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Discrete Order Geometry (DOG): A Study of Its Universal Application System

Author: Zhang Suhang
Luoyang

Abstract

After establishing the definition of Discrete Order Geometry and demonstrating its coupling with the discrete mathematics system, this paper further systematically reviews the applicable scope, application scenarios, and practical value of DOG. Unlike traditional connected geometries, which focus on continuous forms, closed spaces, and adjacent structures, DOG takes non‑connected self‑similar hierarchical structures, discrete scale convergence, and order‑homology determination as its core features, effectively covering numerous natural systems, astronomical systems, mathematical models, and engineering order systems that traditional geometries find difficult to address.

This paper presents empirical application analyses from five dimensions: astronomical celestial bodies, natural rhythms, fundamental mathematical problems, structural modeling, and periodic prediction. Through typical examples, it demonstrates the practical utility of DOG, clarifies its irreplaceable application advantages, and completes a closed loop from theoretical construction to practical implementation, confirming that Discrete Order Geometry possesses universal academic value and practical significance.

Keywords: Discrete Order Geometry; DOG; hierarchical nesting; non‑connected structure; many‑body system; periodic prediction; order modeling

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1 Introduction

The application scenarios of traditional Euclidean geometry and Riemannian geometry are generally limited to objects with spatial connectivity, physical contiguity, and continuous forms. When faced with systems composed of mutually independent, spatially separated units that form a unified whole solely through internal order, these geometries encounter difficulties such as modeling challenges, fragmented regularities, and distorted scale fitting.

Discrete Order Geometry breaks free from the connectivity constraint. Taking hierarchical self‑similarity as its structural basis and continued fraction discrete approximation as its quantitative tool, DOG transcends the limitations of spatial distance and physical connections, extending the scope of geometric research from continuously tangible forms to discretely ordered systems.

This paper categorizes the core application areas of DOG, verifies them with real natural samples and academic problems, and clearly defines its applicable scope and application advantages.

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2 Core Application Area I: Modeling Astronomical Many‑Body Systems

2.1 Nested Celestial Systems in the Solar System

This is the most central and mature application scenario of DOG. Taking the Sun‑Earth‑Moon three‑level nested system as the standard prototype, DOG can be directly extended to the Jupiter‑satellite system, Saturn‑satellite system, and various planetary‑satellite combinations.

· Structural level: Without relying on gravitational differential equations to construct complex dynamical models, DOG directly divides primary and secondary layers according to the DOG hierarchical order theorem, allowing rapid determination of the overall system configuration.
· Periodic level: Using the hierarchical convergence property of continued fractions, discrete truncations are applied to orbital periods, synodic periods, and apsidal precession periods, enabling medium‑ and long‑term celestial prediction.
· Scale level: Irrational physical quantities such as interplanetary distances, orbital radius ratios, and eccentricities are approximated with high precision by rational numbers, avoiding long‑term errors inherent in decimal approximations.

2.2 Concrete Example: Continued Fraction Convergence of the Orbital Period Ratios of Jupiter’s Galilean Satellites and Its Correspondence with DOG Nesting Levels

Taking Jupiter’s three Galilean satellites – Io, Europa, and Ganymede – as an example, their orbital periods exhibit approximate integer ratios well known in astronomy. Here we take the precise period ratios and use continued fractions to demonstrate the correspondence between DOG scale convergence and nesting levels.

· Io orbital period: ≈ 1.769 days
· Europa orbital period: ≈ 3.551 days
· Ganymede orbital period: ≈ 7.155 days

Step 1: Compute the period ratio of Io to Europa

\frac{T_{\text{Europa}}}{T_{\text{Io}}} = \frac{3.551}{1.769} \approx 2.00735

Expand into a continued fraction:

2.00735 = 2 + \cfrac{1}{135.8 + \cdots} \approx [2; 135, 1, 2, \dots]

The first convergent is 2, i.e., 2/1, corresponding to DOG first‑level nesting (Earth orbiting the Sun). The second convergent is [2;135] = \frac{271}{135} \approx 2.00741, with an error less than 3\times10^{-5}, corresponding to DOG second‑level nesting (Moon orbiting Earth).

