How a Coffee Cup = A Donut

Have you ever looked at your donut and your coffee cup and realized you couldn't tell the difference? Maybe not, but this is one of the oldest topology jokes in the book. Today you are going to understand why.

Let's start by diving into what a torus is. A torus is described as a shape with a genus one or a "hole." It is most commonly referred to as a 3d shape. A donut is usually used as an example; however, it is better described as the glaze or surface area. There are three main types of torus: ring, horn, and spindle. A ring torus is the most common but it is important to understand how the other types are made. Imagine a circle on a flat plane. You then rotate that circle around the y-axis resulting in a new 3-d shape. A ring torus's rotation point it not tangent to the original circle, a horn torus has rotation point tangent to the original circle, and a spindle torus has a rotation point inside of the original circle.

Ring, Horn, and Spindle Torus

Even the spindle torus which appears to have no hole is still classified as a type of torus due to the way that it can be formed.

In order to study sets, one important thing in Topology is representing manifolds as something flat or simpler. The torus can also be represented on a flat surface.

An example of this is Pac Man. When Pac Man reaches the end of the board (Blue) he continues on and ends up as the other Blue. The same if he reached the top Red and reappeared at the bottom Red. The arrows represent the sides of the paper that are glued together. It would require stretching and deforming(which we will later discuss), but this is a clear and easy representation way to think of a torus.

Now back to our original problem. The general rule in topology when looking at two different sets is that you can continuously deform shapes without tearing or gluing the object. Both the coffee cup and donut only have one hole.  You can visually and physically change the cup into a donut if you make a coffee cup out of clay then mold the clay around the hole until you end up with a solid ring torus.

There are many more classifications determining and proving that the shape of a coffee cup and a donut is the same topological space that I may dive into in future posts. So next time you reach for your breakfast item, make sure that you have the right one.