Hopf Fibration
Discovered in 1931 by Heinz Hopf, it is an important mapping from the Hypersphere to the 2-sphere and an early example of a Fiber Bundle. Here, points on the 2-sphere correspond to circles on the Hypersphere in 4 dimensions, and projecting those circles back into 3-dimensions gives rise to an intricate structure of space-filling, mutually interlinked circles.
For a mathematical introduction, see this article
The program was written in JavaScript. Source code on github
Rendering is based on Three.js/WebGL
Projected fibers of circles on the two-sphere give rise to interlinked tori in 3-space. Note that one set of points on the base space was chosen to be a semicircle, giving rise to what looks like a sliced torus from this perspective, but it actually homeomorphic to a cylinder.
Note also that R³ has been shrunk into B³ (the open unit ball in R³) here, hence the projected fibers have lost some of their geometric properties.