On the importance of illustration for mathematical research

Abstract

Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical idea into physical form or experience, including hand-made diagrams or models, computer visualization, 3D printing, and virtual reality, among many others. The very process of illustration itself challenges our mathematical understanding and forces us to answer questions we may not have posed otherwise. It can even make mathematics an experimental science, in which immersive exploration of data and representations drive the cycle of problem, conjecture, and proof. Today, modern technology for the first time places the production of highly complicated models within the reach of many individual mathematicians. Here, we sketch the rich history of illustration, highlight important recent examples of its contribution to research, and examine how it can be viewed as a discipline in its own right.

arXiv comment16 pages, 17 figures

References in corpus (7)

• Honeycombs and sums of Hermitian matrices
• A chiral aperiodic monotile
• Subfactors of index less than 5, part 2: triple points
• Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS${}_3$/CFT${}_2$ correspondence
• Spelunking the Deep: Guaranteed Queries on General Neural Implicit Surfaces via Range Analysis
• Direct Construction of Aperiodic Tilings with the Hat Monotile
• Continued fractions over non-Euclidean imaginary quadratic rings