FCFG Demonstration Example: The Periodic Continued Fraction [1,2] and Its Alternating Fractal Skeleton

Author: Zhang Suhang (Luoyang, Henan)

Abstract:

Within the framework of Fractal Continued-Fraction Geometry (FCFG), this paper presents a reproducible demonstrative example. Taking the periodic coefficient sequence {a_k} = [1,2], the corresponding finite-order recursive fractal skeletons are constructed. According to the isomorphism relation of FCFG, the global equivalent similarity ratio of each level of skeleton is given by the finite continued fraction of the same order, and the limiting similarity ratio of the infinite iteration is the limit of the periodic continued fraction, \alpha = \sqrt{3}-1. The paper gives in detail the successive convergent fractions, the geometric recursive construction rules, the point-set visualization scheme, and the numerical computation approach for the box-counting dimension. Compared with the traditional fractal geometry perspective: traditional methods can generate this point set and estimate its dimension, but they do not naturally establish a one-to-one correspondence between iteration levels and continued-fraction convergents; FCFG uses a unified recursive generating sequence to bind the convergent oscillatory behavior on the arithmetic side with the multiscale growth rhythm on the geometric side, embodying the homologous origin of number-theoretic recursion and geometric recursion. This example is intended to demonstrate the computational implementation of the FCFG paradigm, and does not claim to discover a completely new fractal point set.

Keywords: Fractal Continued-Fraction Geometry; FCFG; periodic continued fraction; alternating self-similar set; finite fractal skeleton; box counting

1. Introduction

The core claim of FCFG is that the global equivalent similarity ratio of a finite-order self-similar fractal skeleton is equal to the value of the finite simple continued fraction of the same order; the continued fraction is arithmetic recursion, the fractal skeleton is geometric recursion, and the two are structurally homologous.

The persuasiveness of the framework requires concrete, verifiable examples for support. This paper selects the simplest two-period positive integer sequence a_k \in {1,2}, and walks completely through the entire process from continued-fraction solution, geometric skeleton construction, and visualization to numerical estimation of the fractal dimension.

Conventions in this paper: F_n represents the finite truncated skeleton obtained after n recursions, rather than an infinite complete fractal; r_n is the n-th-order finite continued fraction, representing the global equivalent similarity ratio of that skeleton; n \to \infty is only a limiting extension, and the axioms themselves are established on finite-order objects.

2. Continued-Fraction Side: Convergents and Limit of the Periodic Sequence [1,2]

Consider the periodic continued fraction:

\alpha = \cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cdots}}}}

It satisfies the self-consistency equation:

\alpha = \frac{1}{1+\dfrac{1}{2+\alpha}}

Rearranging gives:

\alpha = \frac{\alpha+2}{\alpha+3}, \quad \alpha^2+3\alpha = \alpha+2, \quad \alpha^2+2\alpha-2=0

Taking the positive root:

\alpha = \sqrt{3}-1 \approx 0.73205

The successive finite continued fractions (convergents) are calculated as follows:

· n=1, a_1=1: r_1 = \frac{1}{1} = 1.0000 · n=2, a_1=1, a_2=2: r_2 = \frac{1}{1+\frac{1}{2}} = \frac{2}{3} \approx 0.6667 · n=3, a_1=1, a_2=2, a_3=1: r_3 = \frac{3}{4} = 0.7500 · n=4, a_1=1, a_2=2, a_3=1, a_4=2: r_4 = \frac{8}{11} \approx 0.7273 · n=5: r_5 = \frac{11}{15} \approx 0.7333 · n=6: r_6 = \frac{30}{41} \approx 0.7317

The sequence oscillates and converges above and below the limit \sqrt{3}-1, which is a typical dynamical feature of periodic continued fractions. Under FCFG, each r_n is directly assigned as the global equivalent similarity ratio of the corresponding skeleton F_n.

3. Geometric Side: Construction of the Alternating Recursive Fractal Skeleton

Seed F_0: the unit line segment on the interval [0,1]. Recursive rules (one-dimensional serial-type skeleton):

4. When a_k=1: take 1 copy of the previous-level skeleton F_{k-1}, with scaling factor 1, without changing the geometric form;
5. When a_k=2: take 2 copies of the previous-level skeleton F_{k-1}, each scaled to 1/2 of the original, and concatenate them end to end within the interval [0,1].

