The Hairy Ball theorem, proved by Henri Poincaré in 1885, demonstrates that it's impossible to create a perfectly smooth hair pattern on a spherical surface.
The theorem states that any continuous pattern of hair (or more technically, vector fields) on a sphere must have at least one point where the hair either sticks straight up or forms a swirl.
The theorem has practical applications. In computer graphics, programmers face this mathematical limitation when trying to generate consistent surface patterns on spherical objects. Meteorologists also see it at work in global wind patterns — the theorem implies there must always be at least one point on Earth where horizontal wind speed drops to zero, essentially creating a cyclone somewhere on the planet.
This limitation doesn't apply to all shapes. As reported in Wikipedia, "a hairy doughnut (2-torus) is quite easily combable." This explains why you can smooth the hair on a ring-shaped surface without creating any cowlicks.
Previously:
• Math, not artificial intelligence, powers new infinite world generator
• Watch: time-lapse of a student with ADHD watching a math video versus watching Star Wars
