An entry-level primer on “rubber-sheet geometry”

When I studied topology as an undergraduate, I always found it difficult to answer the inevitable question from family and friends:

“What on Earth is topology?”

Every time, I would give them a slightly different answer, but I was never really satisfied with any of explanations. If you’ve ever googled topology, you will have no doubt encountered the animation of a doughnut morphing into a coffee mug. Most explanations I gave involved something related to this animation; how in topology a doughnut and a coffee mug are the same or how a sphere and a cube are the same. But an answer like this doesn’t really tell anyone what topology is really about, what is involved in actually working with topology nor explain why it’s even a worthwhile pursuit.

If you take an undergraduate course in general topology, you might struggle with relating what you’re learning to the familiar doughnut coffee mug animation. The purpose of this essay is to establish the basic concept of general topology and show how this concept is related to this familiar animation and other geometrical ideas. Next, we will have a look at why it is actually useful and interesting to consider doughnuts and coffee mugs to be the same things.

In general, something that I find a lot of people struggle with, myself included, is trying to understand exactly how abstract areas of mathematics can actually have real world applications. After we develop an understanding of topological ideas, we’ll take a look at how topology is, perhaps unexpectedly, related to the way we think about real world concepts. Before this, we’ll take a look at the most basic idea in topology. This is the definition that you will be faced with if you ever open a topology textbook or take an introductory topology course.

Topological Space

A topological space is a set of mathematical objects with the most basic form of structure possible. When we talk about structure in mathematics, we often mean that we have the ability to add mathematical objects together, multiply them together, determine how far away the objects are from each other and many other ideas. Obviously, we can do all of these things with the numbers we encounter every day.

But the structure of a topological space is more basic than the ideas of addition, multiplication and distance. In fact, spaces where we have these things are specific cases of a topological space. Meaning that the real numbers are actually a very specific case of a topological space.

The structure on a topological space is called the topology of the space. All the topology is, is a collection of subsets of the set of mathematical objects, known as “the open sets” of the space. Which specific sets that are contained in the topology defines the structure of the space. This might seem very vague and abstract, but that’s because it is. It’s the most abstract form of structure that we have in mathematics.

It’s not necessary to understand this definition fully, just keep in mind that the topology, and the “open sets” inside of it somehow determine the structure of the space. It’s also important to remember that what makes a topological space distinct from another topological space is the sets that we choose to put in the topology of the space. Below I have given a more formal definition of a topological space if you’re interested, but it certainly isn’t necessary to read.

Definition of a topological space
A topological space (X, τ), is a set X with a collection of subsets of X; τ. Such that:
1. X and the empty set are contained in τ.
2. Any union of sets in τ is also in τ.
3. Any finite intersection of sets in τ is also in τ.

So how does this relate to doughnuts and coffee mugs?

Often, topological spaces can be visualised by geometric objects, such as a sphere:

The topological space that represents a sphere is the set of points such that if you were to plot them in three-dimensional space they would make up a sphere, along with a topology. Recalling that the topology defines the structure of the space, it is the topology that is keeping the sphere together. We can imagine the topology as “the thing that is keeping all of the points from falling to the ground”, making the sphere a single object, not just two hemispheres pressed against one another. Now, let’s suppose we have some other topological space visualised as this object below:

Suppose that the sphere above (Fig. 1) is made of play-dough, then we can easily stretch the sphere into this other object (Fig. 2). Being able to do this with a three-dimensional object means that these two objects are topologically the same. This might seem strange, but ask yourself, what is different about these two shapes? They look different, but if we can easily squish or stretch one into the other, are they actually distinct?

