Is Math Big or Small?

Is math big or small?

Summary:

When Illustrating a mathematical idea, the first thing you need to decide is the scale. Is this concept something you can hold in your hand, or something to wander around in? I will reflect on the scale of various analogies used by research mathematicians, such as Thurston’s train tracks and pictures of symplectic manifolds. Topologists use the metaphors of “geography” and “botany” to organize problems in their field. I will argue that geography and botany are flexible analogies, which give a natural scale for mathematical illustrations.

Presented at:

• Rigorous Illustrations - Their creation and evaluation for mathematical research, IHP trimester on mathematical illustration

The post below is a slightly extended version of my talk at IHP workshop on rigorous illustration. You can also watch the video.

Is math big or small?

Imagine a torus. This is not rhetorical, I want you to summon the mental image of a torus.

• How big is your torus?
• Why? Ive always struggled with this question. Every time you illustrate a torus, or indeed any mathematical idea, the first thing to decide is the scale. Scale is more than physical size. In this picture, the word “big” feels large because it dwarfs our friend by the I, whereas “small” feels small because its much smaller than the critter analyzing it. Scale is relative to the viewer.

Is math something to hold in your hand, or something to wander around it? To limit our scope, we’ll look at the scale of mathematical imagery used by researchers. I’m interested in those collective hallucinations which are embedded in the way the community conceptualizes mathematical ideas. In this post, we’ll go through several anecdotes of math, big and small, and discuss how the scale effects the illustrations.

Table of contents:

• Train tracks
• Big math in symplectic topology
• Geography and botany, and the isoperimetric island
• Using geography and botany
• Gallery

Train tracks

This story starts in 1971 at UC Berkeley, which had quite a colorful math department. Back then Berkeley students and faculty still held the spirit of 1960’s counterculture. Administration had just constructed a new math building, Evans hall: big, concrete and brutalist. The poster child for ugly college buildings. Here’s a picture:

This is a thumbnail for a youtube video about such buildings, with Evans hall front and center. The mathematicians were not pleased about the prospect conducting their creative work in an uninspiring box.

After a rabble rousing talk on fascism and architecture, students and faculty grabbed brushes and paint-cans. They staged a paint-in, covering the windowless hallways with mathematical murals. A young grad student named Bill Thurston approached topologist Dennis Sullivan with a sketch from his notebook, and asked “Do you think this would be interesting to paint?” “You bet!” Here is their mural:

Mural from Evans hall

This is a picture of a single simple closed curve in a thrice punctured plane. Thurston built this curve iteratively, starting with a curve enclosing just two of the holes and “braiding” the holes around one another.

The curve gets stretched like taffy, folding over itself to form the curve of the mural. The mechanism is identical to Industrial candy makers. Thurston noticed that the curve quickly limits to sets of parallel strands which he called “laminations”. Sometimes the strands split apart into two groups, other times two groups merge. The mural ordaining the seventh floor of Evans hall showed Thurston’s nascent explorations of laminations, which would later blossom into his very influential theory of pesudoanosov maps. For Thurston’s description of the mural, see the last page of what is a train track.

Thurston encoded the structure of these laminations by collapsing the parallel strands into a single “train track”, a curve which can split or merge like the switches of a railway.

A real set of train tracks

Here’s the train track associated with Thurston’s mural. To build it, we lay “ties” across the lamination, then contract the ties to a central track.

The first picture is a lamination. The second picture is the train track, formed by contracting parallel strands of the lamination.

The term “train track” was coined by Thurston alongside a handdrawn doodle of a little train puttering along these tracks, complete with a white puff of steam.

How might we illustrate the idea of train tracks? Thurston made this task easy by naming it so evocatively, even giving us a sketch. This doodle inspired Conan Wu to draw this lovely piece:

This shows a train track winding around some higher-genus surface. This piece invites the viewer to “ride the train”, following the closed curve and bearing left or right at each switch. The very name, “train tracks” sets the scale. (Thurston famously believes that the best way to view a 3-manifold is by standing inside. Math is big for Thurston). Conan’s piece renders this beautifully, with the train encircling a planet plucked from “Le petit prince”. Here is her piece with an added person for scale.

This is a prime example of Big Math™. We know it’s big because the surface dwarfs the viewer, represented by our little friend. (By the way, the artist Conan Wu is living an very full life after her PhD, including a period as a concept artist and landscape painter at Disney.)

But what if we chose a different scale for our illustration? Here’s my take on the train track picture.

My picture of train tracks

I made the train a toy train, so the whole train track is something you can hold in your hand, or put under a Christmas tree. This is small math. This smaller scale begets different analogies. Instead of a old school locomotive to ride in, it’s a wooden train you push around. Instead of metal tracks, I used those wooden tracks that we played with as a kid. Where Conan’s planetary train track suggests emotions of awe or wonder, the toy one suggests fun and puzzles. Completely different emotional effects, for mathematically identical pictures.

Including the critter for scale, they are omnipotent. Instead of controlling the train as in Conan’s picture, the viewer is invited to piece together the tracks like legos. This emphasizes the combinatorial nature of train tracks. In contrast, in Conan’s picture the viewer is beholden to the shape of the planet and its tracks, Conan’s train revels in geometry.

Which scale should we pick? Is math big or small? I have two conflicting axioms about myself:

• I like small things
• I am a small These are in conflict, for if I am small and surrounded by small things, then I am the same size as my things. Is math one of my little things, or something to immerse myself in? Let’s see what big math has to say for itself.

Big math in symplectic topology

For my day job, I study symplectic topology. I want to tell a story about one of the leaders of the field, Yasha Eliashberg. This will need some mathematical context, which I will try to keep light. Symplectic geometry is, succinctly, the study of symplectic manifolds. These are even dimensional manifolds with extra geometry called a symplectic structure. Here I represent the symplectic structure by the little critters climbing over the manifold.

A symplectic manifold, crawling with symplectic forms.

The field kicked off in the 80s with the introduction of a tool called “pseudo-holomorphic curves”. These are two dimensional surface inside the symplectic manifold, stretched tight like a film of soap.

A pseudoholomorphic curve living inside a symplectic manifold

We understand symplectic manifolds by probing them with these pseudoholomorphic curves. A symplectic topologist makes their living by counting pseudoholomorphic curves, and using these counts to distinguish symplectic manifolds.

Yasha Eliashberg is one such symplectic topologist. Yasha is a 5’3, somewhat squirrelly yet animated fellow. In a conference held in his h