Antennas broadcast radio waves in different directions depending on design choices. Moving on from toy scenarios-the hairy ball theorem actually imposes tangible limitations on radio engineers. 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. 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. A spinning ball rotates around an invisible axis, and the points on either end of that axis don’t move. Upon further reflection, this might seem obvious. Spinning is a continuous motion, so the hairy ball theorem applies and assures a point with no speed at all. Again, we associate a tangent vector with each point based on the direction and speed at that point on the ball. There will always be a point on the surface that has zero velocity. To observe another weird ramification of the theorem, spin a basketball any which way you want. This neat online tool depicts up-to-date wind currents on Earth, and you can clearly spot the swirly cowlicks. 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. (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). 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. 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. This doughnut shape is covered in small lines resembling hairs that are all combed in the same direction, with no tufts resulting. Doughnuts are also distinct from spheres, so a hairy doughnut-an unappetizing image, no doubt-can be combed smoothly. So sadly, math can’t excuse your bedhead. A scalp on its own can be flattened into a surface and combed in one direction like the fibers on a shag carpet. Something that is not equivalent to a sphere is your scalp. (You could mold them all from a ball of Play-Doh without violating the rubbery rules.) 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. If one shape can be smoothly deformed into another without doing these things, then those shapes are equivalent, as far as topologists are concerned. Although that rubber is capable of molding into other forms, it is incapable of tearing, fusing or passing through itself. 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. This claim extends to all sorts of furry figures. Tufts on either side demonstrate the hairy ball theorem. This sphere is covered in small lines resembling hairs that are all combed in the same direction. In full jargon: a continuous nonvanishing tangent vector field on a sphere can’t exist. 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 other words, the arrangement of vectors on the sphere must be continuous, meaning that nearby hairs should change direction only gradually, not sharply. Also, we want a smooth comb, so we don’t allow the hair to be parted anywhere. 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. A vector, often depicted as an arrow, is just something with a magnitude (or length) and a direction. In more technical language, think of the coconut as a sphere and the hairs as vectors. Of course, mathematicians don’t refer to coconuts or cowlicks in their framing of the problem. 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. Juvenile humor aside, the theorem has far-reaching consequences in meteorology, radio transmission and nuclear power. 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. You might be surprised to learn that you can’t comb the hairs flat on a coconut without creating a cowlick.
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