Take your favorite spherical object lying around –an orange, a tennis ball, whatever– and cover it with dots. Now draw a bunch of lines connecting these dots, being careful never to let the lines cross. In effect, you're drawing a polyhedron whose faces have been warped onto the surface of your sphere. Add up the number of dots (vertices), subtract the number of lines (edges), and add the number of faces you've divided the surface into, and you'll always get 2. No matter what dots and lines you chose to draw! The object doesn't even have to be a sphere. If you're comfortable drawing all over the surface of your laptop, you'd find the same thing: V - E + F = 2. This is Euler's characteristic formula.
"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...
