The Hairy Ball Theorem

If you have a continuous vector field on a sphere (imagine combing hair flat on a ball), there must be at least one point where the vector has zero length - essentially a point where the 'hair' must stick up (0:08-1:15). This applies to any attempt at assigning tangent vectors to every point on a sphere.

A vector field assigns a tangent vector to every point on the sphere, where each vector lies in the plane tangent to that point (4:15). The theorem constrains what kinds of continuous vector fields can exist.

When programming airplane orientation along a trajectory in video games, you need to determine the wing direction perpendicular to the nose direction (2:35-4:00). The hairy ball theorem proves you cannot do this continuously for all possible flight directions without glitches.

Radio signals that are identical in every direction are impossible because electromagnetic oscillations must be perpendicular to propagation direction (5:50-6:30). The theorem guarantees at least one direction must have zero signal strength.