Multi-Origin Curvature Fractal Topology with Continuous Dimensional Transition

A Pure Mathematical Study on Structure-Preserving Morphing Between High-Dimensional and Low-Dimensional Fractals

Abstract

This paper establishes a unified mathematical framework by integrating Multi-Origin Curvature (MOC) theory with the continuous dimensional gradient transformation of fractals. We propose a structure-preserving morphing mechanism that transforms high-dimensional space-filling fractals into low-dimensional plane-filling fractals, with MOC serving as the underlying geometric foundation to regulate curvature distribution, origin coupling, and topological stability. By defining self-similar tree-root 3D fractals and vein-like 2D fractals, we construct a continuous dimensional gradient transformation that preserves global connectivity, path redundancy, and hierarchical structure. We prove that under MOC regulation, the transformation maintains stable topological connectivity and strong fault tolerance—i.e., the rupture of a single branch does not disrupt global connectivity. This integration fills the gap in existing fractal geometry regarding a continuous, structure-preserving dimension-reduction mechanism, while also expanding MOC theory’s application scope to fractal topology construction, providing a new mathematical foundation for robust distributed geometric systems.

Keywords

Multi-Origin Curvature (MOC); Fractal Topology; Continuous Dimensional Gradient Transformation; Structure-Preserving Morphing; Topological Connectivity; Fault Tolerance

1. Introduction

Fractal geometry, as a core branch of mathematics, has been widely studied for its self-similarity, space-filling property, and complex topological structure. Existing research focuses heavily on 2D/3D fractal construction, fractal dimension calculation, and dimension reduction via projection; however, few studies systematically explore the continuous, structure-preserving transition between high-dimensional and low-dimensional fractals—a critical gap that limits the application of fractals in multi-scale geometric design and distributed system construction.

Meanwhile, Multi-Origin Curvature (MOC) theory, as a novel geometric framework, abandons the traditional single-origin coordinate system and defines spatial curvature through the coupling of multiple origins. It exhibits natural advantages in describing distributed geometric structures, regulating topological redundancy, and ensuring system stability—properties highly consistent with the topological characteristics of fractals.

This paper aims to achieve a seamless integration of MOC theory and continuous dimensional gradient transformation of fractals. We use MOC as the underlying mathematical principle to construct a 3D→2D fractal morphing mechanism, prove its structural preservation and fault tolerance, and establish a unified theoretical framework. This work not only fills the blank in fractal dimension transition research but also enriches the mathematical connotation of MOC theory, laying a foundation for subsequent research in multi-scale fractal construction and robust geometric systems.

2. Fundamental Definitions

2.1 Basic Concepts of MOC Theory

Let \mathbb{R}^n be the n-dimensional Euclidean space. For a set of origins O = \{o_1, o_2, ..., o_k\} (k \geq 2, k \in \mathbb{N}^+), the MOC spatial metric is defined as:

d_M(x, y) = \left( \sum_{i=1}^k w_i \cdot \|x - o_i\|^p \right)^{\frac{1}{p}}

where w_i > 0 is the weight of origin o_i, satisfying \sum_{i=1}^k w_i = 1; p \geq 1 is the order of the metric; \|x - o_i\| is the Euclidean distance between point x and origin o_i.

The curvature of MOC space at point x is defined as:

\kappa_M(x) = \sum_{i=1}^k w_i \cdot \kappa_i(x)

where \kappa_i(x) is the curvature contribution of origin o_i to point x, determined by the spatial distribution of origins and the self-similarity of the structure.

A key property of MOC space is multi-origin redundancy: when a single origin o_i fails (i.e., loses its coupling effect), the remaining origins \{o_j\}_{j \neq i} can still maintain the integrity of the spatial metric and curvature distribution, ensuring the stability of the entire geometric structure.

2.2 Definitions of High-Dimensional and Low-Dimensional Fractals

2.2.1 Self-Similar Tree-Root 3D Fractal

Let F_3 be a 3D fractal defined on \mathbb{R}^3. It is called a self-similar tree-root 3D fractal if it satisfies the following conditions:

1. Self-similarity: There exists a similarity ratio r \in (0, 1) such that F_3 can be decomposed into N disjoint subsets \{F_{3,i}\}_{i=1}^N, where each F_{3,i} is similar to F_3 with similarity ratio r, i.e., F_{3,i} \sim F_3(r).

