Topology Apology

In lieu of actually delving into this, I’ll just give a short description and let you go look it up on Wikipedia.

There’s a way of classifying or generating seemingly complicated surfaces using a sort of surface (or shape, I guess) that isn’t really that complicated.

The way you do it is this: Take one vector of the polygon, say the top A vector, and glue it head-to-head and tail-to-tail with the other A vector. Now do the same with the B vector. What you literally get is a kind of weird, oriented, triangular surface. In topology-land, of course, you can continuously deform this weird pseudo-prism into something we all know and love, namely a sphere. Or, in the fancy terms, you’re utilizing a homotopy equivalence between the two surfaces. The same thing can be done for arbitrary surfaces, as well, provided they are also closed.

For example, here’s a torus:

This is a neat way to visualize certain surfaces, especially when they cannot be embedded in three-dimensional space without self-intersections (such as the Klein bottle, or the real projective plane).

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