3 Before Infinity

Abstract

This article introduces a novel structural principle known as the "Principle of Triadicity." According to this principle, the number 3—and its geometric counterpart, the triangle—represent the most fundamental concepts in discrete systems, whereas infinity and the circle are limiting, ideal concepts that lack counterparts in physical and computational reality. This claim rests on three theoretical pillars: (1) the isomorphism of 3 and the triangle versus ∞ and the circle, based on Euclidean principles and Archimedes' method; (2) the principle of the dimensional density of 3, based on dimension theory in vector spaces; and (3) the triangular remainder theorem, based on numerical analysis and the finite element method. The article concludes that in any finite system, 3 takes precedence over infinity, and the real world is founded upon 3 rather than infinity.