Buffon noted that the coin would land entirely within a tile whenever the exact center of the coin landed within a smaller square—and that smaller square’s side was equal to the side of a floor tile minus the diameter of the coin used in the toss. He concluded that the probability of the coin landing entirely inside a single tile could be expressed mathematically as the ratio of the area of the tile to the area of the smaller square.
Buffon performed the same experiment using a sewing needle and a checkerboard—hence the name Buffon’s needle. Drop the needle onto the checkerboard, and one of two things happens: Either the needle crosses or touches one of the lines, or it doesn’t cross any lines. (This assumes parallel lines or squares spaced about one inch apart, and the use of a needle one inch long.)
Buffon dropped the needle over and over again, keeping track of how the needle randomly landed each time. His found that the probability that a dropped needle (or tossed coin) would cross a line is approximately 2 divided by π. He divided the number of crossing needles by the total number of needles, and realized that the more times one drops the needle, the closer one would approach the value of the probability—that is, the closer one would come to the value of π.
There are many online versions of this experiment, wherein the player can repeat the “toss” as many times as desired: five hundred, a thousand, even a hundred thousand times. Once again, the more times you repeat the experiment—the more times you roll the dice at the craps table, or spin the roulette wheel—the more closely you will approach the calculated probability. There may be winning or losing streaks in the short term, but the more you play, the more predictable things become. It’s just a quirky little oddity that the value relates to π.
With an infinite number of tosses, the value will be exactly π—that is the limit of that infinite series of tosses. The mathematician Pierre-Simon de Laplace definitively proved this in 1812. This is also the essence of what Mary Malone discovers in The Amber Spyglass. A seemingly random scattering of needles (or yarrow sticks) over a sheet of lined paper can nonetheless give you a very precise number in the end. Such is the power of calculus.
4
The Devil’s Playground
Mechanics is the paradise of mathematical science because here we come to the fruits of mathematics.
—LEONARDO DA VINCI
It is a bright and sunny Sunday afternoon inside Disneyland’s California Adventure theme park. Visitors meander blithely through the broad “streets,” nibbling on ice cream and occasionally pausing for photo ops with life-size characters from popular animated features like Monsters, Inc., The Incredibles , or Lilo & Stitch. They seem oblivious to the ominous shadow cast by the Tower of Terror, looming nearly two hundred feet above the ground, or the screams emanating from within the structure. Blackened scorch marks decorate the crumbling facade, where lightning supposedly struck in 1939, with tragic results.
Of course, nothing in Disney’s many theme parks is real. Those are screams of exhilarated delight, not abject terror, piercing the grim walls. Inspired by the classic TV series Twilight Zone, the Tower of Terror is Disney’s theatrical twist on the classic free-fall ride. We’ve been waiting in line for nearly forty-five minutes to experience those few fleeting moments of thrills and chills.
Inside we encounter the faded glory of a bygone era: Sagging overstuffed furniture, layers of dust, cracked plaster, and glass chandeliers laced with fake cobwebs grace the “lobby.” We gradually shuffle our way to a boarding dock for mock elevators, where an employee dressed as a bellhop ensures we are all tightly strapped into our seats. Our elevator rises midway to the top and stops, and we are treated to Rod Serling’s disembodied voice regaling us with the saga of a dark and stormy night on October 31, 1939, when five hotel guests stepped into an elevator and were launched into . . . the Twilight Zone!
Before we can snicker at the cheesy effects, our elevator makes a sudden gut-churning drop back to the ground floor and then shoots up all the way to the top of the structure (ostensibly the thirteenth floor), the acceleration pushing us into our seats. We pause just long enough to glimpse the rest of the park spread out two hundred feet below, before the elevator car plunges again—one short drop followed by one longer drop, each accompanied by a glorious moment of weightlessness. Then we hurtle back up to the top for one final free fall back down to the “basement,” where the faux bellhop waits to usher us back out into the Southern California sunshine. As the ride ends, Sean turns to me and gleefully exclaims, “Hey! We made a parabola!”
Everyone should visit Disneyland with a physicist in tow, just for the novelty; it’s an entirely new way of looking at the Magic Kingdom. (Motto: “All headgear is improved by the addition of mouse ears.”) I guarantee that nobody else on that ride found their thoughts wandering to calculus and parabolic curves; they were too busy screaming with joy at the free fall. Sean had never been to Disneyland, and I decided it was time to rectify that gap in his cultural development, insisting that it is a slice of Americana that must be experienced firsthand—and besides, what better place to find examples of calculus and classical mechanics in action?
Amusement-park physics is all the rage among high school physics teachers desperate for novel ways to engage their easily distracted young charges. Case in point: Every year, on Physics Day, more than four thousand high school students swarm Six Flags America in Largo, Virginia, armed with homemade accelerometers (devices to measure acceleration) and stopwatches, eager to experience the park’s extreme roller coasters—and perhaps learn a little physics along with the adrenalin rush. So Sean good-naturedly agreed to spend a Sunday at Disneyland, being dragged from one long queue to another, filled with overexcited youngsters, frazzled parents, and purple-haired hipsters with multiple piercings doing their damnedest to look bored and act as though they were Really Just There for the Irony.
