Turning surfaces into polygons

In a previous post I introduced some basic ideas in topology, one of which is to subdivide a surface. Given any surface, we can divide it into “polygons” or “regions” consisting of a certain number of vertices, edges, and faces. This works in all the ways we expect, faces are bounded by edges and edges meet at vertices. Thinking about these kinds of subdivisions we can devise a way to represent particular surfaces by particular polygons.

Look at the torus again and consider two possible subdivisions:

Both subdivisions have one face but the first has two edges and one vertex while the second has three edges and two vertices. Using the first subdivision we can see the torus as a square and using the second we can see it as a hexagon with edges and vertices labelled the same identified:

Doesn’t make sense? Look at glueing the edges that are identified back together to see how we recover our original shape.

Glue the blue edges together to create a cylinder and then glue the red edges together.
Glue the green edges together to create a cylinder, twist 180 degrees, and then glue the blue edges and red edges together respectively.

These types of transformations and representations can be a fun way to better understand and study our surfaces. Think about what we can do for other surfaces. For example, how can you represent the two-torus (like a torus with two holes) as a polygon?

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