Column

Math

Prized Geometric Logic

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 […]

Math

Buses on Quantum Schedules

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 […]

Math

Subtle Logic, Winning Game

Seemingly simple games can serve as thought-provoking exercises in mathematical logic. They can provide deep insights into subtle issues that confront logicians who are interested in the foundations of mathematics. So-called Ehrenfeucht games have proved particularly useful for tackling certain aspects of mathematical logic. They were developed in the 1960s by Andrzej Ehrenfeucht, who is […]

Math

Mayan Mars

The curiously looping movements of the planets relative to the stars have presented all sorts of puzzles to keen, patient observers of the night sky. In 1601, Johannes Kepler (1571-1630) undertook the challenge of deciphering the orbit of Mars and developing a mathematical theory of its motion to fit observations of the planet’s changing position […]

Math

Quirky Video Poker

The lure of easy money brings gullible bettors back again and again to the game of video poker–an immensely popular pastime in casinos and other gambling venues throughout the United States. Most players are bound to lose money, says Todd D. Mateer, a recent graduate of Clemson University, who has studied video poker machines in […]

Math

Appealing Numbers

The ancient Greeks, especially the Pythagoreans, were fascinated by whole numbers. They defined as “perfect” numbers those equal to the sum of their parts (or proper divisors, including 1). For example, 6 is the smallest perfect number-the sum of its three proper divisors: 1, 2, and 3. The next perfect number is 28, which is […]

Math

Immersed in Klein Bottles

“Need a zero-volume bottle? Searching for a one-sided surface? Want the ultimate in nonorientability?” One way to depict a Klein bottle. Computer-generated image by John Sullivan, University of Illinois at Urbana-Champaign Joining the top and bottom of this rectangle produces a cylinder. Matching the arrows of the remaining two sides produces a Klein bottle. One […]

Math

White Narcissus

The elegant, swooping forms carved out of wood by sculptor Robert Longhurst often resemble gracefully curved soap films that span twisted loops of wire dipped into soapy water. Alhough these abstract sculptures bear an uncanny resemblance to mathematical forms known as minimal surfaces, they emerge from Longhurst’s imagination rather than from mathematics. An original design […]

Math

Fibonacci’s Chinese Calendar

In a book completed in the year 1202, mathematician Leonardo of Pisa (also known as Fibonacci) posed the following problem: How many pairs of rabbits will be produced in a year, beginning with a single pair, if every month each pair bears a new pair that becomes productive from the second month on? The total […]

Math

Scheduling Random Walks

Juggling competing demands in a network of feverishly calculating computers drawing on the same memory resources is like trying to avert collisions among blindfolded, randomly zigzagging ice skaters. Example of a graph with one token poised to take a random walk. In this example of dependent percolation, a fickle demon would win (so far), but […]

Math

Scheduling Random Walks

Juggling competing demands in a network of feverishly calculating computers drawing on the same memory resources is like trying to avert collisions among blindfolded, randomly zigzagging ice skaters. Example of a graph with one token poised to take a random walk. In this example of dependent percolation, a fickle demon would win (so far), but […]

Math

Folding Maps

Anyone trying to refold an opened road map is wrestling with the same sort of challenges confronted by origami designers and sheet metal benders. The problem of returning a creased sheet to its neatly folded state gets tougher when you’re not sure if the sheet can be folded into a flat packet and when you’re […]