FREE-FALLIN’
That diverse mix of young and old is exactly what Walt Disney had in mind when he first dreamed up the notion of a “magical park” in the late 1930s. World War II put his plans on hold, but by 1953, he had found one hundred acres just outside Los Angeles where he could build his Magic Kingdom. On July 19, 1955, Disneyland held its official grand opening. It was a disaster. Disney had intended the day to be an exclusive, invitation-only event for a select 6,000 people. But counterfeit invitations were quickly forged and snapped up by eager hordes. People began lining up at the park gates as early as two A.M., and by midafternoon, over 28,000 “ticket holders” had swarmed the park. Vendors ran out of food, and all the rides were overcrowded. A few desperate parents tossed their wailing offspring over the shoulders of bystanders blocking the way, just to get them onto the King Arthur Carousel.
The weather didn’t cooperate either. The mercury hit 110 degrees Fahrenheit, part of a fifteen-day heat wave that baked the greater Los Angeles area that July. Newly laid asphalt hadn’t had time to set, so women’s high heels got stuck in the melting tar, and hardly any of the park’s water fountains worked because of an ongoing plumber’s strike.23 Adding insult to injury, there was a gas leak that forced the afternoon closure of Adventureland, Frontierland, and Fantasyland; only Tomorrowland emerged from the debacle unscathed. But Disneyland proved hugely successful in the long run. By the time the park celebrated its tenth anniversary in 1965, over 50 million people had visited.
Disneyland is much better at crowd management these days, even though lines remain long for the most popular rides. And the Disney empire has expanded and gone global. Within the original park, in Anaheim, there is now New Orleans Square, Critter Country, and Mickey’s Toontown, in addition to the original four “lands.” Florida has Disney World, and there are now Disney theme parks in Paris, Tokyo, and Hong Kong. The California Adventure theme park opened adjacent to Disneyland in 2001; the Tower of Terror can be found in the Hollywood Pictures Backlot section of the park.
Human beings have thrilled to the sensation of free fall for centuries, with occasionally dire
results. Witness the enormous popularity of bungee jumping, which has its roots in the ancient Aztec ritual of the Danza de los Voladores de Papantla; the danza is still practiced today by “Papantla flyers.” In the 1950s, British documentary filmmaker David Attenborough took his BBC film crew to Pentecost Island in Vanuatu, where they recorded several young tribal men who jumped from tall wooden platforms with vines tied to their ankles as a test of courage. It was only a matter of time before extreme sports enthusiasts had the brilliant notion of harnessing themselves to bungee cords and jumping off tall structures for fun (and the occasional profit).24 Bungee jumping quickly spread around the globe, despite numerous accidents and the odd fatality.
For those (like me) who prefer a more sedate form of thrill-seeking, there are mechanical free-fall rides with, shall we say, more rigorous safety constraints. Six Flags Great Adventure introduced one of the first true free-fall experiences in 1983. The L-shaped structure featured a four-passenger car lifted via hydraulics to the top of a 130-foot tower and suspended for a few seconds. At the buzzer, the car would plunge down the drop track and onto the horizontal exit track to end the ride. The latter was necessary because coming to a sudden stop at the end of the drop would most likely cause serious injuries. The deceleration period dissipates all that kinetic energy over a longer period of time so it isn’t transferred all at once to the passengers.
The Tower of Terror is a variation of a “drop tower” ride that gradually has replaced the classic free-fall design since the 1990s, largely because it is closer to a true free-fall experience, and there is less mechanical wear and tear. A gondola or car—in this case, the mock elevator—is propelled upward toward the top of a large vertical structure and then falls back toward the ground. The brakes kick in before impact, slowing the ride, although the Tower of Terror essentially “bounces” its riders a few times before finally coming to rest.
Technically, we enter free fall when there is no longer any force (other than gravity) acting directly on us. Think of tossing an apple into the air. The moment it leaves your hand and you stop applying that upward force, it is in free fall. It continues traveling up, moving more slowly as gravity overpowers its upward motion, has a brief moment of hang time (that period of weightlessness), then begins its descent. Our car in the Tower of Terror follows the same trajectory. It receives an initial push from the hydraulics, but at some point that force is removed and we finish our ascent using pure momentum. That brief, exhilarating period of weightlessness occurs because riders fall at the same rate as their surroundings—in this case, their seats. NASA’s infamous “vomit comet” follows a parabolic trajectory while in flight, taking such extreme lifts and dips that it can achieve about twenty to thirty seconds of weightlessness for every sixty-five seconds of flight.
Thrills aside, the Tower of Terror provides an excellent example of calculus as it applies to classical mechanics. True, our physical motion is straight up and down. But if we plot our change in height (position) over time point by point on a Cartesian grid—both ascending and descending trajectories—and connect the dots, we end up with the telltale parabolic curve that so delighted Sean at the ride’s end. (The same is true for the apple.)
