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 north pole maps to a straight line, while other points map to circles of varying sizes.
Key insight: The north pole fiber is a straight line through the origin (it passes through the projection point at infinity). The south pole fiber is a large circle. All other fibers are circles of intermediate size.
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.