What is a shape?

by Daniel Ranard

Topology jokeMaybe you've heard by now about last week's Nobel Prize in Physics, awarded to three physicists for their work on topological phase transitions. But if you didn't already know what a topological phase transition was, chances are you still don't. When a friend of mine read a few popular articles on the discovery, I asked him if he felt enlightened. “No, it felt like the authors were just free-associating: first they said ‘topological phase transitions,' then they said ‘topology,' and then ‘bagels.'” I sympathize with my friend, but also with anyone trying to explain this year's prize. It's true: you can't explain topological phase transitions without mentioning the underlying mathematics, a field called topology. And when you mention topology, you're tempted to talk about bagels. In fact, not long after the Nobel announcement, a Nobel committee member was waiving bagels and cinnamon buns on screen.

Luckily, I'm not going to talk about topological phase transitions. (I'll leave that to the professionals, like Philip Ball at Prospect.) But I am going to talk about bagels. Or really, I want to focus on the mathematical field of topology, which underpins these discoveries. Topology is the study of shapes. And while shapes are interesting in their own right, topology also demonstrates the unique ways that mathematicians conceive of objects and their properties.

First we can ask, what's a shape? Imagine explaining the concept to an alien whose language doesn't have the word for shape. Let's say our alien hasn't even grasped the basic schema of human perception.

Alien: “What's the ‘shape' of an object?”

Person: “The shape of something is just… how it looks.”

Alien: “So the shape of a basketball is orange and one foot long?”

Person: “Well, you need to ignore the color and the size, but…”

We've already learned something. Mathematicians and physicists are often trying to come up with new properties to describe and classify objects, whether they're talking about physical objects or abstract mathematical constructions. Sometimes, you can come up with a new type of description by asking what's left over in your description once you ignore certain other properties. For instance, the vague property of “how something looks” requires us to ignore exactly where the object is in space: we say that two stop signs look the same, even though they stand on different streets. If we picked up one stop sign and laid it on top of the other, they'd be hard to distinguish. That's what it means to “look the same.” Still, it can be hard to specify exactly what sort of description is left over when we choose to ignore certain properties like color and size.

Alien: “Okay, but how do I describe the shape of a basketball? What's left to say, if I can't say it's big and orange?”

Person: “We just call it a sphere.”

Alien: “A sphere?”

Person: “Yeah, like a basketball, but without properties like color or size. Or, just imagine soccer balls, volleyballs, baseballs, you know.”

There's another lesson: sometimes the best way to describe a property like “spherical” is to list all the familiar objects exemplifying that property. If you find it's hard to pin down the Platonic form of a sphere, just start listing spherical objects. This may sound easy, but it can still be hard to specify exactly when or why two objects share a property. Say you draw two squares on a piece of paper, one larger and tilted, to test your alien friend.

Alien: “I'm ignoring their size, but they still seem like different shapes.”

You: “You need to ignore their orientation, also.”

Alien: “How do I ignore something like that?”

So you cut out the smaller square, explaining how you can rotate and enlarge it to make it match the larger square. Now you have a rule for saying when two objects are the same shape: whenever you can rotate, enlarge, or shrink them to match each other. We say that the rotations and enlargements “preserve” the shape. In a sense, by defining the transformations that preserve a certain property, we pinpoint what that property really is. By describing what is allowed to change, we discover exactly what it is that stays the same.

That's the basic scheme: we specify that two objects have the same property whenever we can transform one into the other, allowing certain types of transformations (like rotations and enlargements) so that we can ignore certain properties of the objects (like the original orientations and sizes) while preserving the salient property (like the shape).

This scheme for thinking about objects and their properties is common in modern mathematics. Following these ideas, we are lead naturally to the study of topology. When discussing geometrical shapes above, we chose to ignore the exact position, size, and orientation of the object. Now we choose to ignore even more: we ignore the distances and angles between points on the object. What's left to say then about the shape? Even when there aren't pre-existing words for the topology of an object, we can still talk about collections of objects with the same topology and give names to each such collection. Following the above discussion, we just need to specify when two objects have the same topology. Again, we specify rules for transforming one object into another. Imagining that both objects are made of playdough, we say that two objects have the same topology if we can bend and stretch one object to match the other. Importantly, we aren't allowed to break or tear the playdough, nor are we allowed to smoosh separate pieces together.

Here's where the famous bagel enters. We can bend a playdough ball into a cube if we pinch the corners, and vice versa. But we can't stretch a ball into a bagel, because we can't tear a hole in the playdough. Got it? Here are more examples: We can roll a ball into a long snake and form it into a horseshoe, but we can't smoosh the ends together to make a bagel. Conversely, we can't bend a bagel into a ball, which would requiring smooshing together the hole in the bagel. But we could roll the bagel into a thin hula hoop, or even into a coffee mug. (Here's a pleasing visualization, as pictured at the top.) So we say that a bagel and a coffee mug have the same “topological shape” or topology, whereas the bagel and the ball have different topologies. Hence the joking adage, “A topologist can't tell a donut from a coffee cup.”

Within mathematics there are many other examples of this type of thinking, in which mathematicians identify new properties and classes of objects. But why go through the mental exercise? Considering new properties can be practical. After all, why does a toddler come to intuitively understand the concept of shapes? Say she's presented with different pegs and holes. Soon she learns that square pegs with different colors are still the same in some sense: they can all be placed in the square hole. Shape is a useful abstraction, because objects with the same shape often share other, more concrete or useful properties. In fact, that's why physicists came to learn about topology from mathematicians. Sometimes, an important physical property like the conductivity of some material will be unchanged when the shape of the material is bent or stretched. In these cases, the conductivity of two objects can only be different when the objects have different topologies.

But most mathematicians do not search for new properties and descriptions in order to serve physics or other practical purposes. For them, these exercises are fun and enlightening; they help one understand the essence of objects and their relationships.