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beyond
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a
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You can read the first 80% of this story for free; if you like it, you can read the rest for $.79 (payable by paypal or credit card.)
[ Read more about author Andrew Burt ]
Whoever says Pi is a constant simply hasn't checked it precisely enough — which may be a good thing!
Δ π
(Delta Pi)
by Andrew Burt
Monday, 06/08/2009
(being the date,
and the first occurrence
of the digits 06082009 in Pi,
starting at the 971,051st decimal.)
Kinsey Stafford noticed it--or, rather, the effect of it--in his office at the Center for Mathematical Studies, Cambridge, England. His chest tightened and he hung his head in grief: His program to calculate the deep digits of pi had suddenly crashed. In a race against time to save his career, he'd just lost precious months' work.Monica Ozawa, at the other end of the video link in Kuala Lumpur, noticed it too. "What in hell was that?" Her program crashed likewise.
The staff at the Rio superconducting supercollider noticed it, thousands of miles away from either.
Two small children playing in Mexico City really noticed it as their cinderblock walls crashed down on them from the resulting earthquake, an intensity XII on the twelve degree Mercalli damage index. The media, still in love with the useless Richter scale, called it as an 8.7. At least two hundred thousand dead, they reported without emotion.
And a junior high school student noticed it in Fertile, Minnesota, doing her Intro to Trig homework. Not that she knew or cared how similar the cosine algorithm in her calculator was to Kinsey's or Monica's pi routines. She knitted her brows: cos(0.5) should not equal 1.000000000 as displayed on the screen. She cleared the entry and retyped, carefully, point, five, cosine. 0.877582562; that looked better. She resumed her calculations, as oblivious as all the others to the monumental discovery mankind had just made.
Sunday, 06/28/2009
(Pi, 617,323rd decimal)
Almost a month later, Kinsey was still blissfully unaware of the role he'd played in the Mexico City carnage. He wasn't cruel or psychotic; had he known, he certainly wouldn't be running the program again. He was, however, narrowing his eyes in disbelief as before."Ozawa," he commanded the screen. Oh, damn, the eight hour time difference--she was going to be pissed off.
"Kinsey?" She craned her neck to check the time. "You forgot again, didn't you." She yawned. "It's three A.M. This is Earth-shatteringly important, right?"
His sheepish grin quickly gave way to his natural excitement. "Sorry, I, uh... Yes. Yes, it is! It's the first time in three weeks that I've had time to get on T. Rex to check the Gauss-Legendre-Stafford convergence, and, uh... Earth-shattering, yes!--sorry, I'm rambling--been up two days--paper deadline--tenure thing--look, have you compared your recent results from your hacked Borwein with your old ones?"
Her eyes were closed as she talked, indicating the depth to which she didn't care. Summing up enormous quantities of ever shrinking fractions as the total homed in on some value, like pi, wasn't the most glamorous work, but she took her shared research into rapid convergence algorithms as seriously as Kinsey. Though, having no career-ending tenure decision riding on it, she didn't feel compelled to work twenty hours a day on it. "No, sorry, we don't have the luxury of a dinosaur system like you society boys."
She smiled at his grimace. She enjoyed poking fun at his swellheaded employers--having chosen not to accept their offer four years earlier, instead opting for the slightly less impressive, but much less stressful, world of Bell-Matsushita Lab's Southeast Asian site. He might have access to T. Rex, a "DiNosAur" DNA-based supercomputer to play with, but she hadn't taken to the high intensity life that the Kinsey Staffords of the world thrived on. With only an old Hyper-Cray, she'd still published twice as many papers as Kinsey since she'd left CMS, a sore point she impishly played on.
He was far too serious lately, with only six months until the curmudgeonly faculty decided his tenure, and it was not looking good. He was brilliant, but never satisfied enough to commit to publication. His algorithmic work was make or break--and 'break' from the CMS meant he might as well look for a job as a dishwasher: CMS only took people they knew they would keep, but not being kept was a scarlet letter. She didn't have that same pressure, which made collaboration with him difficult at times. Like now. "Besides, I didn't restart it until last week; it hasn't reached where it was when it bombed. It's only at a couple trillion digits or so."
