A Unified Continued-Fraction Theory of Rectilinear and Spiral Fractals

Author: Zhang Suhang (Luoyang, Henan)

Abstract

Traditional fractal geometry has long treated two typical self-similar structures separately: piecewise linear fractals have the topological feature of one-directional inward subdivision within a bounded domain, whereas logarithmic equiangular spirals exhibit a smooth, inflection-free, bidirectionally infinite self-similar form. The two systems lack a unified algebraic characterization tool. Within the framework of Fractal Continued-Fraction Geometry (FCFG), and based on the dual transformation rule between finite continued fractions with a leading zero and their reciprocals, this paper establishes a recursive isomorphism unification theory for the two classes of fractal structures. The study shows that the natural bidirectional topology of the logarithmic spiral can fully map the scaling duality between the original value of a continued fraction and its reciprocal: its inward-spiraling convergent structure corresponds to the leading-zero continued fraction, and its outward-spiraling expansive structure corresponds to the reciprocal of the continued fraction; piecewise linear fractals can be regarded as the limiting special case of spiral fractals when the rotation angle degenerates to zero. The two classes of fractals share a unified self-similar dimension law and recursive iteration mechanism. This paper provides a proof of underlying homology between discrete polyline fractals and smooth-curve fractals, offering a new theoretical framework for variable-scale fractal modeling and the geometrization of Diophantine approximation.

Keywords: fractal geometry; continued fractions; logarithmic spiral; bidirectionally infinite topology; recursive isomorphism; dual similarity ratio; FCFG

1. Introduction

The modern research system of fractal geometry mainly contains two classes of self-similar geometric objects. The first is piecewise linear fractals, represented by the Cantor set, the Koch curve, and the Sierpinski gasket, which are generated by finite boundary skeletons iteratively subdivided inward layer by layer, possess only a one-directional convergent structure, and whose expansive dual structures must be constructed artificially. The second is the logarithmic equiangular spiral, which possesses global strict self-similarity, is smooth and continuous with no angular topology, and can simultaneously achieve inward infinite spiral convergence and outward unbounded extension and expansion; it is the standard mathematical model of vortex-like self-similar structures in nature.

For a long time, linear piecewise fractals and smooth spiral fractals have been studied independently under different iteration paradigms, and the homology of their underlying generation mechanisms has not been established. Existing studies have found only a superficial numerical connection between the spiral scaling coefficient and continued fractions in the special case of the golden ratio, without forming a universal algebraic correspondence system, and are unable to explain the topological differences and structural homology of the two classes of fractals.

To address the above problems, this paper, relying on the FCFG fractal recursion axioms, uses finite-order leading-zero continued fractions as a unified algebraic medium, constructs a contraction–expansion dual mapping relation, completes the full-domain unification of piecewise linear fractals and logarithmic spiral fractals, and clarifies the limiting degeneration relation and the dimension invariance law of the two classes of fractals.

2. Basic Definitions and Recursive Duality Axioms

2.1 Leading-Zero Continued Fractions and Their Algebraic Dual Structure

Given any positive integer sequence {a_1,a_2,\dots,a_n}, define the n-th-order finite leading-zero continued fraction:

r_n = [0;a_1,a_2,\dots,a_n] = \frac{1}{a_1+\dfrac{1}{a_2+\dots+\dfrac{1}{a_n}}},\quad 0<r_n<1

Taking the reciprocal of the leading-zero continued fraction eliminates the leading zero term and generates an expansive continued fraction of the same order:

\frac{1}{r_n} = [a_1;a_2,\dots,a_n] > 1

The reciprocal transformation keeps the iteration level n unchanged, forming a strict algebraic dual pairing and providing a number-theoretic basis for the bidirectional scaling system of fractals.

2.2 Comparison of Topological Features of the Two Classes of Fractals

1. Piecewise linear fractals: composed of concatenated straight-line units, with definite boundaries and inflection points, an overall bounded topology, and only an inward one-directional subdivision convergent structure; there is no natural outward-expanding self-similar system, and their expansive dual figures must be constructed separately according to the scaling ratio.

2. Logarithmic spiral fractals: with polar equation \rho=Ce^{k\theta}, the curve is smooth without abrupt inflection points and possesses a bidirectionally infinite topology. As the angle tends to negative infinity, the curve spirals infinitely toward the pole and converges; as the angle tends to positive infinity, the radial scale extends and expands infinitely. A single curve naturally contains two independent self-similar structures.

