English: Finding period relations. Blue numbers refer to synodic cycles; red numbers refer to years. All the seed periods for these processes are known from Babylonian astronomy (Supplementary Tables S5, S6). (a) Linear combinations of Babylonian period relations, which give those for Venus and Saturn from the FCI. (b) Period relations generated by a conventional Parmenides Process, which also give those for Venus and Saturn from the FCI. (c) Iterations of an Unconstrained Parmenides Process. (2p + 2r, 2q + 2 s) is omitted from Iteration 3 because it is the same as 2 x (p + r, q + s). (d) Three iterations of the Unconstrained Parmenides Process. The pairs in colour are those that are factorizable with prime factors < 100. The grey-shaded periods are those that are known from the FCI. Note that for Venus: (1445, 2310) ≡ (289, 462) and (735, 1175) ≡ (147, 235). The same table with errors is shown in Supplementary Table S5. (e) Periods derived from the Unconstrained Parmenides Process for our model of the Antikythera Mechanism and their errors, using our three criteria of accuracy, factorizability and economy. Except for the periods for Venus and Saturn, all the final periods were already known in Babylonian astronomy. The error parameters are defined in Supplementary Discussion S3.