You could be stunned to study you can’t comb the hairs flat on a coconut with out making a cowlick. Maybe much more shocking, this foolish declare with an excellent sillier identify, “the bushy ball theorem,” is a proud discovery from a department of math known as topology . Juvenile humor apart, the concept has far-reaching penalties in meteorology, radio transmission and nuclear energy .

Right here, “cowlick” can imply both a bald spot or a tuft of hair sticking straight up, just like the one the character Alfalfa sports activities in “The Little Rascals.” In fact, mathematicians do not seek advice from coconuts or cowlicks of their framing of the issue. In additional technical language, consider the coconut as a sphere and the hairs as vectors. A vector, usually depicted as an arrow, is simply one thing with a magnitude (or size) and a course. Combing the hair flat towards the edges of the coconut would kind the equal of tangent vectors —people who contact the sphere at precisely one level alongside their size. Additionally, we wish a easy comb, so we do not permit the hair to be parted wherever. In different phrases, the association of vectors on the sphere have to be steady, which means that close by hairs ought to change course solely steadily, not sharply. If we sew these standards collectively, the concept says that any method you attempt to assign vectors to every level on a sphere, one thing ugly is sure to occur: there will likely be a discontinuity (an element), a vector with zero size (a bald spot) or a vector that fails to be tangent to the sphere (Alfalfa). In full jargon: a steady nonvanishing tangent vector area on a sphere cannot exist.

This declare extends to all types of furry figures. Within the area of topology , mathematicians examine shapes, as they might in geometry, however they think about these shapes are made out of an ever elastic rubber. Though that rubber is able to molding into different types, it’s incapable of tearing, fusing or passing via itself. If one form might be easily deformed into one other with out doing these items, then these shapes are equal, so far as topologists are involved. Which means that the bushy ball theorem routinely applies to bushy cubes, bushy stuffed animals and bushy baseball bats, that are all topologically equal to spheres. (You may mildew all of them from a ball of Play-Doh with out violating the rubbery guidelines.)

One thing that isn’t equal to a sphere is your scalp. A scalp by itself might be flattened right into a floor and combed in a single course just like the fibers on a shag carpet. So sadly, math cannot excuse your bedhead. Doughnuts are additionally distinct from spheres, so a bushy doughnut —an unappetizing picture, little question—might be combed easily.

This is a curious consequence of the bushy ball theorem: there’ll all the time be not less than one level on Earth the place the wind is not blowing throughout the floor. The wind flows in a steady circulation across the planet, and its course and magnitude at each location on the floor might be modeled by vectors tangent to the globe. (Vector magnitudes need not signify bodily lengths, reminiscent of these of hairs.) This meets the premises of the concept, which suggests that the gusts should die someplace (making a cowlick). A cowlick may happen in the attention of a cyclone or eddy, or it may occur as a result of the wind blows straight up towards the sky. This neat on-line software depicts up-to-date wind currents on Earth, and you’ll clearly spot the swirly cowlicks.

To look at one other bizarre ramification of the concept, spin a basketball any which method you need. There’ll all the time be a degree on the floor that has zero velocity. Once more, we affiliate a tangent vector with every level primarily based on the course and velocity at that time on the ball. Spinning is a steady movement, so the bushy ball theorem applies and assures a degree with no velocity in any respect. Upon additional reflection, this may appear apparent. A spinning ball rotates round an invisible axis, and the factors on both finish of that axis don’t transfer. What if we bored a tiny gap via the ball precisely alongside that axis to take away the stationary factors? It appears then that each level can be transferring. Does this violate the bushy ball theorem? No, as a result of drilling a gap reworked the ball right into a doughnut! Even doughnuts with unusually lengthy, slim holes flout the foundations of the concept—contradiction averted.

Shifting on from toy situations—the bushy ball theorem truly imposes tangible limitations on radio engineers. Antennas broadcast radio waves in numerous instructions relying on design selections. Some goal their alerts in a selected course, whereas others beam extra broadly. One could be tempted to simplify issues and construct solely antennas that ship equal-strength alerts in each course without delay, that are known as isotropic antennas. There’s only one drawback: a sure hirsute truth from topology mandates that isotropic antennas can’t exist. Image an orb of waves emanating from a central supply. Sufficiently distant from the supply, radio waves exhibit an electrical area perpendicular to the course they’re touring, which means the sphere is tangent to the sphere of waves. The bushy ball theorem insists that this area should drop to zero someplace, which suggests a disturbance within the antenna’s sign. Isotropic antennas serve merely as theoretical beliefs towards which we evaluate actual antenna efficiency. Curiously, sound transmits a special form of wave with out the perpendicular property of radio waves, so loudspeakers that emanate equal-intensity sound in each course are attainable.

Maybe the good software of the bushy ball theorem considerations nuclear fusion energy. Fusion energy carries immense promise to—maybe sometime —assist ease the vitality disaster. It has the potential to generate huge portions of vitality with out the environmental considerations that plague fossil fuels and with far fewer of the radioactive dangers related to conventional nuclear fission reactors. In a nutshell, fusion reactors start by taking a gasoline reminiscent of hydrogen and subjecting it to intense warmth and stress, which rips it into its constituent components to kind plasma. Plasma is a cloud of electrons and different charged particles that bop round and sometimes fuse collectively to kind new particles, releasing vitality within the course of.

There is a basic engineering hurdle when constructing fusion reactors: How do you include plasma that’s 10 occasions hotter than the solar’s core ? No materials can face up to that temperature with out disintegrating into plasma itself. So scientists have devised a intelligent resolution: they exploit plasma’s magnetic properties to restrict it inside a robust magnetic area. Probably the most pure container designs (suppose bins or canisters) are all topologically equal to spheres. A magnetic area round any of those buildings would kind a steady tangent vector area, and at this level we all know what befalls such bushy constructions. A zero within the magnetic area means a leak within the container, which spells catastrophe for the entire reactor. That is why the main design for fusion reactors, the tokamak , has a doughnut-shaped chamber . The Worldwide Thermonuclear Experimental Reactor (ITER) megaproject plans to complete development of a brand new tokamak in France by 2025, and people concerned declare their magnetic confinement system will likely be “the biggest and most built-in superconducting magnet system ever constructed.” That is topology enjoying its half in our clear vitality future.