The generating sequence cycles according to the period [1,2]:

· F_1: a_1=1, form same as F_0; · F_2: a_2=2, two segments of F_1 scaled by 1/2 and concatenated; · F_3: a_3=1, form same as F_2; · F_4: a_4=2, two segments of F_3 scaled by 1/2 and concatenated; · and so on.

Geometric features: the figure is always contained within [0,1]; it densifies whenever a=2, and maintains its form whenever a=1. This belongs to an alternating self-similar set with intermittent densification, unlike classical fractals such as the Koch curve, which are uniformly split at every step. The naked eye can observe the point set gradually densifying, but cannot directly read off the global equivalent similarity ratio and the convergence limit.

6. Box-Counting Dimension Estimation Scheme

This set is an alternating self-similar set, and no simple closed-form Hausdorff dimension exists, so the box-counting method is used for numerical estimation:

D = \lim_{\epsilon\to0}\frac{\log N(\epsilon)}{-\log\epsilon}

where N(\epsilon) is the number of boxes of side length \epsilon required to cover the entire point set.

Numerical experiment steps:

7. Recursively generate higher-order skeletons F_n (e.g., n=8,10,12);
8. Count the number of boxes N(\epsilon) at multiple scales \epsilon;
9. Plot the log-log graph of \log N(\epsilon) versus \log(1/\epsilon); the slope of the linear fit is the dimension estimate.

Note: the traditional self-similar dimension formula \log N/\log s is not applicable here, because the scaling ratio alternates and is not a fixed single-step scaling.

10. Comparison: Traditional Perspective versus FCFG Perspective

· Traditional fractal geometry: one can write the recursive rules, generate the point set, plot it, and estimate the dimension by box counting. However, the iteration steps themselves do not naturally correspond to a set of continued fractions, nor do they express the scaling characteristics of each generation as convergents, nor do they connect the densification rhythm of the point set with the oscillatory convergence of the continued fraction. The point set is merely an independent geometric object. · FCFG perspective: the same generating sequence {a_k} simultaneously drives two chains—the arithmetic recursion of the continued fraction and the geometric recursion of the fractal skeleton. r_n is not only a number, but also the quantitative marker of that level of skeleton; the convergence properties of the continued fraction directly explain the speed and oscillatory rhythm of fractal growth. FCFG does not overturn traditional fractal tools, but adds a precise number-theoretic index to this class of geometric objects.

11. Discussion and Summary

The geometric point set in this example can itself be constructed under the traditional framework; its novelty lies not in discovering a new fractal, but in a complete linked mapping: coefficient sequence \rightarrow successive finite continued fractions (global equivalent similarity ratios) \rightarrow successive fractal skeletons \rightarrow limit irrational number corresponding to the limiting similarity ratio of the infinite fractal.

It is a verifiable sample of the FCFG paradigm, and can also echo the recursive structure of the FCFG-N recursive resistor network, embodying the fact that number theory, fractals, and network topology share the same class of recursive homologous structure.

Future work:

12. Complete numerical box counting for higher-order skeletons and give precise dimension estimates;
13. Try other simple periodic sequences (such as [1,3], [2,2]) and compare different convergence behaviors and fractal forms;
14. Extend this one-dimensional serial skeleton to recursive tiling in the two-dimensional plane.

Appendix: Python plotting code (generating the periodic [1,2] skeleton)

import numpy as npimport matplotlib.pyplot as pltdef fractal_iter(points, a): if a == 1: return points elif a == 2: seg = points / 2 return np.concatenate([seg, seg + 0.5])seq = [1, 2, 1, 2, 1, 2]pts = np.array([0, 1])for ak in seq: pts = fractal_iter(pts, ak)plt.figure(figsize=(8, 2))plt.plot(pts, np.zeros_like(pts), lw=0.7)plt.ylim(-0.1, 0.1)plt.title("FCFG skeleton, periodic sequence [1,2]")plt.axis("off")plt.show()