3. Isomorphic Unification Mechanism of the Two Classes of Fractals

3.1 Continued-Fraction Mapping of the Spiral Bidirectional Structure

Based on the FCFG recursive isomorphism rule, a strict correspondence between spiral topology and the continued-fraction system is established:

1. Inward contraction fractal system: the vortex-line iteration structure of the inner spiral layer approaching the pole, whose global equivalent contraction similarity ratio satisfies

S_n = r_n = [0;a_1,a_2,\dots,a_n]

2. Outward expansion fractal system: the vortex-line iteration structure of the outer spiral layer extending infinitely, whose global equivalent magnification factor satisfies

\tilde S_n = \frac{1}{r_n} = [a_1;a_2,\dots,a_n]

The logarithmic spiral requires no artificial construction of dual figures; it naturally carries the bidirectional self-similar transformation corresponding to the original value and reciprocal of a continued fraction, and is the most fitting geometric embodiment of algebraic duality.

3.2 Invariance of Fractal Dimension

The classical self-similar dimension formula is:

D = \frac{\ln N}{-\ln S}

For the expansive dual structure, the scaling ratio satisfies \tilde S=1/S; substituting gives:

D = \frac{\ln N}{-\ln S} = \frac{\ln N}{\ln \tilde S}

It can thus be proved that the self-similar dimensions of the contraction fractal and the expansion fractal at the same level are completely equal, and the bidirectional dual structure possesses dimensional conservation, verifying the underlying homology of the two scaling systems.

3.3 Limit Degeneration Theorem

When the iteration order n\to\infty, if the continued-fraction sequence converges to an irrational constant \alpha, then the limiting contraction ratio of the spiral is \alpha and the limiting expansion ratio is 1/\alpha.

Under the unified FCFG framework, the core degeneration conclusion can be drawn:

A piecewise linear fractal is the boundary special case of a logarithmic spiral fractal when the rotation angle tends to zero and the rotational degree of freedom completely degenerates.

The bidirectionally infinite spiral topology is the universal form; the linear bounded one-directional topology is the limiting degenerate form.

4. Example Verification: Bidirectional Dual Structure of the Golden Spiral

Select the periodic continued-fraction irrational number as a typical example:

\alpha = [0;\overline{1}] = \frac{\sqrt{5}-1}{2},\quad \frac{1}{\alpha} = \frac{\sqrt{5}+1}{2}

1. Inner-spiral branch of the golden spiral: the iterative contraction ratio is strictly equal to \alpha, matching the periodic leading-zero continued-fraction structure;

2. Outer-spiral branch of the golden spiral: the iterative expansion ratio is strictly equal to 1/\alpha, matching the reciprocal structure of the continued fraction.

This example intuitively verifies the one-to-one correspondence between the bidirectional self-similar structure of the spiral and the dual system of continued fractions, confirming the validity of the unified theory in this paper.

5. Theoretical Extension and Application Expansion

5.1 Unified Adaptation of Variable-Scale Recursive Systems

Fixed-integer continued fractions correspond to uniformly scaled standard linear fractals and standard logarithmic spirals. The Rogers–Ramanujan q-variable-coefficient continued fractions possess hierarchical dynamic scaling features and, under this framework, can correspondingly generate variable-scale polyline fractals and variable-scale bidirectional spiral fractals, capable of precisely characterizing non-uniform self-similar vortex forms in nature such as shell patterns and galactic spiral arms.

5.2 Bidirectional Geometric Interpretation of Diophantine Approximation

Traditional number theory relies only on linear fractals to achieve a geometric explanation of Diophantine approximation. The framework of this paper constructs a bidirectional geometric correspondence: the approximation accuracy of a continued fraction to an irrational number is equivalent to the degree of refinement of the inner spiral's pole-spiraling; the approximation error characteristics of the continued fraction's reciprocal correspond to the topological refinement law of the outer spiral's infinite extension, providing a two-dimensional geometric visualization tool for number-theoretic approximation problems.

6. Conclusion

Based on the algebraic duality rule of continued fractions, this paper completes the systematic unification of piecewise linear fractals and smooth logarithmic spiral fractals under the FCFG framework, yielding the core conclusions:

1. The bidirectionally infinite topology of the logarithmic spiral perfectly embodies the contraction–expansion dual algebraic relation between the original value and reciprocal of a continued fraction, and is the standard geometric model of self-similar dual structures;

2. Piecewise linear fractals and spiral fractals possess completely homologous recursive generation mechanisms and dimension evolution laws; linear fractals are the zero-rotation degenerate special case of spiral fractals;

3. This framework unifies the mathematical paradigms of discrete polyline fractals and smooth-curve fractals, is compatible with fixed-scale and variable-scale recursive systems, and bridges the theoretical barrier between continued-fraction number theory and the two classes of fractal geometry.

This study improves the underlying generation theory of self-similar structures and provides new foundational theoretical support for natural fractal modeling and the optimization of numerical approximation algorithms.

References

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