Part Three: Fractal-Coefficient Coupled Evolution

—— Synchronous Mapping from Numerical Parameters to Graphical Forms

Author: Zhang Suhang (Luoyang, Henan)

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Abstract

Based on the analogical framework and axiomatic system established in the first two parts, this paper formally introduces three fractal coefficients—scaling factor s, iteration coefficient \lambda, and offset coefficient \delta—and provides their number-theoretic counterparts: prime selection, exponent allocation, and prime set expansion rules. By jointly adjusting these parameters, we can generate a continuous evolutionary spectrum from “regular composites” to “disordered hybrid composites” starting from the same initial set of primes. Simultaneously, with the synchronous evolution of fractal patterns (Koch curve, Sierpinski carpet, to random-like patterns), we visually demonstrate the transition process from simple → complex → structurally disordered.

Again, it is emphasized that the terms “disordered” and “chaotic” in this paper refer only to combinatorial explosion of composite factor composition and irregularity of numerical distribution, not to chaos in the dynamical systems sense.

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1. Definition of the Three Fractal Coefficients and Their Number-Theoretic Mapping

1.1 Scaling Factor s

· Fractal side: In each iteration, the entire figure from the previous level is scaled down by a factor of s (typically 0 < s < 1).

· Number-theoretic mapping: Corresponds to the size of primes. Small primes (e.g., 2, 3) produce dense distributions of composites; large primes (e.g., 101, 103) produce sparse distributions. More precisely, s maps to the “scale” of the prime set:

  s \quad \longleftrightarrow \quad \frac{1}{\log p}

  \]  

  or directly take the reciprocal 1/p of the prime p as the scaling factor. The larger the prime, the sparser the composites it helps generate, analogous to a smaller scaling factor in fractals producing finer but sparser structures.

1.2 Iteration Coefficient \lambda

· Fractal side: Controls the number of copies or superposition weight in each iteration (e.g., in the Koch curve, each segment is replaced by 4 segments, \lambda = 4; the growth parameter in logistic maps).

· Number-theoretic mapping: Corresponds to exponent allocation. For a fixed prime p, its exponent a controls the “repetition count” of that prime in the composite. Define:

  \lambda \quad \longleftrightarrow \quad \text{total multiplicity of prime factors } r = \sum a_i

  \]  

  or more finely, the growth rate of the exponent vector (a_1, a_2, \dots). When r is fixed, different exponent combinations yield different “iteration paths.”

1.3 Offset Coefficient \delta

· Fractal side: The translation or rotation angle of the copied figure relative to the original position in each iteration.

· Number-theoretic mapping: Corresponds to the rule for expanding the prime set. The offset coefficient determines whether new primes are introduced:

  · \delta = 0: No new primes are introduced; always use the initial prime set \mathcal{P}_0 (regular composites).

  · \delta > 0: With each increase in iteration level (i.e., increase in total multiplicity r), introduce a new prime not in \mathcal{P}_0 with a certain probability or rule. When \delta is sufficiently large, the set of prime factors is no longer confined to any finite set, entering the realm of “disordered hybrid composites.”

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2. Evolutionary Spectrum: Four Stages in Parameter Space

We fix the initial prime set \mathcal{P}_0 = \{2\} (the simplest case) and gradually adjust the parameters to observe the changes in the generated numbers.

Stage 0: Pure Primes (Initial Figure)

· Number theory: N = p (prime), no iteration.

· Fractal: A line segment or a triangle (initial figure).

· Parameters: r = 1,\ \delta = 0.

Stage 1: Powers of a Single Prime (Regular Composites, Simplest Ordered)

· Number theory: N = 2^a \ (a\ge 2), e.g., 4, 8, 16, 32…

· Fractal: First few iterations of the Koch curve (regular self‑similarity).

· Parameters: r = a,\ \mathcal{P} = \{2\},\ \delta = 0.

· Distribution characteristics: Geometric progression, spacing grows exponentially.

Stage 2: Fixed Set of Two Primes (Richer Regular Composites)

· Number theory: N = 2^a 3^b \ (a,b\ge 0,\ a+b\ge 2), i.e., 5‑smooth numbers.

· Fractal: Sierpinski carpet (regular, infinitely refined).

· Parameters: \mathcal{P}_0 = \{2,3\},\ \delta = 0.

· Distribution characteristics: Density on the number line can be generated recursively; adjacent ratios asymptotically approach \log 2 / \log 3, exhibiting a kind of “quasi‑periodicity.”

Stage 3: Gradual Introduction of New Primes (Transition to Disorder)

· Number theory: Set \delta > 0, e.g., each time the total multiplicity r increases, with probability 0.3 take the next smallest prime not yet used and add it to the set.

  · When r=3 we might get 2^2 \times 5; when r=4 we might get 2 \times 3 \times 7, etc.

· Fractal: Random offsets added to fractal parameters; the figure begins to show irregularities but still retains local self‑similarity.

· Property: The prime factor set is no longer fixed, but the number of distinct primes may remain small (e.g., \omega(N) \le 3). These composites are in a “semi‑regular” state.

