Square tori of varying size in the 3-sphere — only one embeds isometrically as the Clifford torus
Flat tori of every modulus embed isometrically in S3. While most tori have an infinite-dimensional space of isometric embeddings, the rectangular tori have a unique embedding. For the square torus this is the Clifford torus, dividing S3 into two isometric solid tori.
This program numerically computes embeddings of square tori using a spring-and-charge system adapted to the positively curved ambient geometry of S3. The default configuration is the square torus of the correct area, which quickly converges to the Clifford torus. The menu allows modifying the area (via ‘scale’) to see larger or smaller square tori that do not isometrically embed, visualized via stereographic projection.
Controls: Click and drag to orbit, scroll to zoom.
