Geometric Mapping between Continued Fractions with Variable Coefficients and Variable-Scale Fractals: A Case Study of the Rogers–Ramanujan Continued Fraction

Author: Suhang Zhang,

                Luoyang, Henan

Abstract

Classical self-similar fractals are usually constructed with constant scaling ratios. Nevertheless, complex structures in nature often exhibit variable-scale behaviour. This paper investigates the correspondence between the recursive structure of Ramanujan’s continued fractions with parameter q and the geometry of variable-scale fractals. With the framework of iterated function systems (IFS), we argue that the dynamic coefficients within Ramanujan’s q-series can be mapped to layer-wise varying geometric contraction ratios. We further formulate conjectures on fractal generation with varying dimensions controlled by the parameter q, offering a purely geometric perspective for interpreting Ramanujan’s identities and complex natural morphologies.

Keywords: Ramanujan continued fraction; variable-scale fractal; IFS; q-series; variable-dimensional geometry

1. Introduction

Classical fractal geometry is built upon self-similar iteration with fixed scaling ratios (e.g., the Koch snowflake). Meanwhile, number theory contains a rich collection of recursive sequences with “non-constant scaling”. The Rogers–Ramanujan continued fraction possesses an elegant nested structure, whose coefficients vary as powers of q. This paper explores the following question: what topological geometries can arise if this arithmetic recursion with variable coefficients is mapped to IFS iteration in geometry?

2. Construction of the Core Mapping

2.1 Arithmetic Structure of the Rogers–Ramanujan Continued Fraction

The Rogers–Ramanujan continued fraction takes the form:

R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \cfrac{q^3}{1 + \cdots}}}}

Here the partial quotients are no longer constant; they follow a layer-dependent rule a_n = q^{n-1}.

2.2 Geometric Mapping via Variable-Scale IFS

Correspondingly, we define a family of mappings on the complex plane:

f_n(z) = \frac{q^{n-1}}{1 + z}

The contraction rate |f'_n(z)| of each mapping decays or amplifies exponentially with the layer index n.

- If |q|<1, the contraction tends to zero as n\to\infty. The resulting limiting attractor converges rapidly along certain directions and approaches a geometric “freezing” state.

- If |q|>1, the dynamical system exhibits extreme divergence and distortion.

3. Conjectures and Directions for Subsequent Computation

Conjecture 3.1: For |q|<1, the attractor K_q generated by this variable-coefficient IFS is multifractal. Its local dimension is not uniform but varies with spatial location.

Conjecture 3.2: Classical identities for R(q) in modular form theory (such as class invariants) correspond geometrically to phase transitions of the boundary length of this variable-scale fractal at special values of q.

Future work: This paper establishes the mapping scheme from variable-coefficient arithmetic recursion to variable-scale geometry. The next step is numerical computation: plotting the complex limiting sets for q=0.5 and q=0.9, and verifying the multifractal dimension spectrum.