These two objects have the same topology, which means that, even though these two objects are geometrically different, they are topologically exactly the same object. We can stretch our play dough sphere into any strange shape imaginable and all of these shapes are exactly the same shape in the eyes of topology. Okay, maybe not every shape imaginable, there are a few rules with how we can stretch our play-dough:

• We aren’t allowed to rip holes in our play-dough; or
• Take two points of the play-dough and merge them together (we can’t make any doughnut shapes);

If we violate these rules with our stretching, then our two objects are no longer topologically the same. Topologists call this stretching, without breaking our established rules, a homeomorphism, which is just a way of mathematically describing exactly how we mold our play-dough into shapes of the same topologies. So if we can come up with a homeomorphism between two topological spaces, then these spaces have the same topology. This is where the coffee mug and doughnut animation comes in. We can come up with a topological space that describes a doughnut and then, imagining our doughnut is made of play-dough, stretch it into a coffee mug without breaking our rules. So yes, in topology a coffee mug and a doughnut are the same thing, topologically.

Why is a sphere not a doughnut?

Now that we know how we can tell if two objects in topology are the same, let’s now look at how we can tell if two objects are topologically different. There are a number of different properties that topological spaces have that allow us to distinguish them. For three-dimensional objects, such as spheres and doughnuts, the main thing we can use to distinguish objects is the number of holes they have. Two objects are topologically different if one object has more holes than the other. This is because they break our previously established rules for stretching our play-dough. To create a hole, we either have to rip a hole in the play-dough or wrap the play-dough into a doughnut shape and merge two ends together.

Another common way to distinguish between three-dimensional objects topologically, is to imagine walking on them. Consider walking on a sphere for instance. Suppose you start at some point and walk all the way around a great circle on the sphere. Now, after you’ve reached the same point again, rotate 90 degrees in either direction and walk around another great circle. During this second walk around the sphere, you will cross over your first path. This occurs if you do this on any point on the sphere.

This same phenomenon happens on any three-dimensional object that is topologically the same as a sphere. However, on some objects which are not topologically the same as a sphere, there is a way we can do this without crossing the first path. One particular object on which this works is a doughnut, as you can see here:

There are many different properties that are preserved between objects that are topologically the same, but are not necessarily preserved between objects that are topologically different. These topological properties are the main tool used to determine whether two objects are different.

Other topological objects

So far we’ve only talked about topological spaces that can be visualised in 3 dimensions, but a nice thing about topology is that it allows us to easily describe objects that exist in 4, 5 or higher dimensions using the same tools.

One such object that often comes up in topology is the Klein bottle, which if you’re a fan of the YouTube channel Numberphile (or Clifford Stoll!), you will no doubt have seen:

Strictly speaking, we can’t actually observe a true Klein bottle in three-dimensional space, but by allowing it to intersect itself, we can get some idea of what its properties are. In four-dimensional space this object doesn’t actually intersect itself. It’s hard to imagine, but it bends around in the fourth dimension to connect back into itself. The Klein bottle might look like it has an inside and an outside, but you can trace a continuous path from a particular point that goes along the ‘outside’ and ‘inside’ of the Klein bottle, returning to the original point — the 3D representation is the same surface, topologically. Because of this, the Klein bottle has no volume.

An interesting thing about paths on a Klein bottle however, is that if you were to walk along the path described above, when you returned to your original point, you would actually be a mirror image of yourself. This is a topological property of objects that are topologically equivalent (or homeomorphic) to the Klein bottle. So clearly the Klein bottle is not homeomorphic to the sphere or to the doughnut, as no matter how we walk on a sphere or doughnut, we will never be the mirror image of ourselves when we return to our starting point. If an object has this property, they are called non-orientable. Klein bottles are non-orientable and spheres and doughnuts are orientable. Another famous non-orientable surface is the Möbius strip. This object can easily be made with a strip of paper and there are plenty of online tutorials on how to do this.

While the Möbius strip is non-orientable, it is not topologically equivalent to the Klein bottle, but is integral in its construction. A Klein bottle is actually constructed by gluing the edges of two Möbius strips together, attempting to do this in three-dimensional space is actually impossible (you can try it).

Constructing a doughnut from a sheet of paper

A more practical way of thinking about the topology of objects that are hard to visualise in three-dimensional space (such as the Klein bottle), is by considering their gluing diagram. A gluing diagram acts as instructions on how we can construct an object with a certain topology. This construction works by stretching and gluing the edges of a 2D shape.

