Mathematics
h: S³ → S²
h(z₁,z₂) = (2z₁z̄₂, |z₁|²−|z₂|²)
where (z₁,z₂) ∈ ℂ², |z₁|²+|z₂|²=1
Fiber over p∈S²:
h⁻¹(p) ≅ S¹ (a great circle in S³)
Hopf invariant = linking number
of any two distinct fibers = 1
The Hopf fibration is the generator of π₃(S²)≅ℤ. Stereographic projection from S³→ℝ³ maps each fiber to a circle forming nested tori. Any two fibers are Hopf-linked — they thread through each other exactly once.
Rendering fibers...