Understand S¹ fibers over S² and how every point on the 2-sphere corresponds to a circle in 3-space
The Hopf fibration, discovered by Heinz Hopf in 1931, is a mapping from the 3-sphere (S³) to the 2-sphere (S²). The key insight: every point on the ordinary sphere S² corresponds to a circle inside the higher-dimensional S³. These circles are called fibers.
Formally, it is a fiber bundle S¹ → S³ → S², meaning the 3-sphere is decomposed into a family of circles (S¹), one for each point of S². After stereographic projection from S³ to R³, these fibers become circles (and one line) in ordinary 3-dimensional space.
Select different base points on S² and see the corresponding fiber appear in 3D. The small glowing sphere shows the reference 2-sphere, and the colored dot marks the chosen base point. Notice how the south pole maps to a straight line, while other points map to circles of varying sizes.
Key insight: The south pole fiber is a straight line through the origin (it passes through the projection point, which maps to infinity). The north pole fiber is the unit circle, and fibers grow as the base point approaches the south pole.
Watch what happens as a point sweeps continuously around the equator of S². The corresponding fiber rotates smoothly through 3-space, leaving ghost trails behind. Together, all equatorial fibers form a Clifford torus — a flat torus embedded in the 3-sphere.
Watch a point sweep around the equator of S² — its fiber rotates continuously through 3-space. Ghost trails show previously visited fibers.
All fibers over a single latitude circle on S² form a torus in 3-space. Drag the latitude slider to see how this torus changes shape — from a thin tube near the poles to the Clifford torus at the equator.
Key insight: The entire 3-sphere decomposes into a family of nested tori (one for each latitude) plus two singular fibers at the poles. This is the Clifford decomposition of S³.