Mathematical Innovation Abstract

(Purely Mathematical Formulation, Directly Usable in Academic Papers)

This paper introduces a new class of fractal geometric transformations: continuous dimensional gradient transformations from three-dimensional space-filling fractals to two-dimensional plane-filling fractals, and investigates their topological invariances.

1. A class of self-similar tree-root three-dimensional fractals is defined, characterized by spatial filling, multi-branch parallelism, and high connectivity.
2. A class of vein-like two-dimensional fractals is defined, characterized by planar filling, uniform extension, and hierarchical distribution.
3. A dimensional gradient transition transformation is constructed to achieve smooth geometric deformation from three-dimensional fractals to two-dimensional fractals, preserving global connectivity, path redundancy, and branched hierarchical structure.
4. It is proved that the topological connectivity of the system remains stable under this transformation, satisfying the strong fault-tolerance property that the rupture of a single branch does not affect global connectivity.

This transformation fills the gap in existing fractal geometry regarding a continuous, structure-preserving, and engineerable gradient conversion mechanism between 3D and 2D fractals, providing a new fundamental construction for fractal topology and transport networks.

This is a mathematical innovation in the strict sense:

It proposes a structure-preserving continuous dimension-reduction transformation between fractals that has not been systematically studied in the prior literature, and establishes its core topological properties.