Complex-Field Extension of the FCFG Framework: Complex Continued Fractions, Fractal Topology and Phase Rotation

 Author: Suhang Zhang

                (Luoyang, Henan)

Abstract

The prior theory of Fractal Continued-Fraction Geometry (FCFG) establishes a correspondence between truncated continued fractions on the real line and one-dimensional self-similar fractal skeletons. Nevertheless, the real-valued operator f(x)=\dfrac{1}{a+x} can only produce one-dimensional Cantor dust lacking connectedness. To explore richer topological structures, this paper extends the FCFG operator to the complex plane \mathbb{C} and defines the complex continued-fraction mapping f(z)=\dfrac{1}{a_i+z}. We investigate the characteristics of limit sets (Julia-like sets) generated by this mapping as a Möbius transformation on the complex plane, and demonstrate that the phase property intrinsic to complex parameters introduces rotational degrees of freedom for geometric iteration via Euler’s formula e^{i\theta}. This extension lays a theoretical foundation for FCFG to advance from static one-dimensional geometry toward two-dimensional complex dynamical systems.

Keywords: Fractal Continued-Fraction Geometry; complex continued fraction; Möbius transformation; complex dynamical system; Euler’s formula

1. Introduction

Within the earlier FCFG framework, the mapping f_i(x)=\dfrac{1}{a_i+x} (x\in\mathbb{R}^+) rigorously links continued-fraction arithmetic recursion with one-dimensional geometric scaling. For coefficient sequences such as [1,2], however, its real limit set forms a totally disconnected Cantor set, which lacks rich two-dimensional spatial structure.

To describe intricate natural morphologies and dynamical behaviour, rotation and two-dimensional topology must be introduced. This paper generalises the FCFG operator from the real axis to the full complex plane z\in\mathbb{C} and constructs the formalism of “complex continued fractions”, to investigate emergent phenomena of fractal geometry in the complex domain.

2. Construction of the Complex Continued-Fraction Mapping

Define a family of continued-fraction operator mappings on the complex plane:

f_i(z) = \frac{1}{a_i + z}, \quad z\in\mathbb{C},\; a_i\in\mathbb{C}

For a finite coefficient sequence \{a_1,a_2,\dots,a_n\}, the truncated value r_n of the complex continued fraction can be expressed as composition of this mapping chain:

r_n = f_1 \circ f_2 \circ \dots \circ f_n(z_0)

In complex analysis, f_i(z) belongs to the classical class of Möbius transformations. Unlike simple scaling on the real line, complex mappings are conformal: they map circles or straight lines on the complex plane to new circles or straight lines. This supplies the algebraic basis for generating two-dimensional complex topology from one-dimensional arithmetic recursion.

3. Topological Transition: From Cantor Dust to Planar Fractal Sets

When periodic sequences such as [1,2] are substituted into the complex operator, the iterated limit set is no longer confined to a line, but expands into a compact attractor on the complex plane.

In the language of complex dynamics, limit sets generated by such nonlinear IFS exhibit Julia-set-like features:

- Connectedness bifurcation: Minor adjustments to the distribution of coefficients trigger sharp phase transitions of the attractor between fully connected petal-shaped forms and fragmented dust-like clusters.

- Non-integer dimension: The Hausdorff dimension of these complex fractal sets lies strictly between 1 and 2, with boundaries bearing infinitely fine self-similar structure.

[Insert figure here: planar Julia-like fractal generated by the complex continued-fraction operator, with parameters f(z)=1/(1+z) or f(z)=1/(2+z)]

Figure 1: Two-dimensional fractal limit set generated by the complex continued-fraction operator

4. Phase Rotation and the Natural Emergence of Euler’s Formula

Only scaling is accessible on the real line; in the complex domain, by contrast, each complex parameter a_i inherently carries magnitude and phase.

Write the complex coefficient in polar form a_i = r_i e^{i\theta_i}. The Möbius transformation f_i(z)=\dfrac{1}{r_i e^{i\theta_i}+z} then acts geometrically not merely by scaling and reflection, but also by rotation through angle \theta_i.

The underlying mathematical principle governing this continuous spatial rotation is Euler’s formula:

e^{i\theta} = \cos\theta + i\sin\theta

Within the FCFG framework, this yields two core interpretations:

1. When \theta_i=0, the complex continued fraction degenerates back to the real case and produces non-rotational linear fractals.

2. When \theta_i\neq0, iteration proceeds with rotation across the complex plane. Accumulation of these discrete phase deflections over successive iteration levels macroscopically yields spiral fractals widespread in nature, such as logarithmic spirals.

Euler’s formula is thereby no longer treated as an isolated complex identity, but as the intrinsic driving engine that generates rotational degrees of freedom (spin topology) for continued-fraction geometric iteration in the complex plane.

5. Conclusion and Outlook

By extending the FCFG operator onto the complex plane, this paper establishes a structural mapping between complex continued fractions and two-dimensional planar fractals (Julia-like sets). The introduction of phase for complex parameters naturally bridges Euler’s formula and geometric rotation, unifying linear fractals and spiral fractals within the complex domain.

This extension marks the transition of FCFG theory from “one-dimensional geometry of discrete arithmetic” to “two-dimensional topology of complex dynamical systems”. Future numerical experiments may compute connectedness thresholds under varying complex parameters and compile atlases of spiral fractals for different phase angles \theta.