Von Frisch had watched bees dancing on the vertical face of the honeycomb, analyzed the choreographic syntax, and articulated a vocabulary. 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.
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.
"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."
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."
And she does not believe the manifold's presence both in the mathematics of quarks and in the dance of honeybees is a coincidence. Rather she suspects that the bees are somehow sensitive to what's going on in the quantum world of quarks, that quantum mechanics is as important to their perception of the world as sight, sound, and smell.
"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 may be more to the world than there appears to be. Sensitivity to quantum fields is a leap; being aware of a quantum order, and using it to compress normal mathematics, not at all. It reminds me of human transcendence: the idea that by understanding patterns in our reality, you can achieve divinity, without having to go to some Other World.