Comb a hairy sphere however you like — somewhere a tuft must still stand up.
Where's the maths? This is topology — the branch of mathematics about the features of a shape that survive any amount of stretching and bending. Every hair here is a vector: a little arrow with a direction and a length, exactly like a wind arrow on a weather map. "Combing the ball flat" means choosing a non-zero arrow at every single point, all lying flat along the surface and turning smoothly as you move from point to point. The hairy ball theorem says that on a sphere this is impossible: somewhere the arrow must shrink to nothing, leaving a calm point where the hair stands up. Each calm point carries a whole number called its index — how many complete turns the arrows make as you walk once around it. Whatever you do to the hair, those indices always add up to 2 for a sphere, the same 2 that appears in Euler's V − E + F = 2. On a doughnut the total is 0 instead, and a total of 0 can be reached with no calm points at all — which is why a torus really can be combed flat.
Beyond the syllabus: the theorem is topology — enrichment.
A vector field on a surface gives one arrow at every point, lying flat along the surface. Here the direction of the arrow is the way the hair lies and its length is the strength of the flow — hairs are blue where the flow is strong and gold where it is nearly calm.
A zero (a calm point) is a point where the arrow has length 0, so the hair has no direction to lie in and stands upright. Walk a small anticlockwise circle around a zero and watch the arrow turn: the number of complete turns it makes is the index. A source or a sink has index +1, a whirlpool has index +2, and a saddle — flow in along one line, out along another — has index −1.
The counts are not free. The indices of the zeros of any smooth tangent vector field on a closed surface add up to the Euler characteristic χ of that surface (the Poincaré–Hopf theorem). For a sphere χ = 2; for a torus χ = 0.
Since 2 ≠ 0, a field on the sphere can never be zero-free: there is always at least one calm point. So at every instant there is somewhere on Earth where the horizontal wind is exactly calm; a coconut cannot be combed flat; and a hair whorl on your head cannot be brushed away, only moved.
How the app finds them: it evaluates the field on a fine grid, and for each little cell adds up how much the arrow turns as you go once round the cell. If the total is a whole number of turns other than zero, a zero of the field is trapped inside the cell — the cell is marked, the neighbouring marked cells are joined up, and a ring is drawn at the middle with its index.
Controls: drag to orbit, scroll or pinch to zoom, Comb mode then drag on the surface to comb the hair, Reset hair to start again.