The Roman Dodecahedron as a Sexagesimal Computing Device: Group-Theoretic Structure, Physical Encoding, and Computational Signatures in Kepler and Ptolemy

This paper identifies the function of the Roman dodecahedron, a hollow bronze artifact of the 2nd–4th century CE whose purpose has remained unknown since its first recorded discovery in 1739. The identification rests on three independent pillars. The first is algebraic: a theorem proving that 60 is the unique minimal positional base whose digit set carries the structure of a simple non-abelian subgroup of SO(3), realized by the icosahedral rotation group A₅. The Chinese Remainder decomposition Z/60Z ≅ Z/3Z × Z/4Z × Z/5Z is shown to be identical to the subgroup ladder of A₅, mapping onto the three families of symmetry axes of the dodecahedron. The second is textual: systematic extraction of computational passages from Kepler (1596, 1609) and Ptolemy (2nd c. CE) reveals a uniform procedural signature, sexagesimal input, table lookup, and iterated rescaling cascades, isomorphic to the manual operation of the device. The third is physical: the graduated apertures, vertex knobs, and hollow form of over 130 known specimens correspond precisely to the functional requirements of the algebraic structure. A complete reconstruction protocol is provided, with worked examples reproducing Kepler's Mercury computation step by step. Four falsifiable predictions are stated. The silence of the ancient written record, far from being an obstacle, is shown to be a consequence of the Pythagorean tradition of enforced secrecy surrounding dodecahedral knowledge. --- Ξυα Mσςς eva@m0ss.io