Polycentric Geometry: From the Universal Proportion of a Point and a Sphere to Networks of Intertwined Spheres

<jats:p>We introduce and rigorously prove the notion of the universal proportion for a point inside a sphere. This ratio is generalized to polycentric systems—finite sets of intertwined spheres. We develop the formalism of polycentric coding, groups of inversion symmetries, and harmonic networks, establishing connections with conformal and projective geometry.</jats:p>