When a bee finds a source of food, he realized, it returns to the hive and communicates the distance and direction of the food to the other worker bees, called recruits. On the honeycomb which Von Frisch referred to as the dance floor, the bee performs a "waggle dance," which in outline looks something like a coffee bean--two rounded arcs bisected by a central line. The bee starts by making a short straight run, waggling side to side and buzzing as it goes. Then it turns left (or right) and walks in a semicircle back to the starting point. The bee then repeats the short run down the middle, makes a semicircle to the opposite side, and returns once again to the starting point.
It is easy to see why this beautiful and mysterious phenomenon captured Shipman's young and mathematically inclined imagination. The bee's finely tuned choreography is a virtuoso performance of biologic information processing. The central "waggling" part of the dance is the most important. To convey the direction of a food source, the bee varies the angle the waggling run makes with an imaginary line running straight up and down. One of Von Frisch's most amazing discoveries involves this angle. If you draw a line connecting the beehive and the food source, and another line connecting the hive and the spot on the horizon just beneath the sun, the angle formed by the two lines is the same as the angle of the waggling run to the imaginary vertical line. The bees, it appears, are able to triangulate as well as a civil engineer.
Direction alone is not enough, of course--the bees must also tell their hive mates how far to go to get to the food. "The shape or geometry of the dance changes as the distance to the food source changes," Shipman explains. Move a pollen source closer to the hive and the coffee-bean shape of the waggle dance splits down the middle. "The dancer will perform two alternating waggling runs symmetric about, but diverging from, the center line. The closer the food source is to the hive, the greater the divergence between the two waggling runs."
If that sounds almost straightforward, what happens next certainly doesn't. Move the food source closer than some critical distance and the dance changes dramatically: the bee stops doing the waggle dance and switches into the "round dance." It runs in a small circle, reversing and going in the opposite direction after one or two turns or sometimes after only half a turn. There are a number of variations between species.
Von Frisch's work on the bee dance is impressive, but it is largely descriptive. He never explained why the bees use this peculiar vocabulary and not some other. Nor did he (or could he) explain how small-brained bees manage to encode so much information.
One day Shipman was busy projecting the six-dimensional residents of the flag manifold onto two dimensions. The particular technique she was using involved first making a two-dimensional outline of the six dimensions of the flag manifold. This is not as strange as it may sound. When you draw a circle, you are in effect making a two-dimensional outline of a three-dimensional sphere. As it turns out, if you make a two-dimensional outline of the six-dimensional flag manifold, you wind up with a hexagon. The bee's honeycomb, of course, is also made up of hexagons, but that is purely coincidental. However, Shipman soon discovered a more explicit connection. She found a group of objects in the flag manifold that, when projected onto a two-dimensional hexagon, formed curves that reminded her of the bee's recruitment dance. The more she explored the flag manifold, the more curves she found that precisely matched the ones in the recruitment dance. "I wasn't looking for a connection between bees and the flag manifold," she says. "I was just doing my research. The curves were nothing special in themselves, except that the dance patterns kept emerging." Delving more deeply into the flag manifold, Shipman dredged up a variable, which she called alpha, that allowed her to reproduce the entire bee dance in all its parts and variations. Alpha determines the shape of the curves in the 6-D flag manifold, which means it also controls how those curves look when they are projected onto the 2-D hexagon. Infinitely large values of alpha produce a single line that cuts the hexagon in half. Large' values of alpha produce two lines very close together. Decrease alpha and the lines splay out, joined at one end like a V. Continue to decrease alpha further and the lines form a wider and wider V until, at a certain value, they each hit a vertex of the hexagon. Then the curves change suddenly and dramatically. "When alpha reaches a critical value," explains Shipman, "the projected curves become straight line segments lying along opposing faces of the hexagon."
If Shipman is correct, her mathematical description of the recruitment dance would push bee studies to a new level. The discovery of mathematical structure is often the first and critical step in turning what is merely a cacophony of observations into a coherent physical explanation. In the sixteenth century Johannes Kepler joined astronomy's pantheon of greats by demonstrating that planetary orbits follow the simple geometric figure of the ellipse. By articulating the correct geometry traced by the heavenly bodies, Kepler ended two millennia of astronomical speculation as to the configuration of the heavens. Decades after Kepler died, Isaac Newton explained why planets follow elliptical orbits by filling in the all-important physics--gravity. With her flag manifold, Shipman is like a modern-day Kepler, offering, in her words, "everything in a single framework. I have found a mathematics that takes all the different forms of the dance and embraces them in a single coherent geometric structure."
Shipman is not, however, content to play Kepler. "You can look at this idea and say, `That's a nice geometric description of the dance, very pretty,' and leave it like that," she says. "But there is more to it. When you have a physical phenomenon like the honeybee dance, and it follows a mathematical structure, you have to ask what are the physical laws that are causing it to happen."
Researchers have in fact already established that the dance is sensitive to such properties. Experiments have documented, for example, that local variations in Earth's magnetic field alter the angle of the waggling runs. In the past, scientists have attributed this to the presence of magnetite, a magnetically active mineral, in the abdomen of bees. Shipman, however, along with many other researchers, believes there is more to it than little magnets in the bees' cells. But she tends not to have much professional company when she reveals what she thinks is responsible for the bees' response. "Ultimately magnetism is described by quantum fields," she says. "I think the physics of the bees' bodies, their physiology, must be constructed such that they're sensitive to quantum fields--that is, the bee perceives these fields through quantum mechanical interactions between the fields and the atoms in the membranes of certain cells."
There is some research to support the view that bees are sensitive to effects that occur only on a quantum-mechanical scale. One study exposed bees to short bursts of a high-intensity magnetic field and concluded that the bees' response could be better explained as a sensitivity to an effect known as nuclear magnetic resonance, or NMR, an acronym commonly associated with a medical imaging technique. NMR occurs when an electromagnetic wave impinges on the nuclei of atoms and flips their orientation. NMR is considered a quantum mechanical effect because it takes place only if each atom absorbs a particular size packet, or quantum, of electro-magnetic energy.
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