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CRACKIN' GOOD MATHEMATICSAuthor(s): Mike MaySource: American Scientist, Vol. 90, No. 5 (SEPTEMBER-OCTOBER 2002), pp. 415-416Published by: Sigma Xi, The Scientific Research SocietyStable URL: http://www.jstor.org/stable/27857718 .

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SCIENCE OBSERVER

CKACICIN' GOOD MATHEMATICS

Acouple of years ago, Alain Goriely of the

University of Arizona visited Hungary for a mathematical conference. He took a

break to listen to some local bands and watch en tertainers crack bull whips. Watching a whip snake rapidly through the air and then outplay the bands with a thunderous snap, Goriely won dered what created the crack.

The fictional Zorro may have defended himself with a long whip, but a bullwhip was not de

signed as a weapon. The bullwhip?really just a

long whip?was developed as a way to control a herd of cattle or to signal someone out of yelling distance. It consists of a handle followed by a

long, braided section called a thong that tapers down to a fine end, or "cracker." Experts crack

bullwhips in a variety of styles. Most pull the

whip over the head and back, cracking it far in front of them. Others swing it sidearm. Some

swing it their own way. In the early 1900s, some scientists wondered

whether a whip's crack came from a sonic boom. That is, perhaps part of the whip moves faster than the speed of sound, around 750 miles an hour, and the clap of noise comes as the sound barrier is bro ken. Presumably the cracker creates the crack. By the 1920s, high-speed photography revealed that a

whip's cracker can indeed break the sound barrier. The question for Goriely, however, was: How can the relatively slow speed of a whip pulled back and forth generate such high speeds at the tip?

Goriely said, "The first thing I did?after look

ing at the old papers?was buy a whip." He

bought one through the online marketplace eBay for $15. "It was a really crappy whip," Goriely said. Although he bought a book and videotape on bullwhip cracking, he couldn't get a squeak out of his. He moved up to a better whip, one that cost

$70, and started cracking it right away. "You real ize that you need a good whip to make it work," he said. In fact, someone fancying a decent whip can easily pay a few hundred dollars or more. More than a good whip, Goriely needed a good

teammate. He had one in graduate student Tyler McMillen. McMillen said, "We had been doing problems with elastic rods, so we thought a bull

whip would be an interesting problem." Goriely and McMillen started by examining past ap proaches. In general, previous investigators se lected a law of conservation, made an assump tion about the shape of the whip and combined that information to calculate the velocity of the

tip. Different conservation models?say, conser

vation of energy vs. conservation of linear mo mentum?can give different results: The tip ap proaches infinite speed in some models and maintains a constant speed in others. McMillen said, "The problem is: You can't assume the shape of the rod, because the rod obeys a physical law, and you must solve for that shape."

This solution demanded cracking and compu tation. Goriely said, "I was in my backyard crack

ing, and Tyler did all the hard work." McMillen started with what he called a pretty simple model of an elastic rod, basically a rod that can bend as a

whip does. He let the radius of his model vary, so

2002 September-October 415

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that its shape mimicked a whip's taper, and he modeled the unfurling of the whip as a traveling loop. McMillen said, "We derived the equation for the rod with the varying cross section and looked at how this loop would change as you al lowed the cross section to vary. That required some advanced mathematical techniques." Goriely and McMillen found that in fact a loop can stay the same size as it goes down the rod and still accelerate.

The crack happens when the loop reaches the end of the whip and opens. When Goriely and McMillen's model used a tapering that made a vir tual whip's tip just one-tenth the diameter of the

handle, the tip reached speeds 32 times faster than the original speed of the loop. As a result, this mod el bullwhip's tip can break the sound barrier rather

easily. (In real whips, the increase in speed could be much higher, because McMillen said that real

whips often taper even more.) Getting to this an swer required conservation of energy, as well as

conservation of linear and angular momentum. In other words, Goriely and McMillen created a much more realistic picture of the dynamics of cracking.

This work, however, goes beyond curiosity about the sound of a whip. For one thing, whip cracking provided Goriely with a new teaching tool. He said, "I do take it to show the students, because people are very impressed by whip crack

ing." Moreover, Goriely and McMillen's model

provides added insight for the general problem of the motion of waves in complex materials. One

day, investigators might use this work to under stand the motility of bacteria and sperm or the waves along DNA as it unfolds to make RNA. The role of a bullwhip might eventually stretch far beyond herding dogies?all because cracking caught Goriely's attention.?Mike May

ATTACK OF THE PSEUDO-CLONES In bee societies, the majority gains when every

one concentrates on raising the queen's young. Of the approximately 30,000 workers in a hive,

only three or so have functioning ovaries. Work er policing further reduces the number of surviv

ing worker-laid eggs. According to kin-selection

theory, workers are kept in line by the mathe matics of their relatedness: They are more related to the sons of their queen than to the sons of their

half-sisters, and thus the colony's interests sup port the queen and her reproductive success.

Stephen Martin

Parasitic Cape honeybees overwhelm an African honeybee colony with

thousands of their eggs.

416 American Scientist, Volume 90

The Cape honeybee (Apis mellifera capensis), however, flies in the face of those rules. It has evolved its own system?clever parasitism.

When a Cape worker invades a hive, it success

fully evades worker policing by mimicking the host queen's pheromones, which allows its eggs to hatch unimpeded. If that isn't enough, capensis

workers lay diploid eggs?each contains a full set of genetic material, from which hatch anoth er generation of parasitic workers, earning them the moniker "pseudo-clones" from the scientists

who study them. The pseudo-clones reproduce by a process called thelytoky: When the worker bee lays an unfertilized egg, it develops into a

new, genetically identical (barring the occasional

mutation) female bee. In contrast, in other hon

eybee subspecies, unfertilized eggs develop by arrhenotoky into male drones, which are inca

pable of reproduction. The pseudo-clones are obligate parasites and

don't forage; they rely on host African honey bees (A. m. scutellata), or "scuts" as the scientists call them, to gather their food. However, as the number of invaders surpasses the number of

scuts, the amount of incoming food dwindles

and, eventually, the host colony collapses. The entire process takes little more than twelve

weeks, says Per Kryger, a scientist at the Uni

versity of Pretoria in South Africa who has stud ied the bees for years and is currently working to conserve the wild honeybee populations found in South Africa. "The pseudo-clones see

themselves as queens, or princesses maybe," Kryger says.

Before the colony collapses, the Cape bees must catch a ride?perhaps on the truck of an

unwitting beekeeper?to their next host. Unbe

lievably, it may take only a single worker to prop agate the invasion. Using genetic analysis,

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