The 3-sphere S³ — surface of a 4D ball — fibers over the 2-sphere S² with circles S¹ as fibers. Every point on the surface of an ordinary sphere corresponds to a circle on the 3-sphere, and any two such circles are linked. Stereographic-project to 3D and you see a structure that looks like a torus made of circles. Heinz Hopf, 1931 — the first nontrivial example of a fiber bundle.
what you're looking at, and why it's miraculous
Every point on S² (an ordinary sphere) corresponds to a circle on S³ (the 3-sphere). Stereographic projection sends those circles into 3D space. You see them as the colored loops: each one is the pre-image of a single point on the 2-sphere.
Three facts that should not be true but are:
1. Any two fibers are linked — pick any two points on the 2-sphere; their pre-image circles in 3D are interlocked, like two rings of a chain. You cannot separate them without breaking one.
2. Fibers at the same latitude on S² form a flat torus in 3D — the colored bands you see are tori, but in the actual S³ geometry they are flat. The roundness in 3D is an artifact of projection.
3. The fibration is non-trivial — there is no continuous map S³ → S² × S¹ that respects the structure. This means S³ is not "just" a product of S² and S¹, even though every fiber is a circle and the base is a sphere. The "twisting" is captured by an integer (the Hopf invariant), and for the original Hopf map it equals 1.
The Hopf fibration is the smallest non-trivial example of a structure that has become foundational in physics: fiber bundles. Maxwell's electromagnetism, the standard model of particle physics, magnetic monopoles, instantons in Yang-Mills theory, Berry phases, the Aharonov-Bohm effect — all of these live on fiber bundles, and the Hopf fibration is the first place that machinery shows up.