Hairy Ball Theorem — Vector Fields on a Sphere

Field Distortion

Twist A 0.00

Hair density 900

Live Readout

Singularities found

Max field speed

1.00

Sum of winding indices (Poincaré–Hopf, must equal χ(S²) = 2)

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How it works

The Hairy Ball theorem says a continuous field of tangent vectors on a sphere can never be "combed flat" — it must have at least one point where the vector shrinks to zero (a cowlick). This is because every smooth tangent vector field on S² is a special case of the Poincaré–Hopf theorem:

Σ index(zeros) = χ(S²) = 2

Here the field is the tangential gradient of a height function on the sphere, H(x,y,z) = z + A·xy:

∇H = (A·y, A·x, 1)               (ambient gradient)
V(p) = ∇H − (∇H·p) p             (projected tangent to the sphere, p unit)

Hair arrows point "downhill" along −V is not used here — arrows follow V itself (steepest-ascent flow), so hair streams away from maxima and converges at minima. At A = 0 the only critical points are the two poles (climate max at north, min at south) — the classic combed-sphere picture with one cowlick at each pole, index +1 apiece.

Turning up Twist A tilts the height function so a saddle-node pair of extra critical points is born off-axis. At each nondegenerate critical point the simulator fits a local 2×2 Hessian by finite differences in the tangent plane and reads off the index directly:

index = +1  if det(Hessian) > 0   (a local max or min — "source"/"sink")
index = −1  if det(Hessian) < 0   (a saddle)

New singularities always appear as a +1/−1 pair, so however many cowlicks the field grows, the readout above never drifts from +2 — a live check of a theorem that holds for every possible tangent field on a sphere, not just this one.

Tangent field (hair) Index +1 (source/sink) Index −1 (saddle)

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