Hopf Fibration

The Hopf fibration is a specific way of mapping the three-dimensional sphere onto the ordinary two-dimensional sphere so that the collection of points sent to any single point on the target sphere forms a great circle on the three-dimensional sphere, meaning the whole three-dimensional sphere can be decomposed into a continuous family of interlocking circles, one for every point of the ordinary sphere. It was described in 1931 by the German mathematician Heinz Hopf, whose construction gave one of the first and most striking examples of a mapping between spheres of different dimensions that cannot be continuously deformed into a constant map, a discovery that helped found the modern study of homotopy theory in topology. Any two of the circles that make up the fibration are linked with one another exactly once, in the sense used in knot theory, so that the whole structure of interlocking rings gives a natural and much-reproduced visual picture of how points on a four-dimensional space of directions can be woven together in three-dimensional space. The Hopf fibration reappears well beyond pure topology, including in the mathematical description of the quantum state space of a two-level quantum system, known as a qubit, where it explains the geometric relationship between the qubit's underlying quantum description and the more familiar sphere used to visualize its possible states.