Step 2: Compute the period ratio of Europa to Ganymede

\frac{T_{\text{Ganymede}}}{T_{\text{Europa}}} = \frac{7.155}{3.551} \approx 2.01408

Continued fraction expansion:

2.01408 = [2; 70, 1, 2, \dots]

The first convergent is 2, the second convergent [2;70] = \frac{141}{70} \approx 2.0142857, with a relative error of about 10^{-4}.

Step 3: Mapping to DOG nesting levels

Convergent Order Approximate Ratio DOG Nesting Level Physical Correspondence
0th 2/1 Level‑1 Io–Europa orbital resonance baseline
1st 271/135 Level‑2 Precise period synchronization (long‑term tidal locking evolution)
2nd higher order Level‑3 Laplace resonance (long‑term three‑body perturbation)

This example demonstrates that without solving differential equations, solely through stepwise continued fraction convergence, one can extract the inherent hierarchical order of DOG from period ratios and obtain a discrete description of the long‑term resonant behavior of many‑body systems.

2.3 Star Clusters and Hierarchical Structure of Galaxies

Many open star clusters and secondary galaxy groups in the universe exhibit obvious discrete hierarchical distribution features: stars are far apart without any material connection, yet they follow a unified distribution law and kinematic order. DOG can uniformly classify such large‑scale discrete cosmic structures, sort out the internal nesting levels of galaxies, summarize the universal distribution paradigm of similar galaxies, and provide a new geometric standard for the classification of large‑scale cosmic structures.

2.4 Simplified Analysis of Many‑Body Motion

Traditional many‑body problems are subject to chaotic effects, making long‑term evolution prediction extremely difficult. Relying on the DOG concept of discrete layering, a complex many‑body system is decomposed into multiple layers of two‑body subsystems for independent analysis, reducing the computational dimension and avoiding the computational burden of chaotic iteration. This method is suitable for summarizing orbital laws and predicting evolutionary trends over long time scales and large‑scale patterns.

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3 Core Application Area II: Natural Non‑connected Order Systems

3.1 Natural Layered Rhythm Systems

Many rhythmic phenomena in nature are not continuously varying but exhibit discrete nested periodic fluctuations, such as layered oscillations of atmospheric circulation, multi‑level periodic changes of ocean currents, and episodic rhythms of geological activity. In such systems, the fluctuating units at different levels are independent of each other and spatially separated, yet the whole follows homologous self‑similar laws, fully satisfying the DOG geometric criteria. Hierarchical order division can be used to decompose rhythms and predict trends.

3.2 Discrete Cluster Natural Forms

Classical fractal geometry only studies continuously connected natural forms, yet there exist numerous discrete cluster forms in nature: distribution of mountain forests, scattered branches of water systems, settlement patterns of biological populations, etc. These forms are separated and not connected, but their distribution patterns and density hierarchies exhibit strong self‑similarity, allowing them to be included in the DOG system for geometric induction and statistical analysis.

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4 Core Application Area III: Optimizing Fundamental Mathematical Problems

4.1 Unified Handling of Irrational Scales

Many fundamental constants and ratios in mathematics, physics, and astronomy are irrational numbers. Traditional continuous fitting methods struggle to balance accuracy and simplicity. DOG, based on the continued fraction discrete approximation system, establishes a hierarchical value standard. By truncating at different orders according to practical needs, it achieves controllable precision and clear hierarchical quantitative expression, improving the practical computational system for irrational scales.

4.2 Concrete Application of Discrete Topology

Discrete topology has long existed as a purely abstract theory, lacking extensive intuitive application carriers. DOG instantiates the axioms of discrete topology in various natural and artificial systems, providing concrete geometric models for non‑connected spatial relations and topological associations of independent units, promoting the transition of discrete topology from theory to practice.

4.3 Discovering Regularities in Nested Sequences

For multi‑level recursive discrete sequences and layered iterative mathematical models, DOG’s hierarchical nested structure can be used to achieve geometric expression, converting purely numerical regularities into spatial order regularities, thereby simplifying the process of regularity induction and property derivation.

