July 9, 2009
Tunisian desert ants have an uncanny mathematical ability to find their way across the desert, author and math popularizer Keith Devlin remarked in a recent talk at the MAA's Carriage House Conference Center. These tiny creatures, he said, often wander hundreds of meters across the Sahara in search of food. After succeeding, however, they always do something remarkable. They take the shortest routes back to their tiny nests, apparently by calculating and recalculating their positions relative to their starting points, based on their speed and direction of travel.
The desert ants, Devlin said, appear to take advantage of the same process that the ill-fated Apollo 13 astronauts used to plot their course back to Earth: dead reckoning ("deductive" reckoning). This mathematical insight was just one part of Devlin's engaging discourse, "Why Do Golf Balls Have Dimples? (and other mysteries of life that can be explained only with mathematics)."
Galileo, Devlin began, had given the key to the solutions of life's mysteries when he observed (as paraphrased): “To understand the universe you have to understand the language in which it is written, and that language is mathematics.”
To explain why golf balls have dimples, Devlin introduced Daniel Bernoulli and his mathematics of fluid flow from the 18th century. A flying golf ball spins at about 3,000 revolutions per minute, which generates lift. Dimples produce a layer of turbulent air around the ball, which reduces drag. As a result, a dimpled ball can go, at 160 miles per hour, about 2.5 times farther than a smooth ball. Incidentally, dimpled golf balls (336 dimples on U.S. brands; 300 on British brands) were introduced in the 1920s.
Devlin also pointed out that an oft-repeated explanation of what keeps an airplane aloft is often incorrectly attributed to the Bernoulli principle—a pressure differential between the upper and lower surfaces of an airfoil. However, this doesn’t explain why, for example, an airplane can fly upside down. Newton’s third law of motion offers a better explanation, Devlin said. An airplane flies at a slight angle to the horizontal and forces air downward. The resulting reaction force counters the force of gravity and keeps the plane in the sky. For a correct, more complete discussion of lift, Devlin recommended a NASA Glenn Research Center website at http://www.grc.nasa.gov/WWW/K-12/airplane/lift1.html.
Keith Devlin discusses airflow around a plane's wing. Photo by Ryan Miller.
Dimples (or their equivalent) have also been put on some planes (military ones, mostly) to reduce drag. They're also used on expensive bike tires; on bike helmets; and even on Lance Armstrong's cycling attire. Dimples on Armstrong's clothing give him a slight advantage in any race, sometimes making the difference between winning and losing. Hence, the mathematics of dimples, Devlin said, "helped to make Lance Armstrong a champion." See Discovery Channel's "The Science of Lance Armstrong" (2005) for further details.
Two thousand years ago, Greek geometer Pappus of Alexandria noted that bees must know mathematics because they build their honeycombs with great precision and efficiency. The repeating pattern of regular hexagons in a cross-section of a honeycomb, Pappus hypothesized, will hold more honey for a given amount of material than any other geometric shape. His guess, in an essay titled "Sagacity of Bees," became known as the honeycomb conjecture.
Worker bees excrete slivers of warm wax, each about the size of a pinhead, Devlin said. Other workers position the slivers to form vertical, six-sided, cylindrical chambers (or cells). Each wax partition is less than 0.1 millimeter thick, accurate to a tolerance of 0.002 millimeter. Each of the six walls is the same width, and the walls meet at an angle of 120 degrees, producing one of the "perfect figures" of geometry, a regular hexagon.
Devlin then posed several questions. Why don't bees make each cell triangular, square, or some other shape in cross section? Why have straight rather than curved sides in the first place? The problem boils down to finding the two-dimensional shape that can be repeated endlessly to cover a large flat area, for which the total length of all the cell perimeters is the least (so that the area of the honeycomb walls is as small as possible).
It is easy to show that, if you restrict yourself to hexagons, regular hexagons give a smaller perimeter than non-regular polygons, Devlin said. In 1943, Hungarian mathematician L. Fejes Tóth proved that the regular hexagon pattern gives the smallest total perimeter for all patterns made up of any combination of straight-edged polygons. Intuitively, Devlin said, outward bulges would balance inward bulges.
In 1999, Thomas C. Hales, then at the University of Michigan, announced a complete solution of the honeycomb conjecture. That it's taken 2,000 years and 19 pages of advanced mathematics to show that the familiar honeycomb is the most efficient pattern for the storage of honey is a testament to the wonders of nature, Devlin observed.
Moving from bees to birds, Devlin noted the marvels of bird migration, often far out of sight of land with only the sun, moon, or stars or Earth’s magnetic field to guide them. "Their built-in instinctive systems are doing what in human terms are pretty sophisticated calculations,” Devlin said.
In closing, Devlin looked at how skateboarders get the vertical upward force to leave the ground. He also posed the problem of how a rider steers a bicycle through a turn. A video clip revealed that, to steer right, a rider has to first make a slight turn to the left, and vice versa.
Devlin ended his presentation with a nifty problem concerning the area between tire tracks produced by a bike making a turn. The problem can be solved without calculus, Devlin pointed out, and he left it as an exercise for his audience to tackle.—H. Waldman
Keith Devlin is the Executive Director of the H-STAR Institute at Stanford University and The Math Guy on National Public Radio's Weekend Edition. His most recent books are The Unfinished Game: Pascal, Fermat, and the Seventeenth Century Letter that Made the World Modern and The Computer as Crucible: An Introduction to Experimental Mathematics, with Jonathan Borwein. His two dozen other books include Solving Crimes with Mathematics: The Numbers behind NUMB3RS, written with Gary Lorden; The Math Instinct; The Millennium Problems; The Math Gene; Infosense; Mathematics: The New Golden Age; Goodbye Descartes; and The Language of Mathematics.
Listen to Keith Devlin's Lecture