Interactive Explorer

Click points on S² and watch their fibers appear in real time — build your own fibration

Build Your Own Fibration

Click anywhere on the equirectangular map of S² to place a base point. The corresponding Hopf fiber instantly appears in the 3D viewer below. Build up your own collection of fibers and watch the structure emerge.

Try clicking along a latitude line to see a torus form. Click scattered points to see how fibers from distant base points still link with each other.

Point-and-Click Explorer

Equirectangular projection of S² — top = north pole, bottom = south pole

Try this: Click a vertical line of points (same longitude, varying latitude) to see a family of linked circles that grow from a near-line at the top to a large circle at the bottom.

The Complete Picture

For comparison, here is the full fibration with evenly spaced base points. The mathematical structure you built by hand in the explorer above is the same structure — just more densely sampled.

60 fibers

Where to Go from Here

  • Generalizations — There are Hopf fibrations S⁷ → S⁴ and S¹⁵ → S⁸ using quaternions and octonions
  • Physics — The Hopf fibration appears in the Bloch sphere representation of qubits in quantum mechanics
  • Geometry — It connects to the theory of fiber bundles, principal bundles, and gauge theory in physics