Everyday Objects Make Applied Mathematics Tangible
By Paul Davis
Balls in bowls, dropped coins, tapped tea cups, jars of rice rolling (or not) down an incline…demonstrations, questions, a magician’s hands up close on the projection screens, the verbal timing of an improv comic…These are just some of the things that happened at the 2016 SIAM Annual Meeting’s I.E. Block Community Lecture.
Tadashi Tokieda (University of Cambridge and Stanford University) used everyday objects to immerse his audience in such central concepts of applied mathematics as inverse problems, singular limits, symmetry, and finite-time divergence. His overarching success was capturing the excitement of exploring, the challenge of discovery, and ultimately the sentiment that “nothing replaces touching,” a tenet at the spiritual ground-state of applied mathematics.
Practicing applied mathematicians in the audience who might have spent too long digging at the bottom of their own specialized trench were surely energized as Tokieda rekindled the light that led them into this business in the first place. Non-mathematicians who wondered why their mothers, fathers, friends, or partners worked in this quirky discipline obtained answers that they could see and feel. And all of this came from (mostly) simple, everyday objects that showed the audience that they had “as much access to nature” as the speaker.
Finish your tea, then position the cup with its handle at 12 o’clock. Use your spoon to strike the cup first at 12, then at 3, 6, and 9 o’clock. The sound is the same at all four positions. Why? Then strike the cup at 1:30. How does the tone change? Why does it change?
Can you close your eyes and reconstruct the position of the handle from the sounds produced by tapping? Can sound alone distinguish the original cup from a cup with two half-sized handles at 12 and 6 o’clock? Tokieda identified these two questions as inverse problems, then noted with a smile that you could find “lots of science about inverse problems and lots of SIAM members making their livings” by working on them.
Put three jaw-breaker-sized cedar balls into a cereal bowl, then move the bowl on a table to swirl its contents. The balls circulate more or less independently in the same direction. But shift to eight or more balls in the bowl and they seem to coalesce into a solid that moves opposite the swirl. Presto – a phase change with only a few degrees of freedom.
Two apparently identical heptagon-shaped wheels behave completely differently. When set on their edges and given a gentle push, one advances across the table, though hardly rolling smoothly. The other staggers and falls. What nearly invisible difference produces such radically disparate behaviors? The 7-gon that rolls is imperceptibly flawed: its edges are bowed out ever so slightly and its corners are barely rounded, imperfections just sufficient to permit continuous motion. The falling 7-gon has perfectly straight edges and sharp corners – zero defects, a singular limit at which its staggering collapse bears no relation to the unwavering roll seen in the 7-gon that is close to, but short of, this limit.
Tokieda’s table camera focuses on an inclined plane and several cylindrical pill bottles filled with varying amounts of rice. How, he asks, will the rate of descent vary with mass as the bottles roll down the incline? He polls the audience, preparing all for the confrontation between their predictions and the behavior they are about to observe.
The bottle that is two-thirds full moves slowly down the ramp, the rice inside slipping in an avalanche across its free surface while sticking to the wall of the bottle like a viscous fluid.
Preparing to launch the next bottle, Tokieda ceremoniously shoots his cuffs in front of the camera before his magician’s hands place a half-full bottle on the ramp.It stalls. A voice in the audience calls out, “Can I change my vote?” A moment later, a nearly empty bottle rolls right down, the few grains in it sliding easily against the bottle’s wall like an inviscid fluid.
Unlike a professional magician, Tokieda goes on to explain his trick, beginning with a property of granular materials: the sides of a pile of rice slope at a characteristic angle of repose, which depends on the shape, not the size, of the grains. Shallower slopes are stable; steeper ones “landslide” until they attain the gentler angle of repose. Fluids, whether viscous or inviscid, have zero angle of repose: they simply spread into a puddle.
The rolling behavior of a partially-full bottle depends upon the quantity of rice and the relation of the rice’s angle of repose to the angle of the incline: if the center of mass of the rice plus the bottle can position itself—if the rice can landslide—so that the force of gravity acts downhill of the bottle’s point of contact with the incline, then the resulting torque rotates the bottle down the incline. A lot of rice rolls. Intermediate amounts of rice don’t. Very little rice rolls.
Look across the range of bottles from one that is 100% full of rice to one that is totally empty, 0% full; the percentage values that are points of transition between stop and roll, between stationary and moving, are singular limits. Each is yet another instance of the sharp distinctions—like those between viscous and inviscid fluids—that are the meat and potatoes of much of applied mathematics. Here Tokieda displayed them vividly to a large, rapt, laughing audience without demanding the inner vision of a Ludwig Prandtl.
Drop a coin on the floor. It rotates with a clatter that quickly rises in pitch as the coin comes to a stop in finite time. A smattering of physics and some back-of-the-envelope calculus show that the bounce frequency is proportional to an inverse power of \(t_{singular} – t\); the divergence in finite time is apparent.
Chattering magnets, crushed paper balloons that are slapped back to shape, and other everyday items offer more evidence of Tokieda’s central theme: toys provide a rich ecology of phenomena and ideas, near at hand, awaiting our attention and our touch.
His closing exhortation to “explore and discover” provoked sustained applause, then a torrent of questions. To conclude the questions, he reminded the audience of children who set aside a gift to play instead with its wrapping paper. “We are trained to be interested only in the approved topics,” he said. “Make a conscious effort to notice.”
No frivolous toys among Tokieda’s toy models!
To view more images from the lecture, as well as a video of Tokieda's presentation, check out this article.
Paul Davis is professor emeritus of mathematical sciences at Worcester Polytechnic Institute.