An invitation to dimension interpolation
Abstract
A 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 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 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.