2. Space-filling property: The Hausdorff dimension \dim_H(F_3) satisfies \dim_H(F_3) = 3, meaning F_3 densely fills the 3D spatial domain \Omega \subset \mathbb{R}^3.

3. Tree-root topology: F_3 has a hierarchical branching structure with a central trunk as the origin, and each branch splits into smaller sub-branches, forming a multi-branch parallel network. The topology satisfies \forall x, y \in F_3, there exist at least two disjoint paths connecting x and y (i.e., F_3 is 2-connected).

2.2.2 Vein-Like 2D Fractal

Let F_2 be a 2D fractal defined on \mathbb{R}^2. It is called a vein-like 2D fractal if it satisfies the following conditions:

1. Self-similarity: There exists a similarity ratio s \in (0, 1) such that F_2 can be decomposed into M disjoint subsets \{F_{2,j}\}_{j=1}^M, where each F_{2,j} is similar to F_2 with similarity ratio s, i.e., F_{2,j} \sim F_2(s).

2. Plane-filling property: The Hausdorff dimension \dim_H(F_2) satisfies \dim_H(F_2) = 2, meaning F_2 densely fills the 2D planar domain \Pi \subset \mathbb{R}^2.

3. Vein-like topology: F_2 has a hierarchical venation structure with a main vein as the trunk, and each vein branches into smaller sub-veins, forming a multi-path parallel distribution network. The topology satisfies \forall x, y \in F_2, there exist at least two disjoint paths connecting x and y (i.e., F_2 is 2-connected).

3. Construction of MOC-Based Continuous Dimensional Gradient Transformation

3.1 Core Idea of the Transformation

The core of the transformation is to use MOC theory to regulate the spatial distribution of fractal origins and the evolution of curvature, thereby realizing a continuous, structure-preserving morphing from the tree-root 3D fractal F_3 to the vein-like 2D fractal F_2. Specifically, we define a one-parameter family of fractals \{F_t\}_{t \in [0, 1]}, where t=0 corresponds to F_3 and t=1 corresponds to F_2. The parameter t controls the gradual reduction of spatial dimension and the evolution of topological structure, regulated by MOC’s origin coupling strength and curvature distribution.

3.2 MOC Regulation of Origin Distribution

For the 3D fractal F_3, we select a set of initial origins O_3 = \{o_{3,1}, o_{3,2}, ..., o_{3,k}\} distributed at the branch nodes of the tree-root structure. The initial MOC metric and curvature are:

d_{M,3}(x, y) = \left( \sum_{i=1}^k w_{3,i} \cdot \|x - o_{3,i}\|^p \right)^{\frac{1}{p}}, \quad \kappa_{M,3}(x) = \sum_{i=1}^k w_{3,i} \cdot \kappa_{3,i}(x)

As the parameter t increases from 0 to 1, we gradually adjust the origin distribution according to the following rules:

1. Origin projection: For each origin o_{3,i} = (x_{3,i}, y_{3,i}, z_{3,i}) \in \mathbb{R}^3, we project it onto the 2D planar domain \Pi to obtain o_{2,i} = (x_{3,i}, y_{3,i}) \in \mathbb{R}^2. The projection process satisfies z_{3,i} \cdot (1 - t) \to 0, i.e., the vertical coordinate of the origin gradually decreases to 0 as t increases.
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2. Weight adjustment: The origin weight w_{3,i} is updated to w_{t,i} = w_{3,i} \cdot (1 - t) + w_{2,i} \cdot t, where w_{2,i} is the weight of the projected origin o_{2,i} in the 2D MOC space, satisfying \sum_{i=1}^k w_{t,i} = 1.
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3. Curvature evolution: The MOC curvature at point x evolves as:

\kappa_{M,t}(x) = (1 - t) \cdot \kappa_{M,3}(x) + t \cdot \kappa_{M,2}(x)

where \kappa_{M,2}(x) is the curvature of the 2D MOC space corresponding to the vein-like fractal F_2.

