Isoperimetric ratios of toroidal Dupin cyclides

The combination of recent results due to Yu and Chen (2022) [17] and to Bostan and Yurkevich (2022) [2] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In this work we investigate a generalization of the aforementioned results to the case of a rectangular Clifford torus. As in the square case, we find closed-form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to ℝ 3 and show that the corresponding function is strictly increasing. But unlike the square case, we show that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus. These results are obtained using a mixture of differential geometry and computer algebra. Moreover, the computer algebra technique in this work is more streamlined and more direct than in the aforementioned approaches, even for the most important case of the square Clifford torus.