Like many people, I have a few problems with the factual accuracy of The Da Vinci Code, even though I'm much more likely to cut authors some slack when it comes to fiction (so long as they don't make public claims to historical accuracy, as Dan Brown did). But apart from a few overstatements, the depiction of the Golden Ratio (phi) and related mathematical phenomena was reasonably on target. Case in point: the murdered Louvre curator in the beginning of the story leaves behind a coded message for his granddaughter, a sexy French cryptologist named Sophie. It's a string of seemingly random numbers -- 13-3-2-21-1-1-8-5 -- which Sophie unscrambles to reveal the first eight numbers of the Fibonacci Sequence.
You probably already know the drill (c'mon, even if you skipped the book, you at least saw the film, if only to snicker at Tom Hanks' disastrous haircut): after the first two terms, each number is equal to the sum of the two previous numbers, to wit: 1+1=2; 1+2=3; 2+3=5; 3+5=8, and so on, into infinity. But the fun doesn't end there. If you take each number in the sequence and divide it into the one that follows, the answer will be very close to phi -- an irrational number with infinite decimal places, commonly cited in shortened form as 1.618. As you progress through the sequence, dividing one number into the next, the answer will come closer and closer to the value for phi, without ever actually reaching. Try it: 5 divided by 3 is 1.666; 13 divided by 8 is 1.625; 21 divided by 13 is 1.615; and so forth. Plot the Fibonacci Sequence graphically, and you get pretty logorithmic spirals, like this:
I was reminded of the Fibonacci Sequence while ruminating on last week's calculus lectures, expanding on the two central concepts of the derivative and the integral. The key to both lies in breaking something down into the smallest possible increments, an infinite progression of single calculations that nonetheless accumulates and converges towards a single definitive answer. And just like dividing Fibonacci numbers into each other, as those intervals get smaller and smaller, the result gets closer and closer to the actual "answer."
The similarity probably ends there. Both my calculus lectures last week made no mention of the Fibonacci Sequence, and chose instead to once again evoke Zeno and his various paradoxes of motion. To recap the paradox of an arrow flying through the air: if that motion is divided into infinitely smaller and smaller increments of time and distance, in what sense can we say the arrow is "moving" at all? How can a single instant have an actual "speed"? Philosophers can be so tedious sometimes. For all practical purposes, calculus resolves that paradox, and thanks to Michael Starbird's spiffy DVD calculus course, I now have a reasonable grasp of how it does this. (Yay me!)
On the whole, Starbird keeps things reasonably simple. He began with the most basic example. Since the derivative is a measure of change -- how temperature changes, how the Dow Jones average changes, etc. -- the most basic example is motion: change in position with respect to time. So Starbird envisions Zeno driving a car (Jen-Luc points out it should really be a chariot, but we'll grant Starbird some artistic license) along a straight road at a steady unchanging speed, and in the process, running a stop sign. As luck would have it, Officers Newton and Leibniz are on duty and pull Zeno over. Zeno denies he ran the stop sign. As proof, he presents the officers with a time-stamped still photo taken of the car just before its nose passed the stop sign. How can they prove the car was moving at that point, and not stopped? Too bad he got caught by Newton and Leibniz, the co-inventors of calculus.
Here's where the details start to overwhelm the story arc a little, but they are admittedly necessary. That straight road is divided into intervals at every possible distance, even infinitesimally small ones, and at each point there is a traffic camera that takes a time-stamped still photo of the car as its nose passes that point on the road. This makes it a pretty magical road, but we're good at suspending our disbelief in order to achieve a greater truth. Officers Newton and Leibniz can pull out their own photographic evidence: an infinite number of still shots taken at smaller and smaller intervals along that amazing road. So they can show where Zeno was at the two-minute time, subtract where Zeno was at the one-minute time, and from that determine how far the car traveled: one mile. Divide that by the time it took to travel that distance, and you get the car's average speed: one mile per minute.
