Why All Spaces Are Generated by the Infinium: From Orthogonality to Cartesian Coordinates and Onward to All Manifolds (The Geometric Quantum as the Single Root of the Mathematics of Spaces)

It is shown that all known types of spaces and manifolds — from Euclidean ℝⁿ to Kähler, symplectic, spin, and supermanifolds — are generated from a single fundamental object: the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). The key property of the infinium is the orthogonality of its legs, which automatically defines a Cartesian coordinate system. Since all classical spaces are defined through Cartesian coordinates (or their local analogues), they all turn out to be derivatives of the infinium. We give a systematic survey of this generation with explicit formulas and categorical constructions, demonstrating that the mathematics of spaces is one, and its root is a single geometric quantum.