Rolling Rocks and Tumbling Dice; Studies of Tumbling Dice May Suggest New Ways of Looking at Shifting Sand Dunes and Avalanches

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Aquick flick of the wrist sends a pair of dice bouncing, rolling and skidding down a green felt surface. Another toss sends a second set tumbling across the table. But in subtle ways, the latter set-- carefully weighted so that certain faces are more likely to land upward--seems to move a little more erratically than the first pair.

A loaded die tends to bounce around more than a "perfect' die, says graduate student Bradley T. Werner of Caltech in Pasadena. "It appears to move less smoothly,' he says. For the last year or so, Werner has been using computer simulations to study the way loaded and unloaded dice roll.

Werner's research is part of a larger effort to see if the general behavior of granular systems such as rock slides, sand dunes, planetary rings and snow avalanches can be derived from the motions and interactions of the individual particles that make up a given system.

The problem seems immense. A sand dune, for instance, contains trillions of grains. Furthermore, these windblown particles are not randomly placed but highly organized. If it were possible to slice open a sand dune, one would see very fine laminations that consist of alternating layers of larger and smaller grains. Even more curious is the "yell' that certain sand dunes give out when they are kicked.

"There's a whole collection of peculiar things going on,' says physicist Peter Haff, who supervises Werner's work. "You really wonder . . . how the grains are actually rubbing and bouncing off one another.'

One way to study such a complex system theoretically is to begin with a simpler case. But even the motion of a single particle pulled by gravity down a slope is far from trivial to analyze, especially when the particle is allowed to have an arbitrary shape. Friction and energy losses further complicate the picture.

Werner began his study by looking at variously shaped, two-dimensional particles moving down a smooth slope. Initially, he was interested in seeing if sand grains of different shapes could be separated by letting the sand flow down a slope.

Werner's computer simulations showed that these particles, at various times, apparently exhibit three types of motion: sliding, bouncing and rolling. With enough friction, most bouncing particles eventually end up rolling. Sliding particles skid to a stop. Shape doesn't seem to have a great effect.

A field test in a lonely Mojave Desert canyon confirmed that "real' rocks behave similarly. Massive boulders tumbling down a mountainside may bounce at first, but this motion quickly decays into a roll. "We didn't take any measurements,' says Werner, "but we didn't have to roll too many rocks down the slope to see what was happening.'

These results were good enough to encourage Werner to go one step farther with his computer simulations. This time, he chose a cube as his sample particle. A fellow graduate student, who happened to be interested in gambling, suggested that he also look into weighted cubes. This led Werner into a study of loaded dice in action.

His observations, however, took place not at Las Vegas casinos but on the screen of his Macintosh personal computer. Werner wrote a computer program to simulate a cube's motion, carefully applying the appropriate physical laws of motion. His simulations embodied all of the important characteristics of a real, moving die: a nearly rigid particle that loses some energy on each bounce and is slowed by friction.

Werner's aim was to see if he could predict the probability of a particular side landing face up, depending on how the die was weighted. In his initial simulations, he followed the behavior of a square rather than the entire cube. The die rotated in such a way that only four of the faces could turn up.

"I picture the tumbling die as progressing from face to face and continually losing energy until it is finally captured, with one of the faces resting on the plane,' says Werner. …