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. Although topology can accommodate disconnected sets, it always embeds discrete structures into a presupposed continuous background space. Existing geometric systems generally lack a dedicated geometric descriptive framework that takes discrete hierarchical order as its ontology and does not depend 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 constraint of traditional geometry, adopts order isomorphism, hierarchical nesting, and scale recursion as the criteria for determining a geometric system, and uses continued-fraction hierarchical convergence to achieve precise quantification of irrational scales in discrete systems. This paper introduces the concept of a DOG discrete lattice, in which lattice cells are related by hierarchical order functions and scale recursion functions, while Euclidean spatial distance serves only as an observed representation derived from these 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 boundary with Euclidean geometry, Riemannian geometry, and classical fractal geometry. Taking the Sun–Earth–Moon three-level celestial nesting system as an empirical sample, the applicability of the DOG framework is verified. 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 determination 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 commonality is that geometric objects depend on a continuous background space and conventionally take point adjacency and regional connectivity as their form.

In theory, topology permits disconnected sets and separated branches, but topological description only defines whether a set is connected or not. It does not establish ordered hierarchical structures or scale recursion relations among discrete units, and thus cannot characterize the structured order of "nesting across emptiness, orderly arrangement, and hierarchical evolution" that is ubiquitous 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. Its research objects are mostly fragmented structures formed in continuous media (coastlines, clouds, condensation structures, etc.). Its definition of self-similarity strictly depends on invariance under graphical scaling, and cannot directly adapt to physical structures such as celestial many-body systems, which are discrete, separated across emptiness, 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 suffers from chaotic sensitivity and accumulated numerical error;
2. Orbital ratios, period ratios, and eccentricities are mostly irrational numbers, and finite decimal approximations suffer from inherent distortion;
3. Taking 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 takes discrete ordered hierarchy as its ontology and is independent of continuous manifolds, thereby achieving a unified description of both 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 under graphical scaling, retains the generalized self-similar features of homologous arrangement order and isomorphic 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 an optimal rational approximation system for irrational numbers, possessing the unique properties of layer-by-layer truncation, order-by-order convergence, and hierarchical matching. Celestial orbital ratios, period ratios, and perturbation rhythms are mostly irrational scales, and traditional decimal approximation suffers from truncation bias.

DOG introduces the hierarchical convergence mechanism of continued fractions into the definition of geometric scale, achieving a correspondence between discrete structural hierarchy and continued-fraction convergence order, and providing a hierarchical 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 depend on spatial connectivity or a continuous background manifold, and that takes hierarchical nesting order, cross-scale structural isomorphism, and continued-fraction scale recursion as its core criteria to characterize ordered geometric systems composed of discrete independent units.

DOG can define a discrete lattice \mathcal{L}=\{P_i\}, in which lattice cells 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 observed representation derived from these 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; physical adjacency, spatial connectivity, and medium coupling are not required;
2. The "self-similarity" of DOG specifically refers to order self-similarity, rhythm self-similarity, and arrangement-hierarchy self-similarity, which differs from the graphical 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, orderly many-body arrangements, and hierarchically periodic evolutionary systems.

4 Basic Axiom System of DOG

Axiom 1: Order Isomorphism Axiom

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 form a unified DOG geometric system.

Axiom 2: Hierarchical Scale Convergence Axiom

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

Axiom 3: Continuous Special-Case Inclusion Axiom

All traditional geometric structures built on continuous connected manifolds are special solutions of DOG geometry after imposing spatial connectivity constraints. 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—"central primary 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 distances among units, the presence or absence of a medium, and motion rates.

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 dominated by irrational scales, and their long-term evolutionary laws can be characterized order by order by continued-fraction hierarchical convergence sequences. This serves as a parallel descriptive path besides differential dynamics, 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 plus discrete order geometry.

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

The Sun–Earth–Moon system is the most stable and hierarchically clearest natural discrete nesting system in near-Earth space, and it fully matches the theoretical framework of DOG.

6.1 Structural Hierarchy Matching the DOG Nesting Definition

· First-level core unit: the Sun (central gravitational 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, without physical medium connection, and form a stable structure solely through arrangement order and hierarchical nesting, satisfying the DOG order isomorphism axiom. The Sun, Earth, and Moon can be mapped as different nodes of a DOG discrete lattice, and the hierarchical relations among nodes are expressed by order functions.

6.2 Scale Rhythm Matching 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 truncation layer by layer, the multi-level periodic rhythms and long-term perturbation evolution of the system can be quantitatively characterized hierarchically, serving as a parallel descriptive path besides differential dynamical equations.

6.3 Value of the Sample

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 nesting systems throughout the solar system and to large-scale cosmic hierarchical structures. This example is only a geometric-level mapping and does not negate or replace existing celestial dynamics theories.

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, graphical scaling self-similarity;
4. DOG discrete order geometry: discrete, separated across emptiness, hierarchically ordered, rhythmically self-similar; does not depend 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 and introduces the discrete lattice, in which lattice cells are related through hierarchical order functions and scale recursion functions. Taking 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 does not replace classical geometry or celestial dynamics, but provides a purely geometric and number-theoretic parallel descriptive path. Connected geometry is a continuous special case of DOG. DOG expands the applicable boundary of geometric systems and can provide a new theoretical tool for many-body nested structures, orbital rhythm evolution, and hierarchical determination of irrational scales.

Outlook

Future work to be carried out:

1. Construct strict quantitative criteria for order isomorphism; the axiom level is currently still qualitative;
2. Derive specific continued-fraction scale recursion formulas and develop quantitative predictive capability within the DOG framework;
3. Extend high-dimensional multi-level nested DOG configurations;
4. Introduce the DOG discrete order lattice into directions such as fluid topology solving, and carry out interdisciplinary applications;
5. Explore the intrinsic causes that produce DOG-type hierarchical structures in the physical world; this part is only a future research idea and is not ontologically argued in this paper.

References

Omitted

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