How to define your dimension: A discourse on Hausdorff dimension and self-similarity

TL;DR

This paper explores the concept of Hausdorff dimension as a rigorous way to define the 'dimension' of complex objects like fractals, highlighting its ability to assign non-integer dimensions and discussing self-similarity.

Contribution

It provides a detailed review of Hausdorff dimension, addressing foundational issues and illustrating how it captures the complexity of self-similar fractal objects.

Findings

Hausdorff dimension can be non-integer for fractals

Self-similarity often leads to non-integer dimensions

Hausdorff measure is crucial for defining fractal dimensions

Abstract

One often distinguishes between a line and a plane by saying that the former is one-dimensional while the latter is two. But, what does it mean for an object to have $d-$dimensions? Can we define a consistent notion of dimension rigorously for arbitrary objects, say a snowflake, perhaps? And must the dimension always be integer-valued? After highlighting some crucial problems that one encounters while defining a sensible notion of dimension for a certain class of objects, we attempt to answer the above questions by exploring the concept of Hausdorff dimension -- a remarkable method of assigning dimension to subsets of arbitrary metric spaces. In order to properly formulate the definition and properties of the Hausdorff dimension, we review the critical measure-theoretic terminology beforehand. Finally, we discuss the notion of self-similarity and show how it often defies our quotidian intuition that dimension must always be integer-valued.