The Chromatic Tiling Theorem: Scaling Laws and the Separation Dimension of Fractal Partitions

TL;DR

This paper introduces the Separation Dimension to quantify the complexity of fractal partitions and proves a theorem linking the chromatic number of such partitions to their separation properties, advancing understanding of fractal interface geometry.

Contribution

It defines the Separation Dimension and establishes the Chromatic Tiling Theorem, providing a new intrinsic framework for analyzing fractal partitions and their coloring complexity.

Findings

The Separation Dimension quantifies boundary complexity in fractal partitions.

The Chromatic Tiling Theorem bounds the chromatic number by the ratio of maximum to minimum tile size raised to the Separation Dimension.

Existence of Geometrically Regular Partitions on fractals like the Sierpinski Carpet is proven.

Abstract

This paper establishes a rigorous, quantitative link between the combinatorial complexity of a fractal partition and the intrinsic geometry of its interfaces. We introduce the concept of the \emph{Separation Dimension} ($\sepdim$), a novel characteristic that quantifies the Hausdorff dimension of the boundaries between tiles. A natural but flawed approach would be to relate coloring complexity to the fractal's ambient topological boundary. We demonstrate that this extrinsic view is untenable for a vast class of self-similar sets. Instead, our intrinsic framework, centered on the Separation Dimension, provides the correct formulation. We define a new class of well-behaved partitions, termed \emph{Geometrically Regular Partitions (GRPs)}, and prove their existence on canonical fractals such as the Sierpinski Carpet. Our main result, the Chromatic Tiling Theorem, provides a sharp upper bound for the chromatic number ($\chi$) of the graph associated with such a partition, proving that it is bounded by $\chi(\tiling) \leq \Cconst (r_{\max}/r_{\min})^{\sepdim}$. This result demonstrates that the scaling of coloring complexity is governed not by the dimension of the fractal itself, but by the dimension of the separation set induced by the partition. We conclude by proposing the minimization of the separation dimension over all admissible partitions as a new variational problem in fractal geometry.