The Hopf fibration (Heinz Hopf, 1931) is a map η: S³ → S² where each point on S² corresponds to a great circle (fiber ≅ S¹) on S³. Points on S³ are unit quaternions (a,b,c,d) with a²+b²+c²+d²=1; the map sends (a+bi, c+di) ∈ ℂ² to a point on S² via the Hopf map p = (z₁z̄₂ + z̄₁z₂, i(z₁z̄₂ − z̄₁z₂), |z₁|²−|z₂|²). Each fiber is a perfect circle — yet any two distinct fibers are linked! This is the first known example of a map S^n → S^m with m < n that is not null-homotopic: π₃(S²) = ℤ. Drag to rotate.