Cutting up the Two Holed Torus Chantel C. Blackburn Department of Mathematics University of Arizona MATH 520B

Hi there! I’m Mr. Super Torus Guy.

I’m going to show you how to turn a torus into a polygon!

What’s a torus, you say?

Take a close look at my Mr. Super Torus Guy Mask!

Mr. Super Torus Guy here again.

Now we’ve represented the two holed torus by a 12-gon.

We know from class, the two holed torus is represented by an octagon.

Where does the octagon come from?

We will find it by reducing the 12- gon to the desired form by eliminating all but one of the vertices.

By relabeling, we obtain an octogon of the correct form.

At this point, one might ask, “Does it matter HOW we cut up the torus?”

The answer is, “No.”

However, you may need more reduction techniques than will be presented here.

Let’s first demonstrate our main reduction technique, then we will give some examples.

The Symbol of a Diagram

Reduction Technique

Our First Example

Another way to skin a torus…

Yet one more way to skin a torus…

The End Sources: Classification of Surfaces, Richard Koch, November 20, 2005