"Go check it!" His head bounced energetically, eyes wide.
She looked at him sourly; he was far too awake for her tastes. "Can't this wait until our usual link time, tomorrow?"
"No! Right now! Listen, my GLS is at about twenty trillion--and--and--it's changed! Can you say 'Nobel Prize'? My birthday, the digits used to be around fifty million, now they're at two hundred million! And the earthquakes, this must be what's causing them!"
Talking through another yawn, she managed, "Yu-ih nah may-ing en-ih sense."
"Pi, damnit! Pi! It's changed!"
He paced around his office irritably for the half hour it took her to reach her office and call back. Why didn't she work closer to the office? Or have a cot in the office, like he did. Much more efficient that way; he'd have to suggest it.Monica still looked baggy-eyed, but mentally she was wide awake, almost panicked when she finally called. "It can't be. Ok, what do you have at, say, nine hundred billion?"
"Drum roll please! Starting at the nine-hundred-billionth decimal place of pi we have: 11597--"
"No, I have 61630."
"Exactly! And I know you're going to say it's a bug in your program, or mine. It's not. What did you have before? I had, mmm,"--he panned the display on the screen--"38072."
"Hang on, I'm looking."
He drummed his fingers loudly in anticipation.
"Ok, I had 38072 also."
"So? You see?"
"Well, ok, so somehow both our programs became corrupted. Or it could be hardware."
"Oh, c'mon, Monica. I wouldn't call you in a tizzy at three A.M. if I hadn't been thorough, you should know me better than that by now." He gave her a reprimanding grin. He could smell tenure. Better: A Nobel prize. "I've tried five algorithms on four different machines, all twenty combinations. And they all differ, and--get this--they all differ at different places! They all agreed, every time, up to about a hundred billion decimals. The majority agreed out to five hundred billion. In two cases, two agreed out to nearly a trillion. And those two pairs? They agreed on two different values! That's why the programs crashed."
Monica looked distressed. "And you say the timing of your spikes in pi coincide with the earthquakes?" She imagined the seismographs spiking alongside some curve of Kinsey's. She felt the burden of millions of deaths, past and future, pressing on her chest. "Are you sure?"
"Yes, yes," Kinsey waved it off, and straightened up--in his exuberance he'd leaned forward until his face almost touched the screen. He adjusted his tie. "Like it or not, pi has changed. Not only changed, it's fluctuating."
Thursday, 08/06/2009
(Pi, 805,788th decimal)
The core of the earth belched. The number of earthquakes had been increasing, in both frequency and severity. "Just random variations," the geologists said, and joked on the morning talk shows that if they weren't, the quakes would rip the earth's crust apart in less than a year. And of course, they quipped, that wasn't going to happen, was it?Some quakes occurred so deep that they were mere trifles by the time they reached the surface, but not all--this time, bubbles of indigestion bumped the Pacific plate near Wake Island, causing a series of rapid seaquakes that flattened many a fish unfortunate enough to be near the landslides at the rocky bottom. Few people noticed. Instead, they noticed the tsunamis smashing Tokyo and Honolulu. The media endlessly played the miraculously surviving video of a family being lofted by a wave and dashed against a luxury hotel.
The first wave was the worst, carrying the most energy, direct from the first hiccup on the sea floor. But its siblings followed, biggest to smallest, like the waves spreading from a rock tossed into a pond.
The supercollider crew noticed it again. Yet another in their sequence of experiments using precisely timed collisions of supercooled particles, immobilized by microscopic black holes--had failed. Rather, it had succeeded too well, giving results contrary to the laws of quantum mechanics. One of the MIT grad students on a summer internship noticed, Davis O'Reilly, even commented (a little too seriously) on the high correlation of earthquakes to the experiments--and was summarily shunned as if he were a superstitious peasant from the Middle Ages.
