Math’s ‘Hairy Ball Theorem’ Has Surprising Implications

You might be surprised to learn that you can’t comb the hairs flat on a coconut without creating a cowlick. Perhaps even more surprising, this silly claim with an even sillier name, “the hairy ball theorem,” is a proud discovery from a branch of math called topology. Juvenile humor aside, the theorem has far-reaching consequences in meteorology, radio transmission and nuclear power.

Here, “cowlick” can mean either a bald spot or a tuft of hair sticking straight up, like the one the character Alfalfa sports in The Little Rascals. Of course, mathematicians don’t refer to coconuts or cowlicks in their framing of the problem. In more technical language, think of the coconut as a sphere and the hairs as vectors. A vector, often depicted as an arrow, is just something with a magnitude (or length) and a direction. Combing the hair flat against the sides of the coconut would form the equivalent of tangent vectors—those that touch the sphere at exactly one point along their length. Also, we want a smooth comb, so we don’t allow the hair to be parted anywhere. In other words, the arrangement of vectors on the sphere must be continuous, meaning that nearby hairs should change direction only gradually, not sharply. If we stitch these criteria together, the theorem says that any way you try to assign vectors to each point on a sphere, something ugly is bound to happen: there will be a discontinuity (a part), a vector with zero length (a bald spot) or a vector that fails to be tangent to the sphere (Alfalfa). In full jargon: a continuous nonvanishing tangent vector field on a sphere can’t exist.

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This claim extends to all sorts of furry figures. In the field of topology, mathematicians study shapes, as they would in geometry, but they imagine these shapes are made from an ever elastic rubber. Although that rubber is capable of molding into other forms, it is incapable of tearing, fusing or passing through itself. If one shape can be smoothly deformed into another without doing these things, then those shapes are equivalent, as far as topologists are concerned. This means that the hairy ball theorem automatically applies to hairy cubes, hairy stuffed animals and hairy baseball bats, which are all topologically equivalent to spheres. (You could mold them all from a ball of Play-Doh without violating the rubbery rules.)

Something that is not equivalent to a sphere is your scalp. A scalp on its own can be flattened into a surface and combed in one direction like the fibers on a shag carpet. So sadly, math can’t excuse your bedhead. Doughnuts are also distinct from spheres, so a hairy doughnut—an unappetizing image, no doubt—can be combed smoothly.

Here’s a curious consequence of the hairy ball theorem: there will always be at least one point on Earth where the wind isn’t blowing across the surface. The wind flows in a continuous circulation around the planet, and its direction and magnitude at every location on the surface can be modeled by vectors tangent to the globe. (Vector magnitudes don’t need to represent physical lengths, such as those of hairs.) This meets the premises of the theorem, which implies that the gusts must die somewhere (creating a cowlick). A cowlick could occur in the eye of a cyclone or eddy, or it could happen because the wind blows directly up toward the sky. This neat online tool depicts up-to-date wind currents on Earth, and you can clearly spot the swirly cowlicks.

To observe another weird ramification of the theorem, spin a basketball any which way you want. There will always be a point on the surface that has zero velocity. Again, we associate a tangent vector with each point based on the direction and speed at that point on the ball. Spinning is a continuous motion, so the hairy ball theorem applies and assures a point with no speed at all. Upon further reflection, this might seem obvious. A spinning ball rotates around an invisible axis, and the points on either end of that axis don’t move. What if we bored a tiny hole through the ball exactly along that axis to remove the stationary points? It seems then that every point would be moving. Does this violate the hairy ball theorem? No, because drilling a hole transformed the ball into a doughnut! Even doughnuts with unusually long, narrow holes flout the rules of the theorem—contradiction averted.

Moving on from toy scenarios—the hairy ball theorem actually imposes tangible limitations on radio engineers. Antennas broadcast radio waves in different directions depending on design choices. Some target their signals in a specific direction, while others beam more broadly. One might be tempted to simplify matters and build only antennas that send equal-strength signals in every direction at once, which are called isotropic antennas. There’s just one problem: a certain hirsute fact from topology mandates that isotropic antennas can’t exist. Picture an orb of waves emanating from a central source. Sufficiently far away from the source, radio waves exhibit an electric field perpendicular to the direction they’re traveling, meaning the field is tangent to the sphere of waves. The hairy ball theorem insists that this field must drop to zero somewhere, which implies a disturbance in the antenna’s signal. Isotropic antennas serve merely as theoretical ideals against which we compare real antenna performance. Interestingly, sound transmits a different kind of wave without the perpendicular property of radio waves, so loudspeakers that emanate equal-intensity sound in every direction are possible.

Perhaps the coolest application of the hairy ball theorem concerns nuclear fusion power. Fusion power carries immense promise to—perhaps someday—help ease the energy crisis. It has the potential to generate vast quantities of energy without the environmental concerns that plague fossil fuels and with far fewer of the radioactive risks associated with traditional nuclear fission reactors. In a nutshell, fusion reactors begin by taking a fuel such as hydrogen and subjecting it to intense heat and pressure, which rips it into its constituent parts to form plasma. Plasma is a cloud of electrons and other charged particles that bop around and occasionally fuse together to form new particles, releasing energy in the process.

There’s a fundamental engineering hurdle when building fusion reactors: How do you contain plasma that’s 10 times hotter than the sun’s core? No material can withstand that temperature without disintegrating into plasma itself. So scientists have devised a clever solution: they exploit plasma’s magnetic properties to confine it within a strong magnetic field. The most natural container designs (think boxes or canisters) are all topologically equivalent to spheres. A magnetic field around any of these structures would form a continuous tangent vector field, and at this point we know what befalls such hairy constructions. A zero in the magnetic field means a leak in the container, which spells disaster for the whole reactor. This is why the leading design for fusion reactors, the tokamak, has a doughnut-shaped chamber. The International Thermonuclear Experimental Reactor (ITER) megaproject plans to finish construction of a new tokamak in France by 2025, and those involved claim their magnetic confinement system will be “the largest and most integrated superconducting magnet system ever built.” That’s topology playing its part in our clean energy future.

This is an opinion and analysis article, and the views expressed by the author or authors are not necessarily those of Scientific American.