Hairy Ball Theorem
Any continuous tangent vector field on the 2-sphere S² must vanish at at least one point — you cannot "comb" a hairy ball without a cowlick.
Poincaré-Hopf: Σ index(zero) = χ(S²) = 2
Index sum: 2
Counterclockwise spin: zeros at north pole (index +1) and south pole (index +1). Sum = 2.
Current Field Zeros
Deform field button perturbs the field with a small rotation — watch the red zero dot(s) move but never vanish. The index sum stays pinned at 2.
Torus Contrast (χ = 0)
The torus has Euler characteristic χ=0. Its "latitude flow" field has NO zeros — index sum = 0 = χ(torus). Smooth everywhere!