Mathman Explains: Geometry to Topology

 

If you clicked on this link to learn about topology after seeing the word Mathman, you clearly aren’t looking for a serious article that throws a bunch of formal definitions at you. Especially when it comes to higher level concepts in mathematics, there are few sources that attempt to give an intuitive explanation without assuming that the reader is basically a math major.

Luckily for you, Mathman and I assume you only know the following:

1. Basic geometric shapes (circles, squares, triangles, etc)
2. How to describe a shape (How many sides does a shape have, etc)
3. How to change the characteristics of a shape (By rotating a shape, moving the shape around a space, changing the size of the shape, etc)

So, assuming that you remember some basic geometry (even if you don’t, many geometric concepts are reexplained along the way), let’s get back to the question at hand.

What is a topology?

This can be a difficult question to unpack. What does the word even mean? Well, if we look at the root words, topos- means “place, region, or space” and -logy means “study of”. So topology must be the study of places, regions, and spaces.

That doesn’t help us much. What are regions? What are spaces?

Perhaps we should ask our friend Mathman for some help…

So, based on what Mathman just explained, regions and spaces seem to be connected to what we normally think of for geometry. In order to figure out how topology connects to geometry, we need to understand basic geometric properties. Looking at an example, figure 1 shows a triangle in the xy plane. How would we describe this triangle?

Figure 1: Triangle in the xy plane.

Well, we can see that:

1. The triangle has a black outline
2. The triangle has 3 sides
3. One of the angles is a right angle
4. The right angle is the angle closest to the bottom left corner
5. The triangle is near the bottom left of the plane

Only a couple of these properties are geometric properties, which are properties that are true for all congruent triangles. Well what does congruence even mean? How would we go about determining if two triangles are congruent for example? Maybe we should ask Mathman for help, hopefully he is not too busy…

So there are various methods of determining congruence. This is a good guideline for us as we approach this triangle, but we will see later on how we might be able to bend these rules with topology. Looking back at figure 1, we can see that #2 is a geometric property since all triangles have 3 sides. We also know that #3 must be a geometric property as all right triangles have an angle that is 90 degrees.

This seems easy right? You might be thinking, how could all of these properties not be geometric properties? Well, let’s look at #4. This property is essentially stating that all right triangles must also have the right angle at the bottom left corner. However, looking at figure 2 below, we can see that this other triangle must also be a right triangle despite violating #4!.

Figure 2: Similar triangle in the same xy-plane.

So #4 can’t be a geometric property! Similarly, we can show that #1 and #5 can’t be geometric properties! For #1, the outline color of the triangle wouldn’t affect whether or not it is a triangle! Similarly, #5 is not a geometric property since the placement of the triangle wouldn’t affect whether or not it is a triangle! Generally, if you are looking at a potential characteristic that doesn’t focus on the properties of a triangle (like the angles or the lengths of the sides of the triangle), then it wouldn’t be a geometric property.

Secretly, what I’ve done between figure 1 and figure 2 is demonstrate two out of the three ways that we can change a shape without changing the geometric properties.

 
One of the ways I’ve changed the triangle above is performing something called a rotation, where I spun the triangle around until the right angle is in the bottom right corner.

Figure 3: Demonstrating Rotation

Secondly, I performed something called a translation, where I moved the triangle closer to the bottom right corner.

Figure 4: Demonstrating Translation

The last way I could potentially change this shape is performing a reflection, where we can flip the triangle horizontally or vertically.

Figure 5: Demonstrating Rotation

Now that we know what geometric properties are, we can finally learn what a topology is! The main difference between geometry and topology is how we can change the shape. Unlike Geometry, where we can only change a shape by rotation, translation, reflection, and dilation, changing shapes in topology is more like reshaping play doh.

With play doh, you can squish, stretch, twist, and squeeze a shape into another shape! We would say that two shapes are topologically equivalent if they are only formed via the play doh method.

One of the most famous examples within topology is reshaping a donut into a coffee cup!

However, much like with geometric properties, there are still limitations to creating new shapes within topology. For example, “cutting or breaking off” the play doh is only allowed if the two pieces are reconnected “near” each other without adding any additional twists. Formally, we would say that the points we cut off are in the “neighborhood” of each other. So, when we are looking at the shape before and after the cut, the area around the cut should vaguely look the same, much like if you were looking around with beer goggles on! Let’s look at an example:

Here’s an example where cutting doesn’t work! Here in figure 6 is a cylinder with a red line going across one side. If we cut this cylinder in half, turn both sides in opposite directions, and reconnect the cylinders, we get the new cylinder in figure 7.

Figure 6: Cutting and Turning a Cylinder

Figure 7: Final Result of Figure 6

As we can see in figure 7, the red line on the cylinder is disconnected now. Since we have turned the cylinder in such a way that the lines are not within the "neighborhood" of each other, then this violates the rules we’ve established with topologies.

Let’s take a closer look at another example with the cylinder where we don’t violate the rules!

In figure 8, we start out with the same cylinder from figure 6. However, instead of just cutting and spinning the two smaller cylinders around, we are going to twist them. (Remember, we can think of these two smaller cylinders as pieces of play doh!)

By twisting the smaller cylinders, the red line also twists around the cylinder. So, if we twist both cylinders an equal amount in opposite directions, we can reconnect them to get figure 9.

Figure 8: Cutting and Turning a Cylinder

Figure 9: Final Result of Figure 8

As we can in figure 9, the red line on the cylinder twists around (with the red dotted line indicating that the line is on the other side of the cylinder) and isn’t disconnected! So this satisfies our topological properties.

Much like with cuts, another limitation we encounter is that we can’t violate the laws of physics when making a new shape. We can't add or remove dimensions, or get rid of some of the play doh in order to make a new shape! For example, we would never consider a pillowcase and a pillow to be the same object, so we have to apply this same logic to topological structures.

Looking at a specific example, we have a hollow square. Assume that the line that makes up the hollow square is made of a thin piece of string (so it is one dimensional). Is there a way to make this a filled in square?

Spoiler alert, there isn’t. Since we have established that the hollow square is made from a piece of string, we can’t just stretch the edges of the square to try and fill the square in. Since we can’t cut the square without reconnecting it, we can’t just cheat and unfold the hollow square into a line and try to squish it into a filled in square. Why is this true?

We can think of a hollow square as a line segment that is bent into a square, which will make the shape 1D. The filled in square however is in two dimensions. So, as we stated earlier, we cannot violate the laws of physics!

So what are some other examples in different dimensions?

Here’s a few in 1D:

Here’s a few in 2D:

Here’s a few in 3D:

As you can imagine, there must be endless shapes that we can create within topology that are similar to one another. Formally, we would say that these shapes are homeomorphic to each other (which means “similar shape” by its Latin root words).

Overall, topology is a newer field of math that thoroughly explores how objects can be manipulated in different spaces. There are many cool applications of topology, ranging from robotics to stock markets! Even if you aren’t interested in learning more about Topology, I hope that you at least learned something fun and new!