Torus Mug Black Edition

“But wait,” I hear you say, “you’ve misprinted the formula on the mug!”  Quite the contrary, my friend.  If you were to play this same game on the surface of a mug you’d find that the formula changes.  Instead, as you add the vertices subtract the edges and add the faces, you’d always get 0.  Likewise if you tried this on a bagel, or any other surface which has one hole in it.  Or rather, this is true as long as none of the faces you draw end up with a hole.  So you’d have to make sure some of your drawing happens on the handle, for example.  This expression, V – E + F, is a sort of topological fingerprint for a surface. You may have heard that to a topologist, a mug and a doughnut are the same thing, in the sense that one can be continuously deformed to make the other.  The fact that V – E + F = 0 on both these surfaces is one way to make this similarity something a bit more quantitative, and hence something you can actually use for computations and proofs.

You can find a description of Euler’s formula in this video.  My challenge to you is to take the argument in that video, and think through what changes on surfaces with a hole, like a doughnut or a mug.  If you do wrap your mind around it, you’ll find that this is a remarkably clever way to codify the idea of how many “holes” a surface has in it.  And if you feel the need to get empirical with an actual mug, I know of a place where you can buy one…