The Hopf FibrationA way of combing the 3-sphere (the surface of a ball in four dimensions) entirely with circles: every point sits on exactly one circle, no two circles cross, yet every pair is linked like two rings of a chain. Assign each circle a colour by the point it maps to on an ordinary sphere and the circles nest into a family of doughnuts filling all of space. It is the first example of a topologically non-trivial bundle and it shows up in quantum spin, magnetic monopoles and twistor theory. The journey: take the Hopf map (z₁,z₂) → z₁/z₂, lift each target point to its circle e^iα(z₁,z₂) in the 3-sphere, and stereographically project that circle into 3D. Drag to orbit.