Discrete Order Geometry (DOG): A Discrete Hierarchical Geometric Framework Based on Fractal Nesting and Continued-Fraction Scales

Author: Zhang Suhang
(Luoyang, Henan)

Abstract

Euclidean geometry and Riemannian geometry take continuous connected manifolds as their basic carrier, classical fractal geometry mainly studies self-similar structures in continuous media, and topology, although it can accommodate non-connected sets, has always embedded discrete structures in a presupposed continuous background space. Existing geometric systems generally lack a dedicated geometric descriptive system that takes discrete hierarchical order as its ontology and does not rely on spatial connectivity or a continuous background manifold.

Combining the idea of hierarchical nesting structures with the convergence properties of continued-fraction irrational scales, this paper constructs the framework of Discrete Order Geometry (DOG). DOG abandons the connectivity axiom constraints of traditional geometry, takes order isomorphism, hierarchical nesting, and scale recursion as the criteria for determining a geometric system, and uses continued-fraction layered convergence to achieve precise quantification of irrational scales in discrete systems. This paper introduces the concept of the DOG discrete lattice, in which lattice units are related by hierarchical order functions and scale recursion functions, and Euclidean spatial distance serves only as an observational representation derived from the functions, not as the underlying definition of the lattice.

This paper establishes the basic axiom system and core theorems of DOG, and clarifies its relations of inclusion and boundaries with Euclidean geometry, Riemannian geometry, and classical fractal geometry. Taking the Sun-Earth-Moon three-level celestial nested system as an empirical sample, it verifies the applicability of the DOG framework. The study shows that connected geometry is a special case of DOG under continuous constraints, and that DOG can provide a new purely geometric and number-theoretic descriptive path for discrete nested celestial systems, many-body periodic evolution, and hierarchical fixed values of irrational scales.

Keywords: Discrete Order Geometry; DOG; hierarchical nesting; continued fractions; order isomorphism; many-body systems; discrete lattice

1 Introduction

1.1 Inherent Constraints of Traditional Geometric Systems

Euclidean geometry is built on a flat, continuous, connected space, and Riemannian geometry extends it to curved continuous manifolds. Together they form the foundation of modern physics and geometric analysis. Their underlying common feature is that geometric objects depend on a continuous background space and conventionally take the form of point adjacency and regional connectivity.

Topology theoretically allows non-connected sets and separated branches, but topological description only defines whether a set is connected or not; it does not establish ordered hierarchical structures and scale recursion relations among discrete units, and cannot depict the structured order of "separated yet nested, orderly arranged, hierarchically evolving" structures that universally exist in the universe.

1.2 Applicability Limitations of Classical Fractal Geometry

Classical Mandelbrot fractal geometry takes morphological self-similarity and scale-free nesting as its core features, and its research objects are mostly fragmented structures formed in continuous media (coastlines, clouds, aggregation structures, etc.). Its definition of self-similarity strictly depends on invariance under graphic scaling, and cannot directly adapt to physical structures such as celestial many-body systems that are discrete, separated, without medium connection, and possess only arrangement order and rhythmic nesting.

1.3 Defects of Traditional Research Modes for Many-Body Systems

The mainstream study of celestial many-body and hierarchically nested systems relies on differential dynamics, force-field coupling, and numerical iterative integration. This mode has three inherent limitations:

1. Long-term dynamical integration has chaotic sensitivity and accumulated numerical error;
2. Orbital ratios, period ratios, and eccentricities are mostly irrational numbers, and finite decimal approximation has inherent distortion;
3. With dynamical forces as the core, it lacks an independent descriptive perspective based on purely geometric order and structural ontology.

Based on the above theoretical gaps, this paper constructs Discrete Order Geometry (DOG). DOG does not negate existing geometric systems, but supplements them with an independent geometric framework that is independent of continuous manifolds and takes discrete ordered hierarchy as its ontology, achieving a unified description of structural qualification and scale quantification for discrete nested systems.

2 Theoretical Foundations of DOG

2.1 Hierarchical Nesting Structure (Structural Foundation)

DOG inherits the core idea of hierarchical nesting and cross-scale structural homology from fractal geometry, and redefines its scope of application:
It abandons the strong constraint of self-similarity in graphic scaling form, retains the generalized self-similar features of homology in arrangement order and isomorphism in evolutionary rhythm, and allows discrete units without medium, adjacency, or physical connection to form a unified geometric system.

2.2 Continued-Fraction Scale Theory (Quantitative Foundation)

Continued fractions are the optimal rational approximation system for irrational numbers, possessing unique properties of layer-by-layer truncation, stage-by-stage convergence, and hierarchical matching. Celestial orbital ratios, period ratios, and perturbation rhythms are mostly irrational scales, and traditional decimal approximation has truncation deviation.

DOG introduces the hierarchical convergence mechanism of continued fractions into the definition of geometric scale, achieving: discrete structural hierarchy ↔ continued-fraction convergence order correspondence, providing a layered description for the long-term evolution of irrational scales.

3 Core Definitions of Discrete Order Geometry (DOG)

3.1 Basic Concepts

Discrete Order Geometry (DOG):
A geometric framework that does not rely on spatial connectivity or a continuous background manifold, and takes hierarchical nesting order, cross-scale structural isomorphism, and continued-fraction scale recursion as its core criteria to describe ordered geometric systems composed of discrete independent units.

DOG can define a discrete lattice \mathcal{L}=\{P_i\}, in which lattice units are internally related through hierarchical order functions F_{ij}:P_i\mapsto P_j and scale recursion functions; Euclidean spatial distance serves only as an observational representation derived from the functions, not as the underlying definition of the lattice.

