The Infinite Conical Frustum of Number Systems: A Geometric Unification of All Arithmetics

We present a unified geometric framework that embeds all possible number sys-tems into a single three-dimensional structure: an infinite conical frustum (truncatedcone). Starting from two parallel circular bases—one with diameter 1 (bottom) andone with diameter 2 (top)—each containing continuum-many points, we establisha bijection between corresponding points and connect them with oblique line seg-ments. As the frustum’s height extends from 1 to infinity (h ∈ {1, 10, 100, 1000, . . . , googol, . . .} →∞), each oblique segment becomes a complete number line, and the frustum main-tains its geometric identity. Since each base contains 2ℵ0 points, we obtain 2ℵ0distinct oblique segments—sufficient to accommodate every conceivable number sys-tem: R, ∗R, Qp, surreals, and infinitely many others. This construction provides aconcrete 3D visualization where (1) the frustum’s height literally becomes the num-ber line extending to infinity, (2) each oblique fiber represents a distinct arithmeticstructure, and (3) all number systems coexist simultaneously in a unified geometricobject that preserves frustum identity even at infinite height.