Hopf Fibration

The Hopf fibration is a fiber bundle with a two-dimensional sphere as the base space and circles as the fiber space. It is the geometrical shape that relates Einstein’s spacetime to quantum fields. In this model, we visualize the Hopf fibration by first calculating its points via a bundle chart and then rendering the points in 3D space via stereographic projection. The projection step is necessary because the Hopf fibration is embedded in a four-space, yet it has only three degrees of freedom as a three-dimensional shape. The idea that makes this model more special and interesting than a typical visualization is the idea of Planet Hopf, due to Dror Bar-Natan (2010). So, if the base space is a two-dimensional sphere much like the skin of the globe, then we can model the Earth as a sphere and skin the horizontal sections of the bundle. Into the bargain, the Earth rotates about its axis every 24 hours. That spinning transformation of the Earth, together with the non-trivial product space of the Hopf bundle, can be encoded naturally into the visualization. It makes a lot of sense no matter how ridiculous, especially when we try to visualize differential operators in the Minkowski space-time and investigate the properties of spin-transformations. The following explains how the source code for generating animations of the Hopf fibration works (alternative views of Planet Hopf). 🔗
https://iamazadi.github.io/Porta.jl/d...