Kinsey, asleep on his cot, noticed none of it. His programs, hundreds running on hundreds of systems, all forged ahead, having been carefully constructed to survive the vacillations in pi.
Monica, however, felt sickened. Her own grandmother had been in one of the villages. Like some of the other elders, she'd refused to leave her home. None in the village had survived. Monica couldn't unglue her eyes from the scenes of carnage.
It wasn't the devastation, though, that made Monica's jaw go slack as she watched the CNN Disaster Channel, but the pattern the tsunamis made. The rock-in-the-pond ripple pattern--something clicked. The pattern was exactly what she'd been searching for: Ripples, large, then dampened--that was it!
Friday, 09/11/2009
(Pi, 931,656th decimal)
"You still haven't told anyone, right?" Monica asked."No, of course not. I'm not ready for the world to label me the biggest fool since the Indiana Legislature." In 1897, the General Assembly of Indiana had entertained a bill, backed by the State Superintendent of Public Instruction, designating pi to be four. Ever since, they'd been the laughingstock of the mathematical world, a club Kinsey was in no mood to join.
Monica looked her usual, comfortably dressed self, in contrast to Kinsey's ubiquitously rumpled, university-dress-code suit and tie, but her lips were pressed thin, eyes slitted, as she scrutinized the screen. "And look, here, on August 6th, there's another huge spike, another rock in the pond, so to speak. Both your computations of pi and mine went wild right then."
They'd been running pi calculating programs by the millions, carefully logging which algorithm, which system, what time, and how many digits it computed before it disagreed with the "standard" definition--the ten trillion digit value of pi that had gone unchanged for years, despite numerous computations, each confirming its ancestors and computing more trillions of digits. Everyone agreed there was absolutely no practical use for more than perhaps a hundred digits of pi. With no more than that, one could describe any position in the universe within the width of an electron. Kinsey cared only because it was a popular number to test algorithms on.
Yet pi had always had a sort of cult following, such as poor William Shanks, who'd painstakingly calculated 707 digits, by hand, in 1873. Fortunately he hadn't lived to find his years of effort had an error at the 528th digit, or so said D. F. Ferguson in 1948, confirmed by ENIAC when computers had first been applied to the task in 1949.
Kinsey was daydreaming; perhaps Shanks hadn't made a mistake after all. No, chances were he actually had, given the fallible nature of humans posed with tedious tasks. No matter; he shook the cobwebs from his mind and focused on the screen.
"Yes, yes. If you take each big spike, and superimpose an exponentially decaying sine wave starting there, on top of the existing waves--yes, that's it exactly!" The screen showed the strange, complex ripple pattern and below it, a sequence of waves. One new, pristine ripple wave starting for each huge spike in the complex one like a snapshot of a horse race at the finish. When added together, the heights of the simple waves exactly matched the complex pattern--the pattern of the oscillations of precision in pi.
"But I still say we throw out the August 6th data point, where that cheesy Machin algorithm said pi was eighty-seven thousand something." He snorted.
"No, look, that's dead center on that huge spike." She shook her head. Maybe nobody would believe them, but it wouldn't matter whether the crazy value was left in or not. "They dampen out so quickly, it's amazing we had any that show a difference anywhere inside the first million decimals. But we do--we can't ignore them."
"Ye-esss," he protested, "but they'll think we're lunatics to say, 'oh, by the way, on August 6th, 2009, we established an accurate measure of pi to be, yawn, eighty-seven thousand, five hundred ninety-one point zero eight seven,'" he said, emphasizing the most trivial digit for comic effect. "No, we can't announce this until we can describe its exact behavior so others can easily reproduce it."
"That's true," she added, "and maybe it'll go away by then and we can forget it ever happened."
"Hmph." He rubbed his eyes with both hands. He needed to pull something out of this to publish, and he'd spent--wasted--three months now chasing this bouncing pi thing. "Umm. While we're on the subject of really crazy thoughts. You know how the wavelength seems so random, sometimes dampening very quickly, sometimes slowly, without regard to the amplitude?" They both looked at the arrays of "pure" ripple patterns, clearly displaying the issue: They all dampened out to a nearly straight line, but in wildly varying lengths of time; some flattened to zero almost immediately, some took quite a while.
