New Paper (December 20th, 2025): Unity Harmonica Geometric Characterization
A Harmonic Right–Triangle Framework for All Uniform Polyhedra
Abstract
The Unity Harmonica Geometric Theory establishes that every uniform polyhedron (all 31 Platonic, Archimedean, and Catalan solids) is completely generated by a single harmonic right triangle derived from its topological invariants vertices (V) and edges (E). The triangle’s two harmonic factors (x) (constructive full-span) and (y) (asymmetry residual) encode the quadrivium of classical means and four novel differential/logarithmic means, which in turn define a logarithmic spiral whose quadratic expansion (Factor 1) and harmonic contraction (Factor 2) produce all major measurements—vertices, edges, faces, surface area, volume, dihedral angles, and musical intervals—perfectly and without exception. The theory is universal across the 31 uniform polyhedra and extends to novel Electrum-regime solids such as the Alphahedron.
