The simple mathematical concept of a pursuit curve can serve as the starting point for creating wonderfully intricate artistic designs.Pursuit curves can arise in a variety of situations. Suppose, for instance, that four bugs are at the corners of a square. They start to crawl clockwise at a constant rate, each moving toward its neighbor. At any instant, they mark the corners of a square. As the bugs get closer to the original square's center, the new square they define rotates and diminishes in size. In reaching the center, each bug travels on a logarithmic spiral with a length equal to the s...

The simple mathematical concept of a pursuit curve can serve as the starting point for creating wonderfully intricate artistic designs.Pursuit curves can arise in a variety of situations. Suppose, for instance, that four bugs are at the corners of a square. They start to crawl clockwise at a constant rate, each moving toward its neighbor. At any instant, they mark the corners of a square. As the bugs get closer to the original square's center, the new square they define rotates and diminishes in size. In reaching the center, each bug travels on a logarithmic spiral with a length equal to the s...

A pursuit curve is the path an object takes when chasing another object. Such a path might result from a fox pursuing a rabbit or a missile seeking a moving target.More formally, a pursuer must always head directly toward the pursued, and the pursuer's speed must be proportional to or match that of the pursued. Plotting the lines of sight at regular intervals and tracing out the corresponding paths can produce fascinating patterns.Pursuit curves can arise in a variety of situations and may involve more than one pursuer. Suppose that a person stands at each corner of a square traced out on the ...

There's a surprising mathematical ingredient in the sound of many performing artists and recording stars. It manifests itself in the form of clusters of panels hanging on the walls of recording studios, concert halls, nightclubs, and other venues. Sculpted from wooden strips separated by thin aluminum dividers, each panel consists of an array of wells of equal width but different depths.Called reflection phase gratings, these panels scatter sound waves. The result is a richer, livelier sound with an enhanced sense of space. Listeners claim that the panels seem to make the walls disappear. A sm...

For fans of major-league baseball, one of the highlights of the current season is the rate at which Barry Bonds of the San Francisco Giants is hitting home runs.Through June 25, Bonds has hit 39 home runs in 77 games, already setting the record for the most home runs before the all-star break in mid-July. At this rate, he could slug 82 homers by the time the 162-game season ends. That number would easily surpass the record-setting 70 home runs that Mark McGwire of the St. Louis Cardinals hit in 1998.The popular statistics journal known as The Sporting News has gone even further in its trend an...

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...

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...