Based on the equations described in Lawson’s Minimal Surfaces and the Sudanese Mobius Band. This notebook is modified after receiving some help from Dan Piker in this notebook, which is in turn a modification of my original attempt, after inspiration by Dan Piker’s tweet.

We start with a generalization of a Clifford torus in 4D space. below is a parameter of this surface, while applies a four-dimensional rotation before the final stereographic projection into 3D space.

When this is the standard Clifford torus. When it is Lawson’s Klein bottle. A rotation by is applied in the -plane to yield :

The result is then projected stereographically into three dimensions:

The vertex shader receives a surface in coordinates and evaluates the parametric surface function, computing the 4D Clifford torus, applying a 4D rotation parameterized by , and projecting stereographically into 3D.