Topology Is the Geometry Underneath Geometry

​Probably one of the most understated illustrations of anything in science is the classic coffeecup-donut transformation. It's visualized below, but the idea is that the donut can be bent around and twisted such that it becomes the coffeecup without actually breaking the material apart (assuming it's made of something more flexible than fried flour and sugar, of course). A stretch here, a twist there, and a mug it is.

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The property being demonstrated is called homeomorphism and it has to do with topological spaces. Topology is an enormous realm of thinking and it's implicated in everything from algorithms and robotics to astrophysics and biology. But it all comes back to the donut because topology is, generally, the study of shapes and transformations, which is also the study of boundaries and sets. It's inescapable.

While topology is the study of shapes, it's ​not the study of geometry in any usual sense. That is, it doesn't care about distance and volume and angles and coordinates. Instead, it's interested in shapes as shapes are representations of groups or sets. A shape here is a collection of things or properties and so long as that collection is left intact, the shape is intact, no matter how different it looks. The shape of the donut, properly known as a torus, is different than that of the coffeecup but, topologically speaking, we can say the relationship is invariant. The same.

Invariance is everything, it turns out, and our ("our") intuition about this is shallow and primitive. We call a square a square and a circle a circle at our peril, when, in a more complete view of reality, they are more. They both live in two dimensions, for one, and they both divide a two-dimensional plane into two parts, one inside the shape and one outside. That seems like an awfully important similarity, and one that holds no matter how many lines make up the edges of the two shapes and what the angles between them are so long as there are definite insides and outsides. The circle can be homeomorphically transformed into the square, and vice versa.

The mathematician Leonhard Euler provided an even better example than circles and squares way back in 1735, called the Seven Bridges of Königsberg. The starting problem behind his example was to simply come up with a way to cross every bridge linking every island in the city of Königsberg (today Kaliningrad) only one time.

This, he proved, was impossible, but the point was (or is now) to show that the problem had nothing to do with distances between the bridges or their lengths, just that they had the property of connecting two zones. One bridge could stretch from here to the moon and another could be a mere micron in length, but they would be the same—members of the same union. The bridges defined relationships, and it doesn't matter how they did it or what they looked like.

So, the shapes we make in topology are generally sets and these sets could be defined by anything we choose. It's the geometry of whatever, which is huge.

So we can make a topological space be anything. All we need are some rules or axioms relating things to other things and, there it is, a shape. So, our shape is based on some property of the set that doesn't change under transformation, which is a bit like saying that the transformation can be undone or reversed.

The set of people with red hair is the same whether the set is all women or all men or whether it consists of people who are all doing headstands or who are all holding their breath underwater. They have read hair and are so part of a particular shape or, better, a particular space.

We get these spaces/shapes through connections among discrete items. So, if we were to try and morph our donut shape into a circle, filling in its hole, we'd be rearranging it in a way even more profound than if we allowed it to keep its donut shape but stretched it out in every direction for a billion miles. It just needs to keep that hole. Likewise, if we ripped a hole in the middle of a circle, we'd be changing how it's interconnected, fundamentally. Changing a line to a point is changing what it is, while extending the line another billion miles is changing how it is.

Topology is sometimes explained to be a subject of pure mathematics—total abstraction—but it's not hard to see how this extends into the real world. Graph theory, for example, is a way of constructing IRL topographical spaces of things (any things) and relationships (any relationships) in meaningful ways, whether it's in devising better algorithms or uncovering the patterns within biology.

In robotics, topological spaces called ​configuration spaces are used to determine all of the various possibilities for motion. Meanwhile, the universe itself is a topological space (a spacetime) that finds itself deformed in all kinds of interesting ways by gravity, yet remains fundamentally the same big donut, even if it looks an awful lot like a coffeecup from Earth.