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

The Roman dodecahedron, a hollow bronze object known from more than 130 specimens (2nd–4th century CE), has resisted functional identification since the first recorded find in 1739. More than fifty hypotheses have been advanced; none has been established. No ancient text mentions the object. This paper proves that the Roman dodecahedron is a hand-held sexagesimal computing device - the physical realization of the icosahedral rotation group A₅, the only simple non-abelian subgroup of SO(3). Two theorems are proved: Minimality theorem: Base 60 is the minimal base whose digit set admits the structure of a simple non-abelian subgroup of SO(3), realized uniquely by A₅ (|A₅| = 60). Ladder–CRT correspondence: The Chinese Remainder decomposition Z/60Z ≅ Z/3Z × Z/4Z × Z/5Z coincides, coordinate by coordinate, with the coset decomposition along the subgroup chain {e} ⊂ Z₃ ⊂ A₄ ⊂ A₅. The argument rests on three independent pillars: Algebraic (proved): the two theorems above. Textual (documented): a uniform procedural signature — sexagesimal input, table lookup, iterated rescaling (qualium…talium), sexagesimal output - is identified in Ptolemy's Almagest (2nd century), Copernicus' De revolutionibus (1543), Reinhold's Prutenic tables (1551/1585), and Mästlin's computational appendix to Kepler's Mysterium Cosmographicum (1596). Over 130 printed values are verified by recomputation; the procedural identity spans 1500 years. Physical (observed): the 12 graduated apertures, 20 identical vertex knobs, and 60 rotational states of the artifact map bijectively onto the algebraic structure. A full reconstruction of the operating protocol is provided, including three elementary gesture classes (SPIN, ROLL, TILT) and worked examples reproducing Kepler's Mercury computation step by step. Falsifiable predictions are given: interior wear patterns, aperture systematics, replication of historical computations on a physical replica, and ergonomic optimality of the attested size range. The paper is dedicated to Hippasus of Metapontum, drowned for revealing the secret of the dodecahedron. The secret is now a theorem. 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 [email protected]