Attempting to embed the flat hexagonal torus in $\mathbb{R}^3$ — impossible smoothly, but revealing to try
Flat tori do not embed smoothly in R3, but it is interesting to see what happens when one tries. This program starts with a torus mesh whose intrinsic geometry is that of the flat hexagonal torus C/(Z⊕Zω) with ω=e2πi/3, and attempts to find an embedding into R3 via a spring-and-charge system.
The springs encode the flat hexagonal metric, while electric charges provide repulsion to prevent self-intersection. The system converges to a distorted torus that compromises between the flat intrinsic geometry and the constraints of Euclidean 3-space. Properties of the spring-charge system can be modified via the menu in the upper right.
Controls: Click and drag to orbit, scroll to zoom.
