A Generalization of the Hausdorff Dimension Theorem for Fractals

TL;DR

This paper explores the existence and properties of fractals in nature and virtual environments, focusing on the Hausdorff dimension and Lebesgue measure, and extends the theoretical understanding of fractal dimensions.

Contribution

It generalizes the Hausdorff dimension theorem for fractals, proving the existence of a vast class of virtual fractals with specific dimensional properties.

Findings

Existence of beth-two many virtual fractals with specific Hausdorff dimensions

Relationship between Hausdorff dimension and Lebesgue measure for these fractals

Partial answers to the classification of fractals in nature and virtual worlds

Abstract

How many fractals exist in nature or the virtual world In this work, we partially answer the second question using Mandelbrots fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of beth-two of virtual fractals with a Hausdorff dimension of a bivariate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.