Classification of Surfaces

By Ayanna Blake, Lisa Oommen*, Myla Marve, Tamarr Moore, Caylah Vickers, and Lily Zeng.

*Lisa helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

The Girls Talk Math camp is about female high school students from different places who discuss mathematics, mathematicians, and theories. We were split up into groups and were assigned different math topics to learn. Our topic was classification of surfaces, which is listed under the umbrella topic of abstract geometry.

We thought the surfaces project was very interesting and cool to learn about, because it introduced us to college level math and allowed us to understand different parts of geometry. Along with gaining knowledge of surfaces, we also got to learn about other groups topics. Campers presented their topics on the last day and helped us to perceive the significance of the different subjects.

Basics of Surfaces

Before we start off with explaining the basics, we give the definition of a surface, which is an example of a two-dimensional object. When talking about dimensions, basically it’s a way of classifying how many directions of travel an object has.

For example, a line on a piece of paper would be one dimensional because you can only go up or down on that line. A sheet of paper would be two dimensional because you can draw up or down and side to side. A room would be three dimensional because if you imagine throwing ball in the air, it can move up or down, side to side, and forwards or backwards.

When you choose a direction on your surface, go around your surface, and end where you started, if you end up in the same direction as when you started, your surface is orientable, but if you end up in an opposite direction, it’s non-orientable.

Euler Characteristic

One of the topics we learned in our problem set, was the Euler characteristic. The first activity required us to insert the number of vertices, edges and faces of seven different shapes in a table.

After filling in the table, we realized that the number of vertices subtracted by the number of edges and then added to the number of faces, is equal to two for all of these surfaces. The formula is called the Euler characteristic of the surface.

This gives us a way to determine if two surfaces are homeomorphic. Two surfaces are homeomorphic if one if one of them can be bent or morphed into the other without cutting, pinching, or collapsing any part of the shape.

Confused? I know, my group and I had to read over this exercise and ask questions to fully understand this concept. Then, take a look at a solid cube with a hole drilled in the center, called a blocky torus.

Blocky Torus

We found the vertices, edges and faces to detect the Euler characteristic (EC) of the torus. We realized that the Euler characteristic was 0, at that point we moved to a different exercise where we discovered something interesting. We combined two blocky tori and found that the Euler characteristic was -2; then we added another one and found that the EC was now -4.

Blocky 2-Torus

We started to see a pattern that we thought was fascinating and we continued to work. During this process, we learned about the Euler characteristic and how to find it! The Euler characteristic, χ, is the relationship between the amount of vertices, edges, and faces of a surface:  χ(surface) = Vertices – Edges + Faces.

Polygonal Region

The last topic we discuss is to construct a model of any surface using a polygonal region. The polygonal regions are not always squares, it can have a lot of sides. The arrows on each side tells us how to glue each side together, and the direction the arrow pointed at tells us if we should twist the side or not. We can also use words (letters) instead of arrows to help us encode the edges and orientations of the edges. We choose a direction to traverse the perimeter of the polygon, each side is labeled with a correspond letter and a direction (usually counterclockwise); if we traverse it in the opposite direction of the arrow, we use -1 to show it. Keep recording the labels until we return to the starting point. The result of all the letters together is called the word of the polygonal region.