Predicting the geometric shapes of soap bubble clusters can lead to surprisingly difficult mathematical problems.Frank Morgan of Williams College in Williamstown, Mass., recently illustrated such difficulties when he invited an audience of mathematicians, students, and others to vote on which one of a given pair of different representations of the same number of clustered planar bubbles would have a smaller total perimeter. Assembled for a ceremony at the National Academy of Sciences in Washington, D.C., to honor the 12 winners of the 2001 U.S.A. Mathematical Olympiad (USAMO), audience members...

Artist Susan Happersett of Jersey City, N.J., has come up with a novel twist on the venerable Möbius strip: a playful, eye-catching creation she describes as a Möbius accordion.A Möbius strip, or band, is the remarkable one-sided surface that results from joining together the two ends of a long strip of paper after twisting one end 180 degrees. Mathematicians, magicians, artists, and many others have been playing with this intriguing object since its discovery in the 19th century by August Ferdinand Möbius (1790–1868), a professor at the University of Leipzig in Germany.Happersett combines her...

One tradition that flourished 200 years ago in Japan, during its period of isolation from the western world, involved Euclidean geometry. Scholars and others would inscribe geometric problems on wooden tablets, then hang the tablets under the eaves of Shinto shrines and Buddhist temples as offerings. Such a tablet is called a sangaku, which means "mathematical tablet" in Japanese.More than 800 tablets have survived. Many of them feature drawings and problems that concern tangent circles.Here's one example. Suppose three circles are tangent to one another and rest on a base line. Find a relatio...

One tradition that flourished 200 years ago in Japan, during its period of isolation from the western world, involved Euclidean geometry. Scholars and others would inscribe geometric problems on wooden tablets, then hang the tablets under the eaves of Shinto shrines and Buddhist temples as offerings. Such a tablet is called a sangaku, which means "mathematical tablet" in Japanese.More than 800 tablets have survived. Many of them feature drawings and problems that concern tangent circles.Here's one example. Suppose three circles are tangent to one another and rest on a base line. Find a relatio...

Like toy cars chasing each other on a looped racetrack, three stars can, in principle, trace out a figure-eight orbit in space. This newly discovered, mathematically surprising pattern of motion arises from the force of gravity acting on three bodies of equal mass. Their movements are timed so that each body in turn passes between the other two.Newton’s laws provide a precise answer to the problem of determining the motion of two bodies under the influence of gravity. If the solar system consisted of the sun and a single planet, for example, the planet would follow an elliptical orbit. When th...

Like toy cars chasing each other on a looped racetrack, three stars can, in principle, trace out a figure-eight orbit in space. This newly discovered, mathematically surprising pattern of motion arises from the force of gravity acting on three bodies of equal mass. Their movements are timed so that each body in turn passes between the other two.Newton’s laws provide a precise answer to the problem of determining the motion of two bodies under the influence of gravity. If the solar system consisted of the sun and a single planet, for example, the planet would follow an elliptical orbit. When th...

Computer programs can handle all sorts of data, from sums of money in bank accounts to sensor readings from scientific instruments. In many cases, the data are a set of discrete elements, such as temperatures. Moreover, some elements of a set may be larger in value than others, or they may exhibit some other relationship that allows you to rank them or put them in order.In mathematics, such a collection of elements is known as a partially ordered set, or poset. One example of a poset consists of an integer and all its positive divisors (excluding 1). For instance, the positive divisors of 42 a...

Anyone who has waited for a bus in the city has probably casually observed that, after an inordinately long wait, two or three buses often come along at the same time.The question of why such bunching seems to happen has prompted all sorts of speculation. Some claim that bus bunching is actually a rare occurrence, but passengers tend to forget the much larger number of times when a single bus arrives. Others posit that bus drivers simply like to travel in packs.Mathematical models that simulate traffic flow confirm that bus bunching is a real phenomenon. Even though buses leave their depot at...