On the importance of illustration for mathematical research

TL;DR

Illustration plays a crucial role in mathematical research by transforming ideas into physical or visual forms, which enhances understanding, stimulates questions, and can make mathematics more experimental and accessible through modern technology.

Contribution

This paper highlights the historical significance, recent advances, and disciplinary perspective of illustration in mathematics, emphasizing its impact on research and understanding.

Findings

Illustration challenges and deepens mathematical understanding.

Modern technology enables complex models accessible to individual mathematicians.

Illustration fosters experimental and exploratory approaches in mathematics.

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.