Is Math Big or Small?

A mental image, and a question

Imagine a torus. Hold that picture in your mind for a second. Is math something you can hold in your hand, or something you wander around inside? That is the question at the heart of a recent writeup and talk about scale in mathematical illustration — a short, thoughtful riff on how researchers use imagery to think. The post is an extended version of a workshop talk, and you can also watch the video; the author wants us to notice the collective hallucinations that shape how a field imagines its own objects.

Paint cans, taffy, and a tiny doodle

Flash back to 1971 at UC Berkeley. Evans Hall — big, brutalist, famously unpopular — had just opened, and it has been reported that students and faculty staged a paint‑in to liven up its concrete corridors. A young Bill Thurston brought a notebook sketch to Dennis Sullivan and asked, “Do you think this would be interesting to paint?” “You bet!” was the reply. They painted a single simple closed curve in a thrice‑punctured plane; the curve stretches, folds, and limits to parallel strands called laminations. Small moment. Big consequences: those doodles grew into the language of pseudo‑Anosov maps and a new way to think about dynamics.

Collapsing strands into tracks — why scale matters

Thurston’s clever move was to collapse parallel strands into a single “train track” — ties across the lamination are contracted to a central graph that can split and merge like railway switches. The imagery is irresistible: taffy pulled by candy machines, a little doodle of a train puffing along, cartoon steam and all. Artists such as Conan Wu have since leaned into that evocative name to make the idea visible again. Topologists often reach for metaphors like “geography” and “botany” to set scale; those metaphors are flexible, practical tools that tell you whether you’re sketching something to hold or a landscape to explore. Which do you prefer — the pocket map or the atlas? Either way, a good picture can change how you think.

Sources: chessapig.github.io, Hacker News