Hopf fibration: Interactive visualization
The Hopf fibration is a fundamental construction in topology that gives a surprising way of decomposing higher-dimensional spheres. Formally, it is a continuous map from the 3-sphere to the 2-sphere:
\( h: S^3 \to S^2 \)
with the remarkable property that the preimage \( h^{-1}(p) \) of every point \( p \in S^2 \) is a circle \( S^1 \). In other words, \(S^3\) can be written as a disjoint union of circles (called fibers), one for each point on the ordinary 2-sphere.
You can explore Hopf fibrations using the following link: