What is a Torus?

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In my last post on topological objects we saw how to construct a Möbius strip by gluing together two edges of a rectangle with a half twist. In this post I will show you how to construct a torus using a similar type of construction. A torus is often described as a shape which looks like a donut.

To create a torus we will start with a square with arrows on the edges as follows:

Recall that the arrows tell us which sides to glue together and which orientation to do it with. In the picture above the arrows show that we should identify the left and right edges of the square and the top and bottom edges of the square while preserving the direction of the arrows.

Let us do this one step at a time. Note that it does not matter the order in which we do this, we will get a torus either way. If we identify the top and bottom edges of the square first the when get a horizontal cylinder as seen below. Notice that doing this identifies the vertices on the left side of the square and the vertices on the right side of the square so that the left and right edges of the original square are now circles on the left and right of the cylinder.

For the next step we will identify these left and right circles to get the torus as below.

We have drawn a dotted circle in the interior of the torus to show how we glued the two ends of the cylinder together. If we cut the torus along this dotted circle we would be able to unfold the torus and get our cylinder back. We can now see why the torus is sometimes described as having the shape of a donut.

The torus is an important object of study for mathematicians because of several properties that it has. The first important property is that the torus is orientable. This is in contrast to the Möbius strip which is nonorientable. This means we can put a consistent orientation on the torus. The sphere is the simplest orientable surface so the torus is a simple but non trivial example of an orientable surface.

It is easy to see that the torus has a giant hole in the middle of it. The genus of a surface is defined to be the number of holes that it contains. Since a sphere has no holes its genus is equal to 0 and since the torus has a single hole its genus is 1. We can construct a surface of any genus by attaching what are called handles to a torus to increase the genus by one for each handle added. For example if we add one more hole to a torus we get a surface of genus 2 that has two holes and which looks sort of like the number 8.

Another important property of the torus is that it has no boundary. If you remember in the case of the Möbius strip the boundary was just a circle. Also the torus is bounded which means it is contained in a sphere with finite radius. A surface which is bounded and has no boundary is called a closed surface and so we see that a torus is a closed surface.

The torus has many other interesting properties that are harder to describe. For example, we are able to put a Euclidean structure on the torus which allows us to measure the distance between two points on the torus in the same manner as can be done to measure the distance between points in the Euclidean plane using the Pythagorean theorem. This is accomplished by tiling the plane with squares and then realizing the torus as a quotient space of the plane.

In this post we have learned how to create a torus from a square and briefly discussed some of its properties. We have learned that the torus looks like a donut and is a closed orientable surface with genus 1.

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References:

https://en.wikipedia.org/wiki/Torus
http://mathworld.wolfram.com/Torus.html

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All images in this post were created by myself using Inkscape