S³ → S² — fibers are circles in 3D via stereographic projection
Hopf fibration (Heinz Hopf, 1931): the 3-sphere S³ ⊂ ℝ⁴ fibers over S² with fiber S¹.
Every point on S² has exactly one circle (a great circle of S³) lying above it — the fibers are mutually linked Villarceau circles.
Visualized here via stereographic projection from S³ ⊂ ℝ⁴ → ℝ³.
Base points (colored by latitude on S²) sweep a ring; their preimage fibers form beautiful linked tori.
Fundamental in topology: π₃(S²)=ℤ, generated by the Hopf map. Used in quantum mechanics (Bloch sphere = S²).