Hyperspace Flows · 4D
Hopf Fibration
The 3-sphere woven from circles
Every point of an ordinary sphere becomes a circle in the 3-sphere, and no two circles ever touch, yet every pair is linked. Projected into our space they nest as tori of interlocking rings.
- Fibre
- S¹
- Base
- S²
- Total
- S³
Watch it turn through the fourth dimension, hear the chord of its edges, and slice it with our space to see it pass through.
Facts
- Each circle (fibre) corresponds to one point on a 2-sphere.
- Any two fibres are linked exactly once, like two rings of a chain.
- Fibres over one latitude form a torus of Villarceau circles.
- It appears in physics: qubits, magnetic monopoles, light’s polarisation.
History and meaning
Heinz Hopf discovered it in 1931. It showed, astonishingly, that a map from a 3-sphere to a 2-sphere can be knotted in a way that cannot be undone.
Across the dimensions
- Its equator The SphereOne dimension up, the sphere becomes the 3-sphere, woven entirely from linked circles.
Learn it step by step
- Journey Into the Fourth DimensionStep 10: The Hopf fibration
