Discrete Order Geometry (DOG): A Geometric Paradigm Based on Fractal Nesting and Continued Fraction Scaling

Author: Zhang Suhang

Luoyang

Abstract

Traditional geometric systems (Euclidean geometry, Riemannian geometry) and classical fractal geometry have, in their fundamental assumptions, included spatial connectivity, entity adjacency, or medium continuity as prerequisites for the validity of a geometric system. For self‑similar structures that are discrete, non‑adjacent, and lack material connections, existing geometric systems typically classify them as non‑standard geometric objects, leaving a corresponding theoretical gap. Based on the self‑similarity of classical fractal geometry and the irrational‑scale convergence properties of continued fractions, this paper does not require the connectivity axiom and establishes an independent geometric paradigm — Discrete Order Geometry (DOG).

The paper presents the core definition, basic axioms, and three fundamental theorems of DOG. It shows that connected geometries can be regarded as special cases of DOG in continuous space, while discrete ordered structures are more common in the universe. The Sun‑Earth‑Moon three‑body system serves as a natural empirical specimen to verify the feasibility of DOG. This work provides a description of many‑body systems independent of differential dynamics and field theory, offering a new geometric framework for discrete nested structures, many‑body periodic evolution, and the precise quantification of irrational scales.

Keywords: Discrete Order Geometry; DOG; fractal nesting; continued fraction; self‑similarity; three‑body system; geometric paradigm

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1 Research Background and Limitations of Existing Theories

1.1 Basic Assumptions of Traditional Geometry

Euclidean geometry established a flat, connected space; Riemannian geometry extended it to curved, connected spaces. Together they form the main mathematical foundation of modern geometry and physics, but share a common premise: spatial connectivity, point adjacency, and regional continuity. Topology, analytic geometry, and differential geometry likewise assume that disconnected, spatially separated, physically unconnected structures can hardly be incorporated into a unified geometric system. This assumption fits artificial regular figures and continuous natural landforms, but is not fully consistent with many real cosmic structures.

1.2 Scope of Classical Fractal Geometry

Classical fractal geometry, founded by Mandelbrot, takes self‑similarity and scale‑free nesting as its core and breaks through the limitations of regular geometry. However, existing fractal research has mostly focused on continuously connected forms (such as coastlines, snowflakes, leaf veins, and nebular condensation structures). Discrete, materially disconnected structures that nonetheless exhibit hierarchical self‑similarity have not yet been systematically incorporated into standard fractal geometry.

1.3 The Single Approach in Celestial Many‑Body Research

Modern research on the three‑body and many‑body systems relies primarily on Newtonian mechanics, differential equations, and numerical iteration, requiring force analysis, field coupling, and instantaneous dynamical calculations. This approach has inherent difficulties such as chaos, long‑term integration error accumulation, and distortion in approximating irrational scales. In this context, a purely geometric and number‑theoretic description independent of force constraints would have complementary value.

Based on the above analysis, this paper attempts to construct a new geometric paradigm — Discrete Order Geometry (DOG) — using fractal nesting structures and the scale convergence properties of continued fractions.

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2 Preliminaries: Two Types of Tools for DOG

2.1 Fractal Geometry: Structural Foundation

The two core properties of fractal geometry are hierarchical nesting and scale‑free self‑similarity. DOG retains these properties but no longer requires physical connectivity or spatial adjacency between entities, focusing instead on cross‑scale self‑similarity in structural form, arrangement hierarchy, and evolutionary rhythm.

2.2 Continued Fractions: Scale Foundation

Continued fractions provide the best convergent approximation sequences for irrational numbers, with the properties of hierarchical truncation and stepwise convergence. Parameters such as orbital ratios, period ratios, and eccentricities in celestial bodies are often irrational numbers, and continued fractions can provide a stratified quantitative description for such parameters. DOG uses continued fractions to achieve numerical quantification of discrete geometric structures, allowing geometric form and mathematical scale to correspond to each other.

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3 Definition of Discrete Order Geometry (DOG)

3.1 Name and Abbreviation

· Full name: Discrete Order Geometry

· Abbreviation: DOG

3.2 Definition

Discrete Order Geometry (DOG) is a geometric system that takes hierarchical nested self‑similarity as its structural criterion and continued fraction hierarchical convergence as its scale criterion. It does not require spatial connectivity, entity adjacency, or medium coupling; the unity of a system is determined solely by order isomorphism, form homology, and scale recursion.

The definition includes the following points:

1. The validity of a geometric system does not require physical connection, spatial connectivity, or medium coupling.

2. The criteria for classifying a geometric structure as homogeneous are: consistent hierarchical nesting, consistent self‑similar order, and consistent scale recursion.

3. Traditional connected geometries (Euclidean, Riemannian) can be regarded as special cases of DOG under the constraint of continuous space.

4. The widely existing discrete, spatially separated, nested ordered structures in the universe are the primary objects of DOG.

3.3 Applicable Scope

DOG is applicable to: celestial nested systems, many‑body discrete arrangements, natural hierarchical periodic systems, and non‑contact self‑similar order systems. It can describe discrete ordered structures and spatially separated many‑body orders that traditional geometry finds difficult to handle.

