The dimensions of Hausdorff and Mendes France. A comparative study

TL;DR

This paper compares the Hausdorff dimension of fractal curves with Mendes France's fractional dimension of rectifiable curves, using a constructive process to analyze their dimensional properties and relationships.

Contribution

It introduces a geometric process linking self-similar fractal curves and rectifiable curves to compare their dimensions, revealing conditions for their equality.

Findings

Hausdorff and Mendes France dimensions can be linked via a constructive process.

Equality of the two dimensions occurs in strictly self-similar cases.

The study provides a method to compare different fractional dimensions of curves.

Abstract

This paper contains a comparative study of two families of simple curves drawn in the plane. On the one hand, we have the fractal curves on the unit interval, with self-similar structure, which have associated a Hausdorff dimension. On the other hand, we have the opposite: a class of locally rectifiable unbounded curves, which have another "fractional dimension" defined by M. Mendes France. We propose a geometrical constructive process that will allow us to obtain - as the limit of a sequence of polygonal curves - one curve of the first family, by contractive transformations; and another of the second family, by expansive transformations. Thanks to this process of linking curves from both families, we are able to compare their dimensions - our aim in this work -, and to obtain interesting results such as the equality of the latter in the case of strict self-similarity.