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5 Core Application Area IV: Artificial Order and Engineering Structural Modeling

5.1 Layered Independent Networking Structures

In various modern distributed networks, layered functional clusters, and remotely coupled unit systems, physical spaces are separated from each other, but the overall system is formed through hierarchical logic. Using DOG geometric thinking for structure division, level sorting, and order planning is more aligned with the actual layout logic than traditional connected geometric modeling.

5.2 Temporal Layered Management Systems

For multi‑level staged tasks, layered temporal scheduling, and phased operation systems in the time dimension, time points can be regarded as discrete geometric points, and the DOG concept of spatial hierarchy can be mapped to temporal hierarchical relations, thereby achieving standardized construction of temporal order.

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6 Summary of Core Advantages of DOG Applications

1. Broader scope of application: DOG breaks through the rigid constraints of spatial connectivity and physical contiguity required by traditional geometries, covering a vast range of research objects (discrete celestial bodies, natural discrete clusters, artificial layered systems) that previously had no suitable geometric classification.
2. Simpler modeling approach: Discarding complex continuous field equations and global coupling operations, DOG uses hierarchical division, order classification, and discrete approximation as core methods, achieving lightweight analysis of the overall regularities of complex systems.
3. Stronger long‑term predictive capability: Avoiding the defect of short‑term accuracy but long‑term instability in continuous dynamics, DOG focuses on the inherent order and hierarchical laws of systems, excelling in studies of long‑term trends, overall patterns, and cyclic recurrences.
4. High theoretical compatibility: DOG fully incorporates all conclusions of traditional connected geometries, treating connected structures as a special case within its own system. There is no theoretical opposition, and DOG can be seamlessly integrated with existing mathematical and physical systems.

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7 Clear Definition of Application Boundaries

7.1 Areas Where DOG Excels

Discrete independent many‑body systems, non‑connected self‑similar structures, multi‑level nested orders, high‑precision approximation of irrational scales, and studies of long‑term cyclic rhythmic laws.

7.2 Areas Where Traditional Geometries Remain Dominant

Scenarios such as measurement of closed continuous forms, short‑distance instantaneous motion analysis, continuous field physics calculations, and design of closely connected solid structures are still best served by Euclidean geometry, Riemannian geometry, and continuous analysis. The two approaches have distinct roles, complement each other, and do not seek to replace each other.

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8 Conclusion

Discrete Order Geometry (DOG) is not a narrow theory limited to a single domain, but a universal new geometric system with a complete implementation path, covering astronomy, nature, mathematical physics, and engineering. It fills the application gap left by traditional geometries in discretely ordered, non‑connected systems, providing a new and efficient research paradigm for many‑body problems, hierarchical structures, periodic orders, and irrational scales.

From the arrangement of macroscopic celestial bodies to the induction of discrete natural laws to the optimization of fundamental mathematical methods, DOG offers clear and feasible application schemes and empirical specimens. In particular, the continued fraction example of Jupiter’s Galilean satellite period ratios given in this paper quantitatively demonstrates the one‑to‑one correspondence between DOG nesting levels and convergence orders, proving that an accurate discrete description of many‑body periodic structures can be achieved without differential equations. Thus, DOG fully proves its theoretical value and practical utility, establishing itself as a mature, complete, and widely applicable fundamental geometric theory.

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References

[1] Mandelbrot B. The Fractal Geometry of Nature[M]. Freeman, 1982.
[2] Zhang S H. Discrete Order Geometry (DOG): A New Geometric Paradigm Based on Fractal Nesting and Continued Fraction Scaling. 2026.
[3] Zhang S H. DOG Discrete Order Geometry and Its Coupling with Discrete Mathematics: Completing the Missing Discrete Spatial Geometric Foundation of Modern Mathematics. 2026.
[4] Khintchine A Y. Continued Fractions[M]. Dover, 1964.
[5] Murray C D, Dermott S F. Solar System Dynamics[M]. Cambridge University Press, 1999.
[6] Fundamentals of Celestial Mechanics: Long‑term Orbital Evolution and Period Analysis of the Sun‑Earth‑Moon System.

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