## Fun with Homology

There is a theorem (currently being attributed to Wikipedia, but I’m sure I can do better given more time) which states that

*All closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2n-gons) can be glued to make different manifolds.*

*Conversely, a closed surface with non-zero classes can be cut into a 2n-gon.*

Two interesting cases of this are:

- Gluing opposite sides of a hexagon produces a torus .
- Gluing opposite sides of an octagon produces a surface with two holes, topologically equivalent to a torus with two holes.

I had trouble visualizing this on a piece of paper, so I found two videos which are fascinating and instructive, respectively.

**The two-torus from a hexagon**

**The genus-2 Riemann surface from an octagon**

I would like to figure out how one can make such animations, and generalizations of these, using Mathematica or Sagemath.

There are a bunch of other very cool examples on the Youtube channels of these users. Kudos to them for making such instructive videos!

PS – I see that $\LaTeX$ on WordPress has become (or is still?) very sloppy! 😦

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