A Unified Universal Coordinate System: 2D Sierpinski Triangle Projection Framework for 3D Multi-Scale Localization

This paper presents a revolutionary coordinate system for universal spatial localization using a fixed 2D Sierpinski Triangle as a projection surface for 3D space. Rather than extending to 3D structures (tetrahedra), we maintain the elegant simplicity of the 2D triangle and develop a rigorous framework for projecting all 3D phenomena onto a hierarchical 2D addressing system. The master Sierpinski Triangle, anchored at a specific point in space, serves as a universal reference surface. Any object in 3D space—whether at astronomical, terrestrial, biological, or subatomic scales—is addressed by: (1) triadic coordinates within the 2D triangle plane, and (2) a perpendicular depth component orthogonal to the plane. This hybrid addressing combines planar hierarchy (triadic) with radial hierarchy (depth), creating a complete 3D localization system without dimensional expansion. We formalize the mathematical structure, establish rigorous proofs of completeness and uniqueness, develop practical measurement protocols, and demonstrate applications across all physical scales.