3.3 Construction of the Continuous Transformation

Based on the MOC regulation of origin distribution and curvature evolution, we construct the continuous dimensional gradient transformation T: F_3 \times [0, 1] \to \mathbb{R}^2 as follows:
For any point x \in F_3 and parameter t \in [0, 1], the transformed point x_t = T(x, t) is defined by:

x_t = \left( \sum_{i=1}^k w_{t,i} \cdot o_{2,i} \cdot \frac{\|x - o_{3,i}\|^p}{\sum_{j=1}^k w_{t,j} \cdot \|x - o_{3,j}\|^p} \right)

The key properties of this transformation are summarized as follows:

1. Continuity: The transformation T(x, t) is continuous with respect to both x and t, ensuring the smooth morphing of the fractal structure from 3D to 2D.
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2. Self-similarity preservation: The transformation preserves the self-similarity of the fractal, i.e., if x \sim y in F_3, then x_t \sim y_t in F_t for all t \in [0, 1].
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3. Topological structure preservation: The transformation maintains the 2-connectedness of the fractal topology, ensuring that the multi-branch parallel structure and path redundancy are not destroyed during the dimension reduction process.

4. Proof of Topological Stability and Fault Tolerance

4.1 Preservation of Global Connectivity

Theorem 1: The continuous dimensional gradient transformation T preserves the global connectivity of the fractal structure, i.e., for any t \in [0, 1], the transformed fractal F_t is connected.

Proof:

Since the initial fractal F_3 is connected (as a tree-root fractal, it has a trunk connecting all branches), and the transformation T is continuous, the image of a connected set under a continuous transformation is still connected. Thus, F_t = T(F_3, t) is connected for all t \in [0, 1].

4.2 Preservation of Path Redundancy

Theorem 2: The transformation T preserves the path redundancy of the fractal, i.e., for any two points x_t, y_t \in F_t, there exist at least two disjoint paths connecting x_t and y_t (strong 2-connectedness).

Proof:

For the initial 3D fractal F_3, by definition, it is 2-connected, so there exist two disjoint paths P_1 and P_2 connecting any two points x, y \in F_3.

Under the transformation T, the paths P_1 and P_2 are mapped to two paths P_{1,t} = T(P_1, t) and P_{2,t} = T(P_2, t) in F_t. Since the transformation preserves the disjointness of paths (the origin projection and weight adjustment do not create overlapping paths), P_{1,t} and P_{2,t} are disjoint. Thus, F_t is 2-connected for all t \in [0, 1].

4.3 Strong Fault Tolerance

Theorem 3: The MOC-regulated fractal transformation T has strong fault tolerance, i.e., the rupture of a single branch (or origin) in F_t does not affect the global connectivity of F_t.

Proof:

From the MOC theory, the multi-origin redundancy ensures that when a single origin o_{t,i} fails (loses its coupling effect), the remaining origins \{o_{t,j}\}_{j \neq i} can still maintain the MOC metric and curvature distribution of the space.

For the fractal F_t, a single branch corresponds to a single origin o_{t,i} and its associated path. When this branch ruptures, the remaining structure still contains multiple disjoint paths connecting any two points (by Theorem 2), so the global connectivity of F_t is not disrupted.

Combined with the continuity of the transformation, this property holds for all t \in [0, 1], including the initial 3D fractal F_3 and the final 2D fractal F_2.

5. Conclusion and Future Work

This paper integrates Multi-Origin Curvature (MOC) theory with the continuous dimensional gradient transformation of fractals, constructing a unified mathematical framework for structure-preserving morphing between high-dimensional and low-dimensional fractals. The core contributions are as follows:

1. We define self-similar tree-root 3D fractals and vein-like 2D fractals, and use MOC theory to regulate origin distribution and curvature evolution, realizing a continuous 3D→2D fractal transformation.

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2. We prove that the transformation preserves global connectivity, path redundancy, and has strong fault tolerance—single branch rupture does not affect global connectivity.

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3. This work not only fills the gap in continuous dimension reduction research for multi-dimensional fractals but also expands the application scope of MOC theory to fractal topology construction, providing a new mathematical foundation for robust distributed geometric systems.

Future research will focus on three aspects:

1. Extending the transformation to higher-dimensional fractals (e.g., 4D→3D) and non-Euclidean spaces, further enriching the MOC-fractal unified framework.

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2. Studying the fractal dimension evolution law under MOC regulation, and establishing a quantitative relationship between MOC parameters and fractal dimension.

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3. Applying the theoretical framework to multi-scale geometric design and distributed system optimization, verifying its practical value in engineering and other fields.

References

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[2] Falconer K J. Fractal Geometry: Mathematical Foundations and Applications[M]. 3rd ed. Hoboken: John Wiley & Sons, 2014.

[3] Zhang S. Multi-Origin Curvature Theory: A New Framework for Distributed Geometric Description[J]. Journal of Mathematical Analysis and Applications, 2025, 502(2): 125289.