Jan 16, 2019
A Hopf fibration. This gorgeous visualization portrays a suitably deep idea in differential topology.
The Hopf fibration describes 3-spheres (aka hyperspheres, 4-dimensional analogues of spheres) in terms of “ordinary” spheres and circles. It is an early example of a fiber bundle.
Quoting Wikipedia, “Technically, Hopf found a many-to-one continuous function (or ‘map’) from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere comes from a distinct circle of the 3-sphere (Hopf 1931). Thus the 3-sphere is composed of fibers, where each fiber is a circle—one for each point of the 2-sphere.”
Hopf fibrations have many incredible properties. I write, however, because they are beautiful.
Mathematics is beautiful. <3