It might be possible for Zeno to argue with that one counter-example, but not with an infinite accumulation of evidence. Newton and Leibniz offer similar evidence for ever-smaller intervals (eg, at the 1.1 mark, the 1.01, mark, ad infinitum), performing the same exact calculations each time, and the answer is always the same: 1 mile per minute. Ergo, Zeno's car did have a speed at that instant, and that speed was 1 mile per minute. According to Starbird, the evidence doesn't lie in looking at what the car was doing at that very instant, because Zeno was correct when he said that an object can only be in one place at a single moment. Rather, it lies in the accumulation of evidence taken from every possible increment of time (before and after) the critical mark in question.
Ingenious! Also incredibly tedious. Our hats are off to Newton, Leibniz, and untold mathematicians before and after them who repeated the same exact process of calculation, over and over again, until they'd compiled sufficient "proof" that the derivative formula works. It takes a Special Kind of Person to do that. We are not that special. At least I'm not a smartass like this self-proclaimed mathematical genius. (Jen-Luc Piquant, on the other hand....)
Nonetheless, while Starbird's example works, it's a highly idealized, carefully tailored classroom analogy. In the real world, no driver moves at constant steady speed. So Starbird presents the exact same scenario with one crucial difference: the car is now accelerating. It's still necessary to set a few basic parameters, namely, that at each interval of time, Zeno looks at his watch (or mini-sundial) to record the exact minute as he passes that point, and then must square that number. Plus, he must ensure that he's always at the mileage marker that is the square of the time. For example, if the watch reads 1 minute, the mileage marker will be 1; if the watch reads two minutes, Zeno's car will be at mileage marker 4. And so forth. The general principle is that, at every moment, the car's velocity is exactly two times the time.
Starbird goes into considerable detail, but the basic concept is the same: Zeno's car still has an instantaneous speed, based on an accumulation of evidence showing where the car was at all times before and after the moment in question. The same highly repetitive process applies, running the same calculation outlined above for smaller and smaller intervals of time and distance. But there's a crucial difference: instead of getting the same answer each time -- as in the steady state example -- one gets slightly different answers each time. But as the intervals get shorter and shorter, those answers get closer to a convergence: in this case 2 miles per minute. It's never exactly 2, just like dividing one Fibonacci number into the next never exactly equals phi. But it's a pretty darn close approximation, and easy to draw the inescapable conclusion that the instantaneous speed at the moment in question must be 2 miles per minute.
The integral complicates matters further. Essentially it's just the flip side of the derivative: with the derivative, we seek to determine the car's speed based on how its position changed over time. The integral seeks to determine how far the car traveled based only on measurements of its speed at given locations along the same magical straight road. In both cases, however, we're still taking an accumulation of information in tiny parts of the problem to construct a whole answer.
Say Zeno has kidnapped a rival philosopher (let's call her Jen-Luc) who was heckling him in the town square when he was trying to pontificate to his followers (and anyone else in a toga who'd listen). Jen-Luc is bound and gagged in the back seat, able to view just the speedometer, but fortunately she has a video camera on hand, trained on said speedometer, to record the exact speed being traveled. (No doubt the Historical Anachronism Society will object, but again, we claim artistic license and opt for willing suspension of disbelief.) After an hour, Zeno unceremoniously drops his prisoner on the side of the road and drives away. How can Jen-Luc use the information about her speed recorded on the video camera to figure out how far she went, and hence her location? That way she can call me on her anachronistic Cyber-cell phone and tell me where to pick her up.
Assuming a constant speed of 60 miles per hour, and knowing that one hour exactly has elapsed since her perfunctory kidnapping, it's pretty easy for Jen-Luc to conclude that therefore, she has gone 60 miles. (Along a straight road. That probably leads to nowhere.) But once again this is unrealistic. It's far more likely that the car's speed kept changing. No matter, Jen-Luc is tight with Officers Newton and Leibniz, so she knows she can use the time-stamped images on the videotape to see how fast she was going at particular intervals. The shorter the intervals, the better, because the speed is less likely to vary by much over very tiny times and distances.
How do we take the variation into account? By setting limits around the correct answer to get a workable range for determining distance. It's even more labor-intensive than the derivative. First, we do a series of calculations based on the slowest (starting) speed at the start of our journey, breaking that journey into smaller and smaller increments to arrive at a close approximation to the total distance traveled. But this will be an underestimation. So we also need to do the exact same labor-intensive process for the fastest speed the car was traveling over our entire one-hour journey. The resulting approximation will be an over-estimate of how far we went, but at least we know that the correct distance is somewhere in between those two values.
