In 1894, Henry Wente discovered the Wente torus, an immersed torus of constant (but not zero) mean curvature. Such constant mean curvature (CMC) surfaces are counterexamples to Hopf's conjecture that a sphere is the only closed surface with constant mean curvature that is compact [1]. The set of all symmetric tori is in one-to-one correspondence with the set of reduced fractions ; each corresponds to a symmetric Wente torus labeled as [2].

[1] H. C. Wente, "Counterexample to a Conjecture of H. Hopf," Pacific Journal of Mathematics, 121(1), 1986 pp. 193–243. projecteuclid.org/euclid.pjm/1102702809.