On a flat torus of fixed area, the coldest and hottest points of the heat kernel of the Laplace-Beltrami operator can be written in closed form using Gauss' hypergeometric function 2F1 and the elliptic modulus, so the ratio of the two temperatures alone determines the torus geometry. Ramanujan's identity linking 2F1 to Jacobi theta-nulls splits that ratio into the individual extreme temperatures, and the signature-3 theory does the same for the hexagonal torus. A student learns how special-function identities turn an extremal geometry problem into an explicit computation, and how this yields exact conjectured values for Landau's Weltkonstante, including the second lemniscate constant in the rectangular case.
This paper applies hypergeometric functions to compute the heat kernel of the Laplace-Beltrami operator on rectangular and hexagonal tori. It uses Gauss' hypergeometric function and elliptic moduli t…