The Algebraic Deformation of Continued Fractions: A Dimensional-Geometric Framework for ϕ, e, and π

Abstract

This paper presents a minimalist, geometric-arithmetic axiomatic system that views the three fundamental constants of nature—the Golden Ratio phi,, Euler's number, and the circle constant pi—not as isolated analytical objects, but as structural phase transitions within a mathematical deformation. Departing from traditional calculus that introduces these constants through distinct, isolated limit processes (quadratic equations, compound interest limits, circular integrals), this work establishes a rigorous "Geometry-First" approach. The evolution of physical space and its corresponding dimensions maps one-to-one into the stepwise modification of algebraic operations within generalized continued fractions. By formalizing the shift between constants as a necessary dimensional phase transition, the accusation of empirical numerology or overfitting is mathematically refuted. The foundation of this unconventional framework relies on coupling geometric dimensions with increasing degrees of freedom in the coefficient space: Stage 1 (Dimension 0, the Point): Forces absolute mathematical self-reference with no active operations, deterministically yielding the Golden Ratio (\(\phi \)) as the static fundamental symmetry of space. Stage 2 (Dimension 1, the Line): Breaks monotony by injecting linear growth (\(+1\)) into the denominator structure, simulating a continuous flow of building blocks to generate Euler's number (\(e\)) as the base of natural growth. Stage 3 (Dimension 2, the Closed Surface): Transforms linear growth into a cyclic area operation (squaring the odd numerators), while the denominator freezes at the invariant value of \(6\) due to the Euclidean Kissing Number in a flat plane, forcing the system into the perfect curvature of \(\pi \). The framework concludes by deriving a deformed Euler's identity that operates entirely without classical analytical limits, and demonstrates a tetrahedral transformation when transitioning into 3D space, where the infinite fractal continued fraction for \(\pi \) inside a sine function collapses precisely into the rational baseline value of 1/2. This work offers a completely novel, strictly translational perspective on the deep, inherent symmetries of mathematical space.