Prime Fractions and Primordial Nested Fractals

Author: Suhang Zhang

Luoyang School of Mathematics

Abstract

 In traditional number theory, prime numbers are indecomposable integer primitives. Their reciprocals are merely treated as ordinary rational numbers, mostly used in series and convergence calculations, without independent geometric ontological interpretations. Based on the bidirectional evolutionary framework of Fractal Continued-Fraction Geometry (FCFG), this paper proposes that primes act as primordial growth units driving the outward extension of topology, while their reciprocals, defined here as prime fractions, serve as primordial collapse units driving inward structural nesting. A prime-fraction fractal, generated by iteration with a single prime fraction, constitutes an indecomposable primordial fractal within the inward-converging spectrum. All composite-scale fractals can be regarded as superpositions and couplings of prime-fraction fractals. Prime fractions establish a direct pathway connecting prime number theory and fractal topology, furnishing a fundamental specimen for the geometrization of number theory.

Keywords: FCFG; prime numbers; prime fractions; primordial nested fractals; bidirectional evolution; continued fractions

1. Introduction

The core evolutionary law of FCFG is the bidirectional arrow: integers grow outward, and fractions collapse inward. Integers form the topological skeleton of space, while fractions fill its fine, subtle details.

Among integers, primes cannot be factored into products of smaller integers. They are the most fundamental, indecomposable growth atoms in the integer spectrum. Paired with them, the reciprocals of primes — namely prime fractions — occupy an equally special status within the spectrum of fractions.

Conventional fractal research often selects scaling factors arbitrarily, rarely posing the question: what scaling base represents the most primordial nesting unit? FCFG supplies the answer: prime fractions.

2. Bidirectional Pairing of Primes and Prime Fractions

1. Prime p: Belongs to the outward evolutionary branch of integers. As an indecomposable primitive, it represents a pure, composite-free scaling of expansion, the simplest generator for constructing topological skeletons. Skeletons of composite integers arise from the coupling of multiple prime skeletons.

2. Prime fraction \dfrac{1}{p}: Belongs to the inward evolutionary branch of fractions. Its continued-fraction expansion is finite. Under iteration, it performs sustained inward nesting at a fixed, uniform scale. It cannot be decomposed into independent collapse actions governed by two smaller fractions, making it a primordial atom of nesting motion.

Composite numbers and their reciprocals are secondary combinations. For instance, 6=2\times3, and the nesting effect of \dfrac16 is equivalent to the superposition of two sets of nesting rules: \dfrac12 and \dfrac13. It is not an independent primitive, but a composite product.

Only a prime and its reciprocal form a clean, indecomposable pair of bidirectional evolutionary primitives.

3. Prime-Fraction Fractals: Primordial Forms within the Inward Spectrum

Repeated iteration using a single prime fraction as the scaling rule yields a prime-fraction fractal.

The classic Cantor ternary set is essentially a prime-fraction fractal generated with the prime fraction \dfrac13.

For comparison:

- Iteration with \dfrac12 produces a primordial nested structure scaled by one half;

- Iteration with \dfrac15 yields another distinct primordial nested structure scaled by one fifth.

Every prime corresponds to a unique primordial prime-fraction fractal. Its convergence rate, pore arrangement, measure properties and self-similar texture are uniquely determined by the prime itself.

Such fractals follow unidirectional truncated evolution: only inward nesting driven by fractions, without synchronized outward integer growth or rotation. They are linear fractals under FCFG. Linear fractals are not an independent category; they are local truncations of the spiral parent manifold with rotation removed and pure inward collapse retained. Prime-fraction fractals are the most primordial subclass among these truncations.

4. Composite Fractals: Superposition and Resonance of Prime-Fraction Fractals

When multiple distinct prime fractions are introduced simultaneously, interleaved multilayer nesting rules generate composite-scale fractals.

The texture of a composite fractal is essentially the modulation and superposition of multiple primordial prime-fraction fractals. Analogous to colour mixing, prime-fraction fractals are the primary hues, and complex patterns emerge from their blending.

This leads to a corollary: all linear self-similar fractals with finite nesting can be decomposed into combinations of prime-fraction fractals.

5. Conclusion

Within the FCFG picture, primes are no longer isolated special objects in number theory.

Primes stretch outward to build skeletons; prime fractions weave fine fractal textures inward. Every prime carries its own unique language of nested fractals.

Prime-fraction fractals are the primordial cornerstone of the inward-collapse evolutionary branch.

They reveal the hidden geometric form of prime numbers and build a bridge from prime number theory to fractal topology. This is the topological significance of prime fractions.