On optimal cover and its possible shape for fractals embedded into 2D Euclidian space

TL;DR

This paper defines an optimal cover for fractals in 2D space, derives a functional equation for Minkowski dimension, and characterizes possible shapes of optimal coverage based on fractal dimensions.

Contribution

It introduces a new definition of optimal cover for fractals and solves the related functional equation to identify potential coverage shapes.

Findings

Derived a functional equation for Minkowski dimension

Identified possible shapes of optimal coverage for fractals

Provided a theoretical framework linking cover shapes to fractal dimensions

Abstract

In this article a definition of optimal cover for fractal structures is proposed. Expression for Minkowsky dimension is rewritten in terms of functional equation on areas of covers that constructed for different scales.Given the definition, the functional equation is resolved and possible shapes of optimal coverage are defined in correspondence with fractal dimension values.