The Infinite Expanding Sphere of Number Systems: Radial Embedding of All Arithmetics from Origin to Infinity

We present a unified geometric framework that embeds all possible number systems into a single structure: an infinitely expanding sphere.Starting from a sphere of diameter 1, we let the diameter grow through all positive reals:( d ∈ {1, 10, 100, 1000, googol, ....} ∞. At each stage, the sphere’s surface contains (2^N) points, and each point connects to the origin via a unique radial line.As the diameter extends to infinity, each radial line becomes a complete ray from the origin to infinity, providing a “slot” for a distinct number system. The construction unifies three powerful geometric ideas: Radial symmetry — all number systems emanate from a common origin; Antipodal duality — opposite points on the sphere represent dual systems; Infinite extension — the radius grows without bound, making each ray a complete number line. This provides the first geometric model in which algebraic diversity (a continuum of number systems), geometric growth (from finite to infinite), and duality structure (antipodal pairs) are simultaneously realized within a single visualizable object.