Paper: Foundational Axioms and Isomorphism System of Fractal Continued-Fraction Geometry (FCFG)

Abstract

In the existing mathematical framework, continued fractions belong to the domains of number theory and analytic approximation, while self-similar fractals fall under fractal geometry. These two recursively structured systems, despite sharing a common origin, have long been studied in isolation. This paper proposes a core foundational axiom establishing a strict isomorphism between arithmetic recursion (finite-order continued fractions) and geometric recursion (finite-order self-similar fractals): the rational values of n-th order finite continued fractions correspond one-to-one with the equivalent similarity ratios of n-th order finitely generated self-similar fractals. Finite iterations serve as the primary discrete correspondence, while infinite limits yield irrational constants and complete fractal topologies. Based on this isomorphism, a novel interdisciplinary field—Fractal Continued-Fraction Geometry (FCFG)—is established. FCFG provides a unified explanation for the convergence criteria of continued fractions and the dimensional evolution laws of self-similar fractals. It naturally accommodates both classical fixed-scale fractals and Ramanujan variable-coefficient nested continued fractions (a new type of variable-scale fractal), thereby constructing a universal theoretical framework that bridges discrete rational arithmetic and continuous fractal geometry, filling the long-standing theoretical void between number theory and fractal geometry.

Keywords: Fractal Continued-Fraction Geometry; FCFG; Continued Fractions; Self-Similar Fractals; Recursive Isomorphism; Similarity Ratio; Diophantine Approximation

1. Introduction

Since the inception of continued fractions, mathematicians such as Euler, Lagrange, and Gauss have utilized them as core tools for rational approximation of irrational numbers and in the number theory of quadratic fields. Following Mandelbrot's pioneering work in fractal geometry, self-similar geometric bodies like the Koch curve, Cantor set, and Sierpiński gasket have become standard models for geometric irregularity. Both systems are fundamentally constructed through layer-by-layer nested recursive rules:

- Continued Fractions: Arithmetic recursion, where each step nests a fraction a+frac{1}{square}, generating finite-order rational approximations layer by layer.

- Self-Similar Fractals: Geometric recursion, where the geometric body is subdivided at each step according to fixed or variable scaling ratios, generating finite-order fractal skeletons layer by layer.

Over a century of research has only yielded scattered discoveries of mappings between specific continued fraction iterations and complex fractals (Julia sets). A rigorous, universal one-to-one correspondence axiom at the finite-order level has never been established, leaving these two homologous recursive systems confined to separate disciplines. Grounded in the construction of finite positive integer orders, this paper proposes an isomorphism axiom and formally establishes Fractal Continued-Fraction Geometry (FCFG), achieving a global unification of the arithmetic structure of continued fractions and the geometric structure of self-similar fractals.

2. Core Fundamental Definitions and the First Axiom of FCFG

2.1 Definition of Finite n-th Order Simple Continued Fractions

For a sequence of positive integers {a_1, a_2, dots, a_n}, the n-th order finite continued fraction is defined as:

r_n = frac{1}{a_1 + frac{1}{a_2 + frac{1}{dots + frac{1}{a_n}}}}

where r_n in mathbb{Q} is a rational number, n in mathbb{N}^+, requiring no infinite limit.

2.2 Definition of Finite n-th Order Self-Similar Fractals

Let F_n be a finite-order self-similar geometric body generated by n rounds of geometric scaling recursion. The initial prototype undergoes n successive proportional subdivisions and nested replications to form a finite fractal skeleton. The global equivalent similarity ratio S_n is defined as the characteristic constant representing the overall scaling transformation of F_n.

2.3 Foundational Axiom of FCFG (The First Axiom of the Discipline)

Isomorphism Axiom: For any finite positive integer n, the value of the n-th order continued fraction r_n is strictly equal to the equivalent similarity ratio of the corresponding n-th order self-similar fractal: r_n = S_n.

Corollary 1: Continued fractions serve as the arithmetic and algebraic representation of self-similar fractals, while self-similar fractals act as the geometric embodiment of continued fractions; arithmetic recursion and geometric recursion share a common origin and are structurally isomorphic.

Corollary 2: In finite-order scenarios, rational continued fractions correspond one-to-one with finite self-similar fractals possessing rational similarity ratios.

2.4 Extension via Infinite Limits

As n to infty, if the sequence {r_n} converges to an irrational number alpha, the corresponding sequence of finite fractals {F_n} converges to a complete infinite self-similar fractal F_infty, with alpha serving as the global limit similarity ratio of F_infty.

Note: Under infinite iterations, the fractal measure (area, length) tends to degenerate. Therefore, the measure of the geometric body is not equivalent to the value of the continued fraction; however, the similarity ratio that determines the essential fractal topology is always precisely characterized by the continued fraction.