Stage 4: Fully Disordered Hybrid Composites

· Number theory: Set \delta to its maximum value—allow arbitrary selection from the entire set of primes, and introduce new primes without bound as r grows. For example, N = 2 \times 3 \times 5 \times 7 \times \dots \times p_k (primorials), or randomly generated products of many primes.

· Fractal: Completely random iterative “fractional Brownian motion” patterns, losing any discernible self‑similarity.

· Distribution characteristics: Intervals on the number line have no simple formula; the density of the complement (regular composites) tends to zero; asymptotically, almost all composites are disordered hybrid composites (since the set generated by any fixed finite prime set has density zero).

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3. Coupled Evolution Table (Complete Correspondence)

Stage Number‑Theoretic Parameter Setting Typical Examples Fractal Parameter Setting Fractal Pattern Characteristics

0 (initial primitive) Prime p 2, 3, 5, 7 Initial figure (no iteration) Simple line segment / triangle

1 (single power) \mathcal{P}=\{p\},\ a\ge 2,\ \delta=0 4, 8, 9, 27 Fixed scaling factor s=1/2, iteration coefficient \lambda=4 Koch curve (regular)

2 (fixed multi‑prime set) \mathcal{P}_0=\{2,3\},\ \delta=0 6, 12, 18, 24, 36 Fixed offset \delta=0, multiple scaling Sierpinski carpet

3 (limited expansion) \mathcal{P}_0 expands slowly, bounded 2^2\times5,\ 2\times3\times7 Small‑range random offset Locally regular, globally slightly irregular

4 (fully disordered) No fixed prime set, \omega(N)\to\infty Primorials, random products of many primes Large random offset Noise‑like pattern (no self‑similarity)

Note: The boundary between Stage 3 and Stage 4 is not mathematically absolute, but can be rigorously distinguished by whether there exists a finite prime set containing all prime factors.

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4. Numerical Experiment Example (Simulated Data)

Below are sequences of composites (first 20) generated with fixed parameters, visually demonstrating the evolution from “regular” to “disordered.”

Regular composites (\mathcal{P}_0=\{2,3\}, sorted, first 20):
6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162
→ Adjacent ratios fluctuate between about 1.2 and 1.5, but overall predictable.

Disordered hybrid composites (contain at least one prime factor >3, same magnitude range, mixed):
10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 51
→ Irregular intervals, evident combinatorial explosion (e.g., 30 = 2×3×5).

If we plot both types on the number line, regular composites form a sparse but recursive “skeleton,” while disordered composites fill the gaps as a “chaotic background.”

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5. Visualization Concept for Synchronous Evolution with Fractal Patterns

Although this paper cannot generate actual images, we can describe the mapping textually:

· Stage 1: Koch snowflake – boundary everywhere non‑differentiable but regular. Corresponds to 2^a on a logarithmic scale forming a straight line.
· Stage 2: Sierpinski carpet – holes distributed regularly. Corresponds to the lattice of points (a,b) for 2^a 3^b.
· Stage 3: Sierpinski triangle with random gaps – locally regular, globally offset. Corresponds to sequences that restrict the number of distinct primes but allow a few new ones.
· Stage 4: Random fractional Brownian motion trajectory – no repetition at any scale. Corresponds to sharp jumps in values near primorials.

Readers may imagine: as the parameter \delta gradually increases from 0, the fractal pattern slowly develops “defects” from perfect symmetry, eventually becoming completely disordered; simultaneously, the recurrence relations of the composite sequence gradually break down, and the prime factor entropy \omega(N) tends to infinity.

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6. Conclusion and Summary of the Series

This three‑part series has accomplished the following:

1. Part One established a homologous analogy between the generation of composites from primes and fractal iteration, and clarified the distinction between “regular composites” and “disordered hybrid composites.”
2. Part Two extracted three axioms, provided formal mathematical formulas (factorization, hierarchical recurrence, power construction, disorder measures), and strictly distinguished the non‑dynamical‑system semantics of the term “chaos.”
3. Part Three introduced the mapping of scaling factor s, iteration coefficient \lambda, and offset coefficient \delta, demonstrating a continuous evolutionary spectrum from primes to regular composites to disordered hybrid composites, contrasted with synchronous changes in fractal patterns.

Final conclusion: The constructive evolution of numbers (primes combining into composites) and the growth of fractal patterns (primitive figures iterating into complex shapes) are highly consistent in structural logic. The only difference is that fractals often involve continuous transformations and dynamical systems, whereas number theory is a discrete combinatorial system. This analogy not only provides intuitive pedagogical value but may also inspire new visualization methods for number‑theoretic distributions.

All uses of “disordered,” “chaotic,” etc. in this series have been clearly defined in their combinatorial sense and have not been conflated with dynamical systems chaos. Readers are encouraged to perform further numerical experiments based on this framework, such as plotting histograms of composite distributions under different parameters, or generating corresponding fractal patterns for comparison.

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Acknowledgments
Thanks to the mathematics enthusiasts in Luoyang, Henan, for their inspiring discussions.

References
[1] Zhang, S. (2026). Number Theory–Fractal Analogy Series: Parts One and Two.
[2] Mandelbrot, B. B. (1982). The Fractal Geometry of Nature.
[3] Literature on the distribution of smooth numbers (de Bruijn, 1951).

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(End of the series)