An invitation to dimension interpolation

TL;DR

This paper explores the concept of dimension interpolation, which unifies various fractal dimensions into a continuous framework, providing a coherent geometric understanding of fractal complexity.

Contribution

It introduces the idea of dimension interpolation to reconcile differing fractal dimension notions as boundary points of continuous families.

Findings

Demonstrates how classical fractal dimensions can be viewed as boundary points

Provides a unified geometric framework for understanding fractal complexity

Motivates new perspectives on measuring fractal structures

Abstract

A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions of \emph{fractal dimension}. However, even for the simplest examples, the different definitions of dimension may completely disagree about the answer. In this expository article I will examine this phenomenon and use it to discuss and motivate \emph{dimension interpolation}. Dimension interpolation views these classical notions as boundary points of continuous families of dimensions, thus transforming isolated numerical answers into a coherent geometric picture.