Integers and Fractions from the Perspective of FCFG

 Author: Suhang Zhang

Luoyang School of Mathematics

Abstract

In traditional number theory, integers and fractions are merely different numerical forms within the set of rational numbers. Within the framework of Fractal Continued-Fraction Geometry (FCFG), however, their distinction goes far beyond arithmetic magnitude, revealing two opposite directions of topological evolution. This paper proposes that fractions correspond to inward collapse of fractals from large to small scales, while integers correspond to the outward growth of topology from small to large scales. This "law of bidirectional evolution" redefines the geometric nature of rational numbers and unveils the dynamical generation mechanism of discrete skeletons and continuous embedding.

1. Introduction: Breaking the Illusion of Static Quantities

In arithmetic, 3 and 1/3 differ only in value. Geometrically, by contrast, recursively subdividing 3 and iteratively replicating 3 lead to two entirely distinct universes. Continued fractions serve as the key: arithmetic recursion and recurrence map to two inverse physical processes in geometry.

2. Fractions: Inward Collapse from Large to Small

1. Arithmetic structure: The layered nesting of a fraction in continued fraction form 1/(a+1/(b+\dots)) constitutes an infinite recursive process in which denominators grow and the value converges toward a limit.

2. Geometric mapping: This matches the generation of classic fractals such as the Cantor set. Large-scale shapes are scaled down and nested infinitely within smaller regions — a process of scale contraction and inward convergence.

3. Topological characterization: Fractions yield convergent attractors, static and centripetal. Truncated at finite orders, they produce conventional linear fractals.

3. Integers: Outward Growth from Small to Large

1. Arithmetic structure: The recurrence sequence of integers (1,2,3\dots), or the partial quotient sequence (a_1,a_2,a_3\dots) in continued fractions, accumulates and extends step by step.

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2. Geometric mapping: This corresponds to the growth of an iterated function system (IFS) or tree branching. Starting from an initial point (unity), the structure branches and expands outward to form an extensive network or skeleton. This is a process of scale expansion and outward extension.

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3. Topological characterization: Integers act as divergent generators, dynamic and centrifugal, forming the primary skeleton of topological space.

4. Bidirectional Synthesis: Emergence of Linear Fractals and Spiral Fractals

1. Linear fractals as truncated unidirectional structures: A geometric system with only finite inward fraction-style nesting, or only linear integer-style extension, yields flat, bounded linear fractals.

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2. Spiral fractals as bidirectional resonance: When integer-driven skeletal growth and fraction-driven fine-scale nesting operate simultaneously with rotation, infinite spiral fractals emerge. Outward expansion arises from integers, and inward collapse from fractions.

5. Conclusion: The First Law of Geometric Evolution in FCFG

Integers and fractions are not merely numerical classifications. They form the two-way arrow of fractal geometric evolution.

Integers are generators of spatial skeletons (small to large);

Fractions are embedding operators for fine spatial details (large to small).

This law supplies the fundamental dynamical engine for all discrete fractals and continuous topological manifolds.