Isometric embedding of the flat hexagonal torus in the 3-sphere with adjustable parameters
Flat tori of every moduli embed isometrically in the three-dimensional sphere S3; generically in an infinite-dimensional space of ways. This program numerically computes an embedding of the hexagonal torus C/(Z⊕Zω) using a spring-and-charge system on a mesh, adapted to the ambient geometry of positive curvature.
The charge interaction uses the geodesic distance on S3 (via arccos of the inner product of unit vectors in R4) rather than Euclidean distance, and vertices are reprojected onto S3 after each integration step. The menu at the top right modifies the strength of the electric charge and its interaction distance, to help remove self-intersections. The result is stereographically projected to R3 for visualization.
Controls: Click and drag to orbit, scroll to zoom.
