By the American Mathematical Society
Transcribed from the AMS interview with Richard Evan Schwartz on Playing with Shape and Form.
Playing with Shape and Form gives a playful and intuitive introduction to topology, favouring simple and crisp illustrations over long explanations and formal definitions. The material includes some of the classic results one would see in undergraduate courses on topology and graph theory, but presents these results in a way that readers without any knowledge of mathematics will be able to understand.
What are the key concepts introduced in Chapter 2, and why are they important to the overall theme of the book?
In Chapter 2, the key concept is the idea of a path. A path is something drawn on a piece of paper or a plane which connects two points – A and B. There is a path, but in another case, the path could be more complicated. The reason why paths are important is because they are the simplest thing you can do after drawing points. The next thing you can do is draw paths between them. Paths are the building blocks for more complicated objects like networks of paths. You might, for example, have a polygon shape where the paths are the things that go around the edge. For most of the other objects that come up in the book in later chapters, paths are the building blocks.
The second idea in Chapter 2 is the idea that a path without any crossings doesn’t separate the plane in any way. For example, if you draw a path and have points on either side, you can connect them and avoid the path. The second chapter is setting up paths as non-separating things. What this property implies is that if you have any number of points given in a particular order, you can connect them without any crossings. This non-crossing property is a consequence of this non-separating property. I focus mostly on connecting points in order without crossings, as I had some experience talking to kids about this and I found that they liked this as a game – being able to connect points in order even if they were in very complicated positions, without any crossings. At first it seems really surprising, and it’s also a maze-like puzzle, and so I thought it would be fun to present it this way.
Why is the order in which we connect the points important?
I’m going to start with a sports analogy. Imagine you’re playing tennis, but you change the rules and remove the net. It would be way too easy and it wouldn’t be an interesting game. The same would apply with basketball if you brought the hoop down to chest-level. The idea is that in sports, the rules are set up to get an interesting game. In terms of connecting points, imagine I draw a bunch of points and I give them numbers. Now, imagine I’m supposed to connect them without any crossings but I don’t care about the order. This is something you could easily do by sweeping right. You could always do this, no matter how the points were arranged. If you didn’t care about the order, you could connect them by one of these zig-zag paths which sweeps to the right.
But let’s say you’re forced to connect them in a given order. Then it becomes a more interesting question and it’s no longer so obvious how to do it. In order to connect these points in order, I have to draw a more interesting path and there’s some amount of planning. The idea of connecting points in order was to get a more interesting construction. A better game, so to speak.
What’s one key takeaway that you hope readers take away?
The key takeaway that I hope people get is this idea that in topology, general inter-relationships between points and curves is more important than exact position. When people are playing around with drawing these things, it will occur to them that it doesn’t matter so much where the points are. The fact that you can always do something tells you that the exact positions don’t really matter.
Imagine that the whiteboard is made of honey and is mixable with points embedded in it like poppyseeds. I could mush the plane around and the points would start moving around. If I had a path that connected them without any crossings, then as I moved it around, I wouldn’t create any crossings. Now, imagine that I’ve got these points in there in a crazy order and I start stirring things up and moving them until they’re all in order on a straight line so that I can connect them up in an obvious way. If you imagine undoing what I did, the line segment that connects them up would gradually swirl around, leaving a very complicated non-crossing path that connects them. If you start to think about the plane as more flexible than you would in geometry, then for all intense and purposes, you might as well think of the points as being in some direct order. This is activating the topological part of your brain.
This may sound strange, but we do this all the time in real life. Let’s say, you’re talking to your friend or want to recognise your friend. You don’t care about the fingerprints of your friend or the number of hairs they have on their head. In order to really recognise people, you have to forget detail. In almost every aspect of your life, you have to forget. This topological way of thinking and of selectively forgetting details is one of the underlying things I want people to take away.
Find out more about topology in Playing with Shape and Form.
Watch the full AMS interview with Richard Evan Schwartz here.