four centuries to soak into Mary Rose’s timbers—no wonder it’s taking decades to get it out. All the wood in Titanic has gone—we don’t have to bother about it. Our problem is rust; and there’s very little at that depth, thanks to the cold and lack of oxygen. Most of the wreck is in one of two states: excellent—or terrible.”
“How many of these little . . . microspheres . . . will you need?”
“About fifty billion.”
“Fifty billion! And how will you get them down there?”
“Very simply. We’re going to drop them.”
“With a little weight attached to each one—another fifty billion?”
Parkinson smiled, rather smugly.
“Not quite. Our Mr. Emerson has invented a technique so simple that no one believes it will work. We’ll have a pipe leading down from the surface to the wreck. The water will be pumped out—and we’ll simply pour the microspheres in at the top, and collect them at the bottom. They’ll take only a few minutes to make the trip.”
“But surely—”
“Oh, we’ll have to use special air locks at both ends, but it will be essentially a continuous process. When they arrive, the microspheres will be packaged in bundles, each a cubic meter in volume. That will give a buoyancy of one ton per unit—a comfortable size for the robots to handle.”
Marcus Kilford turned to the long-silent archaeologist.
“Dr. Thornley,” he asked, “do you think it will work?”
“I suppose so,” she said reluctantly, “but I’m not the expert on these matters. Won’t that tube have to be very strong, to stand the enormous pressure at the bottom?”
“No problem; we’ll use the same material. As my company’s slogan says, ‘You can do anything with glass’—”
“No more commercials, please!”
Kilford turned toward the camera, and intoned solemnly, though with a twinkle in his eye: “May I take this opportunity of denying the malicious rumor that Mr. Parkinson was spotted in a BBC cloak room, handing me a shoe box stuffed with well-used bank notes.”
Everyone laughed, though behind the thick glass of the control room the producer whispered to his assistant: “If he uses that joke once again, I’ll suspect it’s true.”
“May I ask a question?” said Dr. Thornley unexpectedly. “What about your . . . shall I say, rivals? Do you think they’ll succeed first?”
“Well, let’s call them friendly competitors.”
“Indeed?” said Kilford skeptically. “Whoever brings their section up to the surface first will get all the publicity.”
“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 he 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 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 ma
p 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.