Family of rhombic flat tori isometrically embedded in the 3-sphere, parameterized by vertex angle
Flat tori of every modulus embed isometrically in the three-dimensional sphere S3; generically in an infinite-dimensional space of ways. This program numerically computes embeddings of rhombic tori — those whose fundamental domain is a rhombus with a given vertex angle at the origin — using a spring-and-charge system on a mesh adapted to the ambient geometry of positive curvature.
The menu at the top right allows selecting the particular torus by changing the rhombus angle, as well as adjusting the strength of the electric charge to help remove self-intersections. The spring system enforces the target flat metric while the charge prevents the surface from passing through itself in S3, visualized via stereographic projection into R3.
Controls: Click and drag to orbit, scroll to zoom.
