Exact coordinates for vertex-minimal symmetric paper tori

Construct exact coordinates for an eight-vertex embedded paper torus of modulus exactly i or exactly \rho, thereby giving an explicitly specified realization of the square or hexagonal flat torus.

Background

The existence proof constructs eight-vertex paper tori of the square modulus i and hexagonal modulus \rho through the implicit function theorem, but the resulting configurations are not given by explicit coordinates. The appendix supplies only numerical coordinates for more inflated examples, rather than exact coordinates. Thus an explicit exact realization remains unresolved.