"Yeah, did you find something?"
"Well, maybe. I let it run a multi-way correlation with every variable I could find. Only one came up significant." He frowned, skeptically.
"And? And?"
"Er, it doesn't make any sense, but... ok... it was nk, the number of observations. The more observational test runs we do, the longer the value of pi wobbles in the early decimals. The fewer test runs, the sooner the variation drops to the distant digits."
She shook her head. He was grasping at straws. She knew the feeling. Maybe she should suggest that she could put in a good word for him at the Labs, and he could come work in Kuala Lumpur if he wanted. On the other hand, she couldn't stomach him having a cot next door; or, worse, if they ended up sharing an office... "No, that's because you have more points, to plot a more accurate curve."
"I thought so too, but it's not. The wavelength is wider with more observations, even at an absurdly tight 99.99% confidence level. It's truly bizarre. I've never heard of anything like that."
Monica's eyes grew wide. "Wait a minute--I have!" She swallowed hard. "What do you know about quantum mechanics?"
Friday, 10/02/2009
(Pi, 551,412nd decimal)
Although nobody among the supercollider staff had taken O'Reilly's comment seriously, they nonetheless regretted that he'd said it. He regretted it now, too. Rebuffed, he'd half-jokingly, half-vindictively mentioned it in an on-line chat session. Now, thanks to some self-appointed vigilante lurkers who overheard, there was an organized protest outside the lab, demanding they shut down their "dangerous," "defense-related," "weapons research" experimentation. "No more quakes." "End cruelty to particles.""'Nuke the black hole bomb', indeed," the lab's director said as he stared sidelong out the window, holding the blinds out just enough to see. He sighed. Black holes might be useful for many things, but if anyone were working on making weapons out of them, it certainly wasn't the Rio collider people, as their precariously low funding showed. Maybe they should be, mused the director. "So, we're all agreed, reluctantly?" The group murmured their assent, heads nodding slowly; when the National Congress jangled the purse strings, they listened. "All right then. Black hole related research is shelved for sixty days minimum, and we're closed for business until further notice."
Kinsey knew Monica would really be pissed this time. She'd specifically told him not to disturb her, that she wanted to be fresh for the news conference tomorrow. Then again, she might be pissed if he didn't tell her as soon as possible. He teetered on the edge of indecision. "Arrrgh! Oh, to hell with it! Ozawa!" The screen blinked, in dialing mode."Damnit, Kinsey! I told you--"
"I know, I know, but... look!"
He pulled up the oscillation map of pi that they'd perfected for tomorrow's demonstration. All the recent patterns showed no variation. Pi, the constant, was constant again. Flat lines filled the screen instead of ripples. "I don't know about you, but I don't think I want to demonstrate the 'conclusive' evidence of the variability of pi when the little bugger won't vary."
"Oh, shit!" She slumped back into her pillows. "Well, maybe it'll resume by tomorrow. I'll call you when I get up." She signed off.
He paced the room, wondering what career move to make now that he was sunk.
"Kinsey! Kinsey, wake up!" He'd left his end of the connection open, so her call re-entered the old session; Monica could see his head resting on the keyboard. "Kinsey! I figured it out! Wake up!"He tilted his head back to see the screen. "What? Oh, must have dozed off. Up three days. What's new? What time is it?" Suddenly he remembered, he jerked upright--"Oh God, the press conference!"
"Relax, that's not for four more hours. If we want to go on with it. We won't have anything to show, and I can tell you why."
"Go on..."
She teased him with suspense: "And, it explains, exactly, the amplitude spikes where each new ripple begins."
"Get on with it, will you?"
"Earthquakes." She paused, for dramatic effect. "And black holes, and supercolliders." Another pause, to see if he'd made the connection himself.
"Will you please get to the point!"
