---

Analysis of the Continued Fraction and Geometric Mapping of the Natural Constant e within the Fractal-Continued-Fraction Geometry (FCFG) Framework

Author: Suhang Zhang (Luoyang, Henan, China)

Abstract

Within the framework of Fractal-Continued-Fraction Geometry (FCFG), this paper investigates the infinite continued fraction structure of the natural constant e and its corresponding iterative geometric mapping characteristics. Based on Euler's classical regular continued fraction expansion of e, we substitute its structured partial quotient sequence into the core FCFG iterative operator f(x) = \frac{1}{a+x}. By constructing a one-dimensional Iterated Function System (IFS), we analyze the hierarchical geometric skeleton, iterative stability, and topological scaling behavior generated by this ordered increasing sequence. The results indicate that, compared to periodic continued fractions and disordered random sequences, the quasi-periodic progression structure of e generates a fractal limit set with highly regular nested properties and smooth scale evolution. This paper establishes a standard geometric characterization paradigm for structured non-periodic irrational numbers within FCFG, completing the correspondence spectrum between continued fraction number theory and fractal geometry.

Keywords: Fractal-Continued-Fraction Geometry (FCFG); Natural constant e; Continued fraction partial quotients; Iterated Function System (IFS); Geometric topological mapping.

1. Introduction

Fractal-Continued-Fraction Geometry (FCFG) establishes a one-to-one correspondence between real number continued fraction expansions and self-similar fractal structures. For any real number, its continued fraction partial quotient sequence \{a_i\} can generate a finite-level geometric skeleton step-by-step through the iterative operator f_i(x) = \frac{1}{a_i+x}, converging to a unique fractal limit set.

Existing works primarily discuss the strictly self-similar fractal structures corresponding to periodic continued fractions (quadratic algebraic numbers). To complete the applicable spectrum of FCFG, it is necessary to introduce classical number-theoretic constants that are non-periodic but highly structured as standard examples.

The continued fraction expansion of the natural constant e possesses a globally ordered, recursively evolving, and non-chaotic structure. It is one of the few irrational sequences in mathematics whose overall recurrence pattern can be precisely written out. In this paper, we fully incorporate the continued fraction sequence of e into the FCFG iterative system and analyze its unique dynamical and topological features.

2. Algebraic Structure of the Continued Fraction of e

According to Euler's classical result, the continued fraction expansion of the natural constant e is:

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, \dots] 

Its partial quotient sequence strictly satisfies the recurrence paradigm:

· First term: a_0 = 2

· Subsequent terms follow a (1, 2k, 1) cycling progression, for k = 1, 2, 3, \dots

This sequence exhibits three key characteristics:

1. Non-periodic: It does not satisfy the conditions of a periodic continued fraction and therefore is not a quadratic algebraic number.

2. Highly structured: The overall pattern is completely predictable and non-chaotic.

3. Slow growth: The increment exhibits linear growth, without jumps or explosive divergence.

In the FCFG system, this implies that its corresponding geometric iterative rules possess an inherent structural capacity for ordered hierarchical evolution.

3. Geometric Mapping Construction under the FCFG Framework

For any continued fraction partial quotient sequence \{a_i\}, define the iterative mapping family:

\mathcal{F} = \{f_1, f_2, \dots, f_n\}, \quad f_i(x) = \frac{1}{a_i+x} 

Substituting the ordered sequence of e step-by-step yields the hierarchical iterative rules:

· Level 1: a_1 = 2

· Levels 2~4: 1, 2, 1

· Levels 5~7: 1, 4, 1

· Levels 8~10: 1, 6, 1

· And so on.

This iteration generates a one-dimensional nested geometric skeleton on the positive real axis \mathbb{R}^+. Because the partial quotient sequence follows a stable progressive paradigm, the iteration compression ratio changes smoothly with the hierarchy, forming an ordered multi-scale self-similar structure distinct from chaotic sequences.

4. Analysis of Topological and Dynamical Characteristics

4.1 Iterative Stability and Boundedness

Although the partial quotients of e tend to infinity, the growth rate is extremely low and uniform, minimizing the perturbation of the compressive transformation at each iteration level. Within the finite truncation levels, the overall iterative orbit consistently remains stable, convergent, and free of oscillatory divergence.

4.2 Scaling and Dimensional Characteristics

Unlike periodic continued fractions that generate strictly self-similar fractals, or random disordered sequences that generate uniform Cantor dust, the progressive ordered structure of e is expected to yield a limit set whose Hausdorff dimension exhibits a unique continuous scaling evolution. This possesses hierarchical differentiation, ordered nesting, and asymptotic smoothness.

This is the first time FCFG has characterized the geometric morphology of a quasi-periodic progressive irrational number, filling the intermediate spectrum between periodic structures and chaotic disordered structures.

5. Conclusion

This paper formally incorporates the classical continued fraction structure of the natural constant e into the FCFG system, constructing its corresponding one-dimensional fractal iterative limit set through a standard iterative operator.

Core conclusions:

1. The continued fraction sequence of e is a structured non-periodic progressive sequence, generating a stable, ordered, and multi-level nested special fractal topology in FCFG.

2. Its geometric morphology is distinct from both the periodic fractals of quadratic irrationals and the chaotic fractals of irregular irrationals, constituting a novel intermediate standard paradigm.

3. This example further demonstrates that the continued fraction expansion of all real numbers can uniquely correspond to a dedicated class of fractal geometric topologies, validating the universality of the FCFG framework.

Future work will precisely calculate the fractal dimension of this limit set and conduct a comparative FCFG topological study of three typical constants: e, \pi, and \sqrt{2}.

References

[1] Euler, L. De fractionibus continuis dissertatio, 1744.

[2] Zhang, S. Fractal-Continued-Fraction Geometry (FCFG): Core Foundations and System Construction, 2026.