Topological Manifolds

Something I doubt many people outside of mathematics know about, is that there is a whole genre of mathematical proof that relies just on pictures and diagrams. Of course, they rest on some pretty strict technical definitions, but an image or a drawing can be a proof if everything in it is well defined.

I am as fascinated by the diagrams in manifold theory as I am by the mathematics itself. Manifolds are the generalization of "surfaces" in three-dimensions (spheres, planes, donuts etc). There are quite a few ways to talk about manifolds. You can think of them as purely abstract "topological" things that just describe how a space is connected (i.e. a line versus a circle). You can also think of them as very tangible things with shape and curvature that you can travel across and measure distances on.

One way to think about a surface or manifold is to think about it "embedded" in another space. You can, for example, think about a regular old sphere in 3D. It's something you can reach out and touch. But you can also think about it as a space in itself-- a surface that you can walk around on with no awareness of the ambient space. In this model, you walk forward in any direction and you eventually end up right back where you were. There are quite a few spaces in fact, where you might walk around, moving only forward, and end up where you started, but pointing in a different direction.

There are quite a few beautiful metaphors in manifold theory and dynamical systems theory that make me think about the complex ways in which change occurs-- the ways in which ideas, concepts that we might usually think of as fixed can transform into new things. Topology gives an interesting structure to daydreams about phenomenology and ontology. I'm mostly interested in manifolds for their application to robotics, but it sort of gets my wheels turning about categories of identity, ideology, etc. etc. as well. If that's not your cup of tea, it's NBD.

You have to be pretty interested in this stuff to stick with it, so I don't recommend trying to study it unless you have a good reason, but I thought I'd share a few thoughts about it, and it never hurts to go to the Strand and just leaf through some crazy math textbook for pictures.