Jen-Luc has the patience of Job (plus, she really wants to come home) and will keep doing this unbelievably repetitive process at smaller and smaller intervals. She will get ever-finer approximations of the likely distance traveled; that is, the range becomes smaller and smaller, converging towards a single answer without ever reaching it exactly -- just like with the derivative. Jen-Luc can't determine her location via any single division of the interval of time; she gets it via an infinite number of increasingly improved approximations. The power of the method is this: as long as she knows both her position and velocity at any given moment in time, she can determine where she was at every other moment in time in her short journey. That's how scientists can make such impressively accurate predictions about classical physical systems.
Clearly we learned a lot last week, although in fairness, these are just the tiniest baby steps in the course; until we get start performing applied calculations, we can't truly be said to be "doing" calculus. But we do have a greater appreciation for the importance of position and velocity in classical mechanics, and why Newtonian laws break down at the subatomic scale. See, even though the calculations are a series of approximations, at some point at the macroscale (things the size of Zeno's car, for example) the intervals become so small that the difference between approximations is really pretty trivial. I can probably locate Jen-Luc if I know she's somewhere between 1/4 inch and 1/2 inch on the side of the straight road to nowhere. But at tiny scales, even tiny differences become pretty darn important. And the Uncertainty Principle tells us that were Zeno's car the size of a subatomic particle, the more accurately we measure its position, the less we know about its velocity, and vice versa. So the Newtonian calculations break down from lack of precise information. Fortunately, others came to Newton and Leibniz's rescue and founded quantum mechanics. (Sure, it's all about probabilities and averaging things out a la Feynman's sum-over paths, but the predictions still work incredibly well.)
Does your head hurt yet? Ours did. After an entire week wrestling with such heady ideas, we were in need of a stiff drink. And apparently we're not the only ones. We learned from fellow physics blogger Ben Schumacher that there is a well-known mnemonic for memorizing the first few digits of another irrational number, pi: "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." The trick is to count the letters in each word to get 31415926.... Schumacher went one step further in his online research and found some interesting tidbits via Wikipedia. Check out his post (linked above) to read a pi-mnemonic poem that parodies Edgar Allan Poe's "The Raven," extending to 740 places. Truly ingenious.
But we still needed a drink. Fortunately we have a growing collection of physics-themed cocktails at our disposal, plus a few recipes from a recent article in Wired on molecular mixology entitled "Better Drinking Through Chemistry" (see our prior post on the topic here). Lacking a class-IV laser, we can't even attempt caramelaserizing red wine with a vanilla bean. And we are disinclined to special-order cola-flavored Pop Rocks from Japan to make a dehydrated rum and coke. But at some future date we're definitely going to try our hand at a Jellied Gin and Tonic:
1 frozen lime
2 oz simple syrup
1-1/4 tsp citric acid
1/4 tsp bicarbonate of soda
1/4 tsp confectioner's sugar
1-1/2 sheets of sheet gelatin
1 oz gin
2 oz tonic water
Freeze lime and cut into thin chips (even infinitisemally smaller and smaller chips if you want to be all calculus-like about it). Coat slices in syrup and 1 tsp citric acid, then bake at 150 degrees until crisp. Mix bicarbonate of soda, sugar, and remaining citric acid. Soften sheet gelatin in cold water for two minutes. Warm gin and add gelatin. Pour into shallow baking pan lined with plastic wrap, add tonic, and refrigerate for two hours. Cut into 1/2 inch cubes. Put cube onto lime chip, sprinkle on sugar-soda-acid mixture, and serve.
Sure, it's pretty labor-intensive, but once you've made all those little cubes, you can sit back and ponder things like this: In 2004, scientists at the Technische Universitat Wien in Austria actually managed to measure the shortest possible interval of time to date: 100 attoseconds. According to this article at BBC News, stretching 100 attoseconds to the point where it lasts one second, that one second would last 300 million years on the same scale. That's close enough to an infinitesimally small segment of time for me. Even Zeno might be impressed.