Least Action

Nontrivializing triviality..and vice versa.

Archive for October 2016

Fun with Homology

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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 n non-zero classes can be cut into a 2n-gon.

Two interesting cases of this are:

  1. Gluing opposite sides of a hexagon produces a torus T^2.
  2. 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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Written by Vivek

October 20, 2016 at 21:57