Bobdude161 said:
But interesting drawings, RacerX. Was this out of boredom as mine were?
I guess it would be more along the lines of a
mental workout.
The type of visualization that I was doing back when I was in school was far more complex than what I have shown in the images I posted. But the general technique is the same so doing these drawings helps to keep me able to work with them.
I don't think anything I currently do either for a living (computer consulting and web design) or as a hobby (studying fringe computer platforms) is as mentally taxing (or rewarding) as when I was doing mathematics. And as I'm heading back to school this year to finish up (something I had meant to do long ago), I've been spending more and more time getting back up to speed on the research I was doing when I, well, took a break.
In my paper on
Contour Diagrams and Regular Homotopy of Orientable Surfaces I showed a very complex deformation of a surface... which required 10 illustrations similar to those I posted to show the steps. Back then (1995) it took me four months to work out the steps of the deformation and then draw the surface at each step (that isn't counting the two months I lost when I made an error and had to start over).
But it is something I enjoy doing, though it can be pretty hard to do at times (trying to maintain a visualization of a complex surface in your head while also deforming it). Originally I started the drawings of them more as a way to check my work, but they also proved to be very good visual aids.
Sadly this type of illustration really can only be done with embeddings and immersions of surfaces (2-manifolds) in Euclidean 3-space. When working with higher dimensional manifolds mapped into higher dimensional Euclidean spaces, the ability to
draw what you are thinking gets much harder. But the intuition that you build up in these lower dimensional examples is very helpful.
There are a number of good references for this type of visualization of mathematics. One good online source is a series of images on
Conway's Zipper Proof.
Again, that is most likely more than most people wanted to know on the subject.