"Ok! Ok! Sit back and get comfortable. First, all the earthquakes--you hopefully noticed there has been a sudden unexplained increase in the number of them recently, and a lot of particularly huge ones?"
"Oh, I guess so. I don't pay attention to the news. But--ok, stipulated."
"Then you probably didn't notice the brouhaha in Rio de Janeiro about the supercollider. The salient facts are: They're conducting experiments smashing tiny particles into each other, as colliders do. What's unique is they've got the particles down to absolute zero when they collide. They use magnetic fields and some sort of layers of microscopic black holes of diminishing size, the larger ones guiding the smaller ones; extremely precise. At the lowest level, a lattice of teeny tiny ones holds a single particle, and holds it absolutely still--at absolute zero, in fact. Then they accelerate the lattice toward another one. The two lattices carefully slide through each other, their mutual gravitational effects keeping the particle extremely precise in spatial position, so that the two particles inside collide dead on, still at absolute zero with respect to the lattices.
"So: Note that they have the energy of the particles at nearly zero, with respect to the lattices.
"Next, the timing. The black hole 'tunnel' is precisely aligned, so they know the timing exactly. So: Note they now have both the time, and energy, of impact precisely known.
"Here's the killer part: Heisenberg's Uncertainty Principle. You know it?"
Kinsey shook his head. "Just the name. I'm math, not physics, remember?"
"Ok. Most people think of it as saying you can't know both the position and momentum of a particle exactly, that the more precisely you know one, the less you know the other. Well, the same thing applies to time and energy. The usual way to express that is delta-t delta-E is greater or equal to h-bar over two." She wrote on the screen, t E r /2. "This says the uncertainty in the knowledge of time multiplied by the uncertainty in the knowledge of energy is greater or equal to /2. And, hold onto your hat, h-bar is h, Plank's constant, over two pi. Do you see it? If you isolate pi, you have pi r h/(2 t E)."
Kinsey rubbed his cheek in contemplation. "You're saying, if I understand this, that these physicists got the uncertainty in both time and energy very small, yes? So small, that it no longer could be greater than what we normally consider a constant value, so one of the constants had to change?"
"Yes! It makes perfect sense if you think about one thing: What is pi, anyway? What does it mean?"
"The ratio of the circumference of a circle to the diameter."
"Right--it's a physical value, relating to the shape of the universe. When they smashed those particles together, instead of the energy and time uncertainties being high, they'd accidentally--at least, I think it was accidentally--constrained them so they had to both be low. This pushed the uncertainty somewhere else, into the curvature of the universe. All of mathematics shudders to remain consistent. In theory, the value of every fraction has to bounce to keep in step."
With a mocking blandness, as one might announce that the sun was expected to rise yet again tomorrow, he said, "And that's what caused the earthquakes, the changing curvature of the universe? Of course."
"No, not actually. That would be consistent within itself, i.e., everywhere changing the same way. So, even though pi changed, it normally wouldn't matter. No, we caused the earthquakes."
"What's that now?" Kinsey had a smile of severe disbelief on his face.
"It's the Uncertainty Principle again. One implication of it is that the act of measuring something very precisely alters the uncertainty in other types of measurement. We were measuring pi. That altered other measurements of it, in effect, altering pi itself."
He threw up his hands. "Now I know you've gone off the deep end."
"No, wait, hear me out. Say that's true, then we'd see this all the time, right? Well, what if we don't, because the uncertainty in pi is normally so small we never detect it? What if the uncertainty is normally way past the trillions of digits we've measured; say it was in the trillion trillion trillionth place, or somewhere so far down that it would strain the limits of computation even to store the digits? Not that it is, but it could be. That would be consistent, i.e., some formulation of the Uncertainty Principle involving the magnitude of numbers we can actually physically compute. Remember, when it comes to irrational numbers, by definition, we already know we can never compute them exactly, we approximate them out to some number of digits that works for whatever purpose we need, or we use a symbol for them, like pi, e, and so on."