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4 Basic Axioms of DOG

Axiom 1: Order Isomorphism Axiom

For a set of discrete, physically unconnected independent units, if they exhibit cross‑scale hierarchical self‑similarity in morphology and evolutionary rhythm, they can constitute a unified DOG geometric system.

Axiom 2: Scale Hierarchy Axiom

All irrational scale parameters (e.g., ratios, periods, eccentricities, angular velocity ratios) within a DOG system can be hierarchically converged and quantitatively described via stepwise truncation of continued fractions, without relying on finite decimal approximations.

Axiom 3: Connectivity as a Special Case Axiom
All traditional geometric structures that satisfy connectivity are special solutions of DOG in continuous space. The DOG system is compatible with the conclusions of traditional geometry.

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5 Three Fundamental Theorems of DOG

Theorem 1: Discrete Self‑Similar Nesting Theorem
For a discrete arrangement system with a three‑level structure of "central primary unit – secondary orbiting unit – tertiary nested satellite unit", if it satisfies scale‑free self‑similarity, it belongs to the DOG geometric configuration. This conclusion holds regardless of spatial distance, presence or absence of a medium, or differences in orbital velocities.
Corollary: The Sun‑Earth‑Moon system, the Jupiter‑Galilean satellite system, the Saturnian satellite systems, and galactic cluster hierarchies can all be regarded as instances of DOG geometry.

Theorem 2: Continued Fraction Scale Matching Theorem
The operational ratios, periodic rhythms, and structural scales of a DOG discrete nested system are predominantly irrational numbers. The hierarchical convergence orders of the continued fraction correspond to the nesting levels of DOG and can be used to describe the long‑term evolution of many‑body systems.
Significance: This method does not rely on differential equations and can avoid error accumulation in long‑term numerical simulations.

Theorem 3: Geometric Paradigm Inclusion Theorem
Connected geometry is a finite special case of DOG in continuous space; DOG is a geometric paradigm with broader applicability. The connectivity constraint in traditional geometry is an artificially imposed condition, not a necessary attribute of cosmic geometry.

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6 Natural Empirical Specimen: The Sun‑Earth‑Moon Three‑Body System

This paper takes the Sun‑Earth‑Moon system as a natural empirical specimen for DOG, for the following reasons.

6.1 Structural Conformity to the Discrete Self‑Similar Nesting Theorem

· Level‑1 parent unit: Sun
· Level‑2 orbiting unit: Earth (orbiting the Sun)
· Level‑3 sub‑nested unit: Moon (orbiting the Earth)
The three are spatially discrete without physical connection, yet the nesting pattern and orbital arrangement exhibit self‑similarity, satisfying the Order Isomorphism Axiom.

6.2 Scale Conformity to the Continued Fraction Scale Matching Theorem
Quantities such as the Sun‑Earth distance, Earth‑Moon distance, orbital period ratio, synodic period, and orbital eccentricity are all irrational numbers. Using stepwise truncation of continued fractions, one can describe phenomena such as eclipse cycles, tidal cycles, precession, and long‑term orbital perturbations, without solving differential equations.

6.3 Empirical Value

· Authenticity: Based on long‑term astronomical observational data, it can be checked and verified.
· Uniqueness: A stable three‑level discrete nested system with clear hierarchy in near‑Earth space, providing a repeatably testable specimen for DOG.
· Generalizability: The method can be extended to other planetary‑satellite systems and to broader cosmic hierarchical structures.

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7 Positioning of DOG within the Geometric Framework

The introduction of DOG leads to the following landscape of geometric systems:

· Euclidean geometry: flat, connected, applicable to artificial and terrestrial continuous structures.
· Riemannian geometry: curved, connected, applicable to continuous spacetime structures.
· Discrete Order Geometry (DOG): discrete, ordered, applicable to cosmic nested structures.

Each corresponds to a different domain of application, together forming a more complete geometric picture.

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8 Conclusion

Based on the self‑similar nesting property of fractal geometry and the irrational‑scale convergence property of continued fractions, and without requiring spatial connectivity, this paper has established the basic framework of Discrete Order Geometry (DOG). It has presented the definition, axioms, and theorems of DOG, and has used the Sun‑Earth‑Moon three‑body system as a natural empirical specimen to verify the feasibility of this geometric system. DOG provides an independent geometric and number‑theoretic path for describing discrete nested structures, many‑body periodic evolution, and irrational scales. This work helps to fill the gap left by traditional geometry regarding discrete ordered structures, constituting a complementary dimension of geometric systems.

References (examples)

[1] Mandelbrot B. The Fractal Geometry of Nature. 1982.
[2] Riemann B. On the Hypotheses Which Lie at the Foundations of Geometry. 1854.
[3] Khintchine A Y. Continued Fractions. Dover, 1964.
[4] Celestial mechanics literature: Long‑term orbital evolution data of the Sun‑Earth‑Moon system.
[5] Foundations of topology: Spatial connectivity and neighborhood axiom systems.