How does this work? We begin with our starting velocity. It is not 0, because we are specifying our starting velocity at the moment we enter freefall, not at the start of the ride (when it would be 0). We need a speedometer to tell us how fast we are traveling at that moment, and that number becomes our starting velocity (a constant). Let’s imagine the Tower of Terror tracks acceleration for us. We can take an integral of our acceleration to get our velocity, essentially adding the acceleration—in this case, the gravitational constant—at each moment in time.
The result, when we graph it out, is a straight downward-sloping line. This is our velocity function. We can use that to determine our position (height) by taking another integral, adding together how far we traveled at each point in time. Plot each position as a function of a time and you get a pretty parabola. Now that we have a position function, finding our height at any given point in time is a snap.25
Many years ago, I went to Six Flags New Jersey with a group of friends, and we all went on the Devil Dive—a cross between bungee jumping and a really big tire swing. Three of us were strapped into one big harness and lifted to the top of a 200-foot tower. That might not sound very high unless you happen to be one of the people hanging precariously at the top of it; then terra firma seems a very long way down. One of my cronies had just enough time to nervously remark, “Um, maybe this wasn’t such a good idea after a—AUUGHH!” The buzzer sounded, the catch released, and we plummeted, screaming, toward the ground.
Just as we were about to hit the ground, the harness caught and swung us outward in a sweeping pendulum motion, moving through space along the trajectory of an arc of a circle. We swung back and forth like a three-person pendulum, until we slowed down sufficiently for the ride operators to grab us and release us from the harness.
The Devil Dive gives us a double dose of Galileo. First, there is the free fall. It is roughly the same problem outlined above, except in this case our position as a function of time forms only half a parabola, because we don’t enter true free fall until we begin our descent. Our acceleration is −32 feet per second per second at any time (t) after our drop begins. (The sign is negative because we are falling and our height is decreasing.) We can take an integral to get our velocity, and then integrate the velocity to get our position function, just as we did before.
Second, there is the pendulum motion at the end of the ride. An oft-told anecdote from Galileo’s youth tells of the seventeen-year-old future scientist growing bored during Mass in a drafty cathedral in Pisa. He noted a chandelier hanging from the ceiling swaying in the breeze. Sometimes it barely moved; other times, it swung in a wide arc. This proved more interesting to the teen than the priest’s sermon, and he began timing the swings with his pulse, with a surprising result: It took the same number of beats for the chandelier to complete one swing, no matter how wide or narrow the arc. Granted, the chandelier moved faster during wider arcs, but it completed its arc of motion in the same amount of time. The same motion can be seen in playground swings and the arc we make at the end of our Devil Dive. But there is a twist: Don’t be misled by that arc-like motion. If we plot our changing position with respect to time during this portion of the ride, we get a periodic sine wave. The fact that the pendulum swings in predictable periods is why it became the basis for the pendulum clock.
There is another relevant curve called the Witch of Agnesi, named after eighteenth-century mathematician Maria Gaetana Agnesi. The eldest of twenty-one children,26 Agnesi was known in her family as the Walking Polyglot because she could speak French, Italian, Greek, Hebrew, Spanish, German, and Latin by the time she was thirteen. Agnesi had the advantage of a wealthy upbringing; the family fortune came from the silk trade. And she also had a highly supportive father, who hired the very best tutors for his talented eldest daughter and insisted she participate in regular intellectual salons he hosted for great thinkers hailing from all over Europe.
The young Maria delivered an oration in defense of higher education for women in Latin at the age of nine; she translated it from the Italian herself and memorized the text. Contemporary accounts suggest that Agnesi loathed being put on display, even though her erudition earned her much admiration. One contemporary, Charles de Brosses, recalled, “She told me that she was very sorry that the visit had taken the form of a thesis defence, and that she did not like to speak publicly of such things, where for every one that was amused, twenty were bored to death.”
De Brosses admired her intellectual prowess greatly, and was horrified upon learning that she wished to become a nun. She did become a nun, but not before spending ten years writing a seminal mathematics textbook, Analytical Institutions, published in 1748—the first surviving mathematical treatise written by a woman. She was also the first woman to be appointed a mathematics
professor at a university (the University of Bologna), although there is no record she ever formally accepted the position. She died a pauper in 1799, having given away everything she owned.
But her work lives on. One of the curves featured in Analytical Institutions is the Witch of Agnesi. Agnesi dubbed it la versiera, a nautical term meaning “a rope that turns a sail”—an allusion to the motion by which the curve is drawn.
At some point, a harried English translator misinterpreted the word as l’avversiera, “she-devil” or “witch.”
What does this have to do with the pendulum motion of the Devil Dive? Among other things, this curve describes a driven oscillator near resonance—a swinging pendulum that is being poked or prodded to keep it in motion, for example, like someone pushing a child on a swing. When the rate of prodding matches the rate of the pendulum’s swing, it is said to be in resonance. If the rate of prodding is very, very close to the rate of the swing, the amplitude (height) of the swing, plotted as a function of frequency, forms the Witch of Agnesi. We’ve already seen that the physical motion of a pendulum forms an arc, while plotting its position as a function of time gives us a periodic sine wave. So had someone (or something) been pushing us during the pendulum phase of the ride at almost the exact same rate as our swing, the Witch of Agnesi would describe our amplitude as a function of forced frequency (rate of prodding).
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 9