3.2 Key Points of Core Definitions

1. The necessary and sufficient condition for the establishment of a DOG geometric system is order isomorphism and hierarchical nesting, without requiring physical adjacency, spatial connectivity, or medium coupling;
2. The "self-similarity" of DOG specifically refers to order self-similarity, rhythm self-similarity, and arrangement-level self-similarity, which differs from the graphic scaling self-similarity of classical fractals;
3. Euclidean connected geometry, Riemannian curved connected geometry, and classical continuous fractal geometry can all be incorporated as special cases of DOG under continuous constraints;
4. The main application domain of DOG: discrete nested systems in the universe, many-body ordered arrangements, and hierarchically periodic evolving systems.

4 Basic Axiom System of DOG

Axiom 1 Axiom of Order Isomorphism

If several spatially discrete, mutually independent units without medium connection possess generalized self-similar features of consistent cross-scale hierarchical nesting structure and consistent evolutionary rhythmic arrangement, then they can constitute a unified DOG geometric system.

Axiom 2 Axiom of Layered Scale Convergence

All irrational structural scales, motion ratios, and periodic rhythm parameters in a DOG system can be hierarchically approximated through continued-fraction stage-by-stage truncation, and their convergence order corresponds one-to-one with the system's nesting hierarchy, enabling scale description without long-term accumulated error.

Axiom 3 Axiom of Inclusion of Continuous Special Cases

All traditional geometric structures built on continuous connected manifolds are special solutions of DOG geometry after imposing spatial connectivity constraints, and the DOG system is fully compatible with existing classical geometric conclusions.

5 Three Core Theorems of DOG

Theorem 1 Discrete Hierarchical Nesting Theorem

A discrete system possessing three or more levels of hierarchical arrangement of "central main unit—secondary orbiting unit—satellite nested unit" and satisfying cross-scale order isomorphism belongs to a standard DOG geometric configuration.
The determination of this configuration is independent of the spatial distance between units, the presence or absence of a medium, and motion speed.

Corollary: The Sun-Earth-Moon system, planet-satellite systems, galaxy cluster hierarchical structures, and nested orbital resonance systems are all natural DOG geometric instances.

Theorem 2 Continued-Fraction Scale Matching Theorem

The structural ratios and evolutionary rhythms of a DOG discrete nested system are mainly irrational scales, and their long-term evolutionary laws can be described stage by stage by continued-fraction layered convergence sequences, serving as a parallel descriptive path besides differential dynamics and avoiding chaotic error and numerical drift caused by differential iteration.

Theorem 3 Geometric Paradigm Inclusion Theorem

Connectivity is not a universal prerequisite of geometric systems, but an additional constraint of continuous medium systems; DOG breaks through the limitation of connectivity and forms a complementary geometric system of continuous geometry + discrete order geometry.

6 Empirical Sample: The Sun-Earth-Moon Three-Level Nested System

The Sun-Earth-Moon system is the most stable and clearly hierarchical natural discrete nested system in near-Earth space, and fully matches the DOG theoretical framework.

6.1 Structural Hierarchy Matches the DOG Nesting Definition

· First-level core unit: the Sun (central gravitational main body of the system)
· Second-level orbiting unit: the Earth (stably revolving around the Sun)
· Third-level nested unit: the Moon (secondary nested revolution around the Earth)

The three are spatially separated and without physical medium connection, forming a stable structure solely through arrangement order and hierarchical nesting, satisfying the DOG axiom of order isomorphism. The Sun, Earth, and Moon can be mapped as different nodes of the DOG discrete lattice, and the hierarchical relations among nodes can be expressed by order functions.

6.2 Scale Rhythm Matches Continued-Fraction Convergence Characteristics

Key parameters such as the Sun-Earth orbital ratio, Earth-Moon orbital ratio, solar and lunar eclipse cycles, synodic periods, and orbital eccentricity perturbations are all irrational scales and cannot be precisely expressed by finite decimals.
Through continued-fraction layer-by-layer truncation, the system's multi-level periodic rhythms and long-term perturbation evolution can be quantitatively described in layers, serving as a parallel descriptive path besides systems of differential dynamical equations.

6.3 Sample Value

This system has complete observational data, stable structure, and clear hierarchy. It can serve as a standard test specimen for DOG geometry, and can also be extended to planetary-satellite nested systems throughout the solar system and large-scale cosmic hierarchical structures.

7 Paradigm Boundaries Between DOG and Traditional Geometry

1. Euclidean geometry: flat, continuous, connected, suitable for artificial regular continuous forms;
2. Riemannian geometry: curved, continuous, connected, suitable for continuous spacetime manifolds;
3. Classical fractal geometry: continuous media, graphic scaling self-similarity;
4. DOG discrete order geometry: discrete, separated, hierarchically ordered, rhythmic self-similarity, not relying on a continuous background manifold.

The four are complementary, compatible, and mutually non-contradictory. DOG provides a dedicated geometric descriptive tool for discrete ordered nested structures.

8 Conclusion

This paper constructs the complete conceptual framework, axiom system, and core theorems of Discrete Order Geometry (DOG). DOG abandons the mandatory connectivity constraint of traditional geometry, introduces the discrete lattice, and lattice units establish connections through hierarchical order functions and scale recursion functions; with hierarchical nesting order and continued-fraction irrational scale convergence as its dual core, it establishes a geometric descriptive system adapted to discrete nested systems in the universe.

DOG provides a purely geometric and number-theoretic parallel descriptive path. Connected geometry is a continuous special case of DOG, and DOG expands the applicable boundaries of geometric systems, and can provide a new theoretical tool for many-body nested structures, orbital rhythm evolution, and hierarchical fixed values of irrational scales.

References

Omitted