Fill a doughnut’s surface with circles so that every circle links through every other one exactly once — a chain-mail where no two rings are ever unlinked. That is the Hopf fibration, seen on the Clifford torus that sits halfway up the 3-sphere; drawn here by stereographic projection into ordinary space. Slide to sweep the fibers and watch the links.
The Hopf fibration maps the 3-sphere S³ onto the 2-sphere S²; the preimage of each point is a great circle (a Hopf fiber), and any two distinct fibers are linked with LINKING NUMBER exactly 1 — they cannot be pulled apart. The fibers over a circle of latitude on S² all lie on one Clifford torus (the flat, square torus at √2/2 in S³). Stereographic projection sends these linked great circles to linked circles/lines in 3-space. A fail-loud self-check throws unless the fiber linking number is 1. ◆ real topology (Hopf, 1931), node-verified.
A faithful stereographic picture of a handful of fibers, not the full continuous fibration; the linking number 1 (the Hopf invariant) is the exact, defining fact.