He pursed his lips. "Something like saying, irrationals are not rational simply because they're a kind of rational that would require more digits or other storage space than is physically possible in the universe? That we consider them to have an infinite number of non-repeating decimals only because (a) we can't store all of them and (b) they're changing anyway? And that the actual amount of storage space in the universe implies how much variation there could be in a given value? Infinity is just 'more than we can compute'? I'll be damned, but it almost makes sense. It would imply there's a maximum integer, after which the entire universe would be unable to store a representation for anything larger, but... I don't know, that's pretty queer. But you still haven't explained the earthquakes."
"That's easy, from those assumptions. Here we are, measuring pi--and make no mistake about it, the formulas we use, they do measure it, or attempt to, exactly--and then there are these physicists who push off a huge uncertainty onto it. We bump into that uncertainty, yet not just once, but multiple times, once for each running program. The first earthquake happened when we were both running our algorithms, and we suddenly pinned the universe down to two values of pi, both valid; something else had to give. I don't know what, but I'm guessing something magnetic, which affected the Earth's core, causing, hell, I don't know anything about geophysics, but some kind of burp in the core. I'm thinking it's localized, perhaps around the midpoint of the triangle formed by the three events. Pi may vary with distance from the events, and at the intersection it gets into a paradox. I don't know what caused it, but I do know that the timings all match up.
"I called down to the lab and got a schedule of when they ran that experiment. It coincided with every one of the five recent catastrophic earthquakes, plus the sixty or so others that were too deep to be noticed except by seismographs, but had as much energy. And the severity of the earthquakes coincided with the number of simulation programs we were running, and how far down into pi they were. Look for yourself." She displayed a graph, a logarithmic curve with a stripe of data points almost right on it above "two," looking like the profile of a 40's Chevy with a hood ornament. It sliced right through dozens of other points scattered off to the right, values from where they'd been running hundreds of simultaneous programs: A very precise fit. "The more programs we run, the worse the earthquakes. But, you can see it's pretty bad just at two."
He rested his chin on his hands, elbows on the table. "Euclid. Newton. Gauss. Stafford and Ozawa. We'll be famous for the entire rest of civilization. My God. I'd say the press conference is back on!"
She didn't look as joyous as he expected. In fact, she looked downright disconcerted. "Try this one. Einstein. Atomic Bomb. Even though he did nearly nothing to produce it, he's associated with it in some almost infamous way. I may not be an outspoken peace activist like he was, but I sure don't want this on my head."
He looked puzzled, eyes searching around the room as he tried to intuit what she was suggesting. Finally: "Damn. I see what you mean."
She nodded knowingly.
"But," he continued, "we don't know that anyone could learn to target it and send earthquakes anywhere they want; though I admit just the underlying disruption that caused the quakes, whatever its nature, probably has military uses." He shrugged. "I don't see that it matters. We can't sit on a fundamental discovery in both mathematics and physics. We can't forget it ever happened. Someone else will re-discover it. We might as well get the credit." In fact, he needed the credit. Weapons would always be built on top of harmless scientific discoveries; he couldn't be held responsible.
Monica's ethics ran deeper; she couldn't erase the images of small crushed bodies in Mexico City. "So, what do you want to announce at our press conference? We've found out how to alter the curvature of space, now be good and don't mess around; in particular, don't make a weapon out of it? And please forgive us for accidentally killing thousands of people, but we'd like to kill a few more while we validate this theory? You don't even know what the potential is--who's to say some brainless twit wouldn't wipe out the whole planet just playing with it?"
She had a point. Of course she had a point. Several. All very valid. And if he accepted them, it meant the end of his career. No, they were scientists. They had a duty to inform the world. He had to persuade her, counter her arguments about the millions they'd already killed, injured, and made homeless. He had to get tenure.
They argued and argued until, moments before they would go live on the video press conference, Kinsey shouted "Eureka!"
Friday, 12/21/2009
(Pi, 932,401st decimal)
The spin doctors at Bell-Mat had...
Copyright © by Andrew Burt . All rights reserved unless specified otherwise above.
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