3. Verification via Classical Examples (Demonstration of FCFG Application)

Example 1: Golden Ratio Continued Fraction and Golden Fractals

The continued fraction r_n = [1; 1, 1, dots, 1] (n-th order) converges to alpha = frac{sqrt{5}-1}{2} as n to infty.

The corresponding n-th order finite golden fractal has a layer-by-layer scaling ratio exactly equal to r_n. The global similarity ratio of the limit fractal is the golden ratio constant, perfectly satisfying the axiom.

Example 2: Unit Rational Continued Fraction Corresponding to Fixed-Scaling Fractals

For r_1 = frac{1}{k} (k in mathbb{N}^*), the corresponding first-order fractal is generated by scaling and replicating the prototype by a factor of frac{1}{k}, yielding a similarity ratio S_1 = frac{1}{k} = r_1.

4. Three Core Research Domains of the Discipline (Territorial Division of FCFG)

4.1 Fundamental Theory Domain: Finite & Infinite FCFG Analysis

1. Finite Order: Relying on the isomorphism axiom, the scaling rules of fractal geometry are used to derive continued fraction recurrence formulas and rational approximation errors. Conversely, the algebraic operations of continued fractions are used to calculate the similarity ratios and local refinement degrees of finite fractals.

2. Infinite Order: The convergence of continued fractions is logically equivalent to (\iff) the topological convergence of infinite fractals. The convergence rate of continued fractions is determined by the decay law of the layer-by-layer similarity ratios of the fractal, replacing traditional analytic inequality tests. The irrational limit of a continued fraction \iff the characteristic constant of a complete self-similar fractal.

4.2 Number Theory Intersection Domain: FCFG and Constant Continued Fractions / Ramanujan Continued Fractions

Infinite continued fractions of common constants (e, \pi, \zeta(3), etc.) correspond to dedicated complete self-similar fractals, where the constant serves as the limit similarity ratio of the fractal. Scattered numerical identities are thus transformed into geometric propositions regarding the isomorphism of two fractals and the equality of their similarity ratios.

Ramanujan Variable-Scale Continued Fraction Subclass (Core Expansion Direction of FCFG): In the Rogers-Ramanujan continued fraction, the scaling parameter at each layer varies dynamically with q^k, corresponding to variable-scale self-similar fractals (distinct from classical fixed-similarity-ratio fractals). This pioneers a new research branch in variable-dimensional fractals, systematically encompassing all of Ramanujan's nested continued fractions and q-series identities.

4.3 Applied Engineering Domain: Practical Implementation of FCFG

Embedded Numerical Algorithms: Finite-order continued fractions equal finite fractal similarity ratios. Integer fractional operations replace floating-point calculations, enabling low-cost, high-precision approximation of irrational constants in numerical control and precision optical engineering.

Natural Fractal Modeling: Natural structures such as leaf veins, coastlines, and polymer chains are predominantly variable-scale fractals. Ramanujan-type variable-coefficient continued fractions achieve high-precision fitting via FCFG, compensating for the deficiencies of classical fixed-scale fractal fitting.

Chaotic Cryptography: Linear iterations of continued fractions generate fractal chaotic boundaries. Leveraging FCFG theory, novel chaotic encryption and post-quantum cryptographic construction schemes can be designed.

5. Disciplinary Positioning and Innovations

Innovation 1: Establishes a strict isomorphism axiom at the finite order, bridging continued fractions and self-similar fractals at the discrete finite level for the first time, distinguishing it from past scattered connections limited to infinite iterations in complex dynamical systems.

Innovation 2: FCFG serves as an overarching meta-theory. Classical fixed-similarity-ratio fractals and traditional elementary continued fractions are merely special cases of this theory, while Ramanujan variable-coefficient continued fractions give rise to an entirely new subclass of variable-scale fractals.

Innovation 3: Reconstructs the geometric connotation of Diophantine approximation: The precision of rational approximation by continued fractions equals the degree of local geometric refinement of the corresponding fractal, transforming number-theoretic approximation problems into fractal geometric topology problems.

6. Conclusion and Future Research Directions

This paper completes the foundational establishment of Fractal Continued-Fraction Geometry (FCFG). With recursive isomorphism as its core axiom, it unifies the arithmetic system of continued fractions and the geometric system of self-similar fractals, carving out an independent and comprehensive interdisciplinary research territory. Future research will proceed in three steps:

Refine the recurrence theorems and similarity ratio operation rules for finite-order FCFG.

Systematically rewrite classical continued fraction convergence theories and fractal dimension calculation formulas utilizing the FCFG framework.

Specifically construct the sub-theory of Ramanujan variable-scale FCFG, integrating its vast array of continued fractions and q-identities entirely into the fractal geometry explanatory system.