Base 60 and the Icosahedral Factorization of Coordinate Space

This work establishes a minimality property of the positional base 60 grounded in the classification of finite subgroups of SO(3). Among these subgroups, the icosahedral group A₅ ≅ I of order 60 is the unique simple non-abelian member. Two notions of admissible group structure on a digit set are formulated — one requiring realization as a subgroup of SO(3), the other requiring only a faithful irreducible three-dimensional real representation — and shown to be equivalent for simple non-abelian groups. The minimal admissible base is b = 60, attained uniquely by A₅. The digit set {0, …, 59} is placed in bijection with A₅, and digit sequences are read as walks in the Cayley graph, equivalently as paths in the tessellation of S³ by the 120-cell. Doubling the base to 120 corresponds to the spin lift 2I ⊂ Spin(3) and yields an integer-exact encoding of discrete orientations: the encoding replaces floating-point quaternion composition with finite-group multiplication and applies to systems with intrinsic icosahedral symmetry, including icosahedral quasicrystals, viral capsids, and icosahedral molecular clusters. The relation to the abelian Chinese Remainder decomposition Z/60 ≅ Z/4 × Z/3 × Z/5 is recorded: the two structures are inequivalent readings of the same digit set, reflecting the non-isomorphism of Z/60 and A₅. The companion paper developing the abelian CRT structure is published at doi.org/10.5281/zenodo.19779500