Understand S¹ fibers over S² and how every point on the 2-sphere corresponds to a circle in 3-space
Every point on an ordinary sphere has a whole circle hiding behind it in four dimensions. Pick points, sweep them around, and watch those circles fill space.
Pick a base point on S² from the menu and watch its fiber appear. The colored dot on the small sphere marks your point. Notice that the south pole gives a straight line, while other points give circles of different sizes.
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.
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 a point sweep around the equator of S² while its fiber turns through space, leaving ghost trails behind. Drag to orbit and see the trails close up into a torus.
Watch a point sweep around the equator of S² — its fiber rotates continuously through 3-space. Ghost trails show previously visited fibers.
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.
Drag the latitude slider and watch the torus of fibers change shape — from a thin tube near the poles to the Clifford torus at the equator.
All fibers over a single latitude circle on S² form a torus in 3-space.
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.
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