Before we consider the gluing diagram of a complex shape, let’s first consider the gluing diagram for a more simple shape; a doughnut:

Imagine the square in the diagram is made of play-dough. Next, imagine stretching the square to attach or “glue” opposite edges. When we glue these edges together, we need the arrows to be pointing in the same direction. So we stretch the above diagram out as follows:

The next diagram is similar to Figure 7 except the two red arrows are now in opposite directions. This means that we need to twist the object so that the arrows are pointing in the same direction before we glue the edges together:

The first step in the above gluing diagram is to stretch the square so the the two blue lines meet and we construct a cylinder, just like the first step for building the doughnut. Except now, the two red arrows are pointing in opposite directions to each other, whereas the red arrows for the doughnut gluing were facing in the same direction. This means that we have to somehow twist one end of the cylinder so that the arrows are pointing the same direction before gluing them together. As you might imagine, this is physically impossible. Hence the surface that results from this gluing diagram is physically impossible as well. In fact, it’s a physically impossible surface we’ve already seen, the Klein bottle!

Gluing diagrams are an easy way to see whether an object is orientable or not. We can imagine that walking on a gluing diagram works similarly to how the world in Pac-Man works. When Pac-Man reaches one side of the world, he appears coming out of the other side. If we imagine Pac-Man moving on a gluing diagram, when he enters one side, he will emerge from the other side of the same colour, with the arrow determining his orientation.

Suppose Pac-Man enters the right-hand side of the torus gluing diagram, then he will emerge from the left hand side unchanged. This is how the topology of the normal Pac-Man world works.

Now consider if Pac-Man enters the right-hand side of the Klein bottle gluing diagram. Then Pac-Man will emerge from the left hand side as a vertical flip of himself:

So gluing diagrams allow us to easily consider some topological properties of objects that would otherwise be difficult to work with.

Why is topology useful?

Topology is actually surprisingly useful in the world of statistics. An emerging research area in statistics is topological data analysis. Useful data has some sort of structure, in the form of patterns or trends, and data analysis is essentially the process of uncovering this structure. Finding structure in data can often depend on how we view it, that is, what statistical tests are used, which variables we compare to other variables and what visual representations we use.

From topology, we know that things that look completely different can actually have the same structure. This idea can apply to data as well, as data can appear completely different depending on how we view it, even though we are always dealing with the same data.

In topological data analysis, the structure of data is treated topologically. We know that a topological property is a particular feature of an object that is preserved through objects that are topologically the same. So when performing topological data analysis, we search for certain properties that persist in the data after viewing it in a variety of different ways. We can think of this as being analogous to stretching our data like it’s play-dough. By doing this, we can determine the real structure of the data and distinguish it from any dependence on how the data was observed.

This is just one of many applications of topology in the so called ‘real world.’ Other applications of topology also involve determining whether things that look different are actually the same. This is very important in situations where different people might choose to represent the same information in different ways. A few examples of things that have various different representations include, molecular structure, geographical maps, DNA structure and knots in ropes or string.

It can be hard to see initially, but topology is the foundation for most areas in mathematics. Defining exactly how topology is ‘used’ is quite difficult, as it’s so ingrained in the way mathematics works that often we don’t even notice we are using it. Only relatively recently has topology been considered and researched independently to other areas of mathematics and new research and applications of topology continue to arise.

Epilogue

I hope that after reading this that you can now appreciate that a doughnut and a coffee mug being the same is a much deeper and useful idea then it might initially have seemed. If you’re interested in digging deeper into topology, here are some recommended sources:

1. How Is Topology Applicable to the Real World?
2. Some more in depth notes on Gluing Diagrams by Maia Averett
3. Sidney A. Morris’ Free General Topology Textbook: Topology Without Tears

Disclaimer: All graphics in this story without a source were created using Wolfram Mathematica.