1æ´:(*****+ Michelle Imber: B3D, Chapter 7

B3D: Chapter 7

Michelle Imber

Chapter 7: Configuration Spaces

Chapter 7, I hath not forgotten thee, though the woes of thesis do verily get me down...

Now then, on to configuration spaces. I was very curious to finally read this chapter after hearing everyone complain all week long about how complicated it is. It seems natural that an instructional or informational text should grow increasingly complex as the reader nears its end. Why, then, did this chapter seem so much startlingly harder?

At the beginning of the chapter, the term configuration space is defined more by example than by precise definition. I think that if the definition were fleshed out a little more, that would make the concept of a configuration space easier to handle from the start. I had to read that paragraph over a couple of times, because I kept feeling like some vital piece of information was missing. The concept itself isn't too bad, though.

I liked the idea that we are all moving through spaces which are higher than three dimensions, if we define the space in terms of the number of parameters which need to be specified to accurately describe something within it. Certainly, though, keeping track of all the parameters needed to specify an arm movement or the motion of a pendulum requires really being on your "mental toes"--there's a lot to think about. There is more of a "mental energy" requirement from the reader, in order to logically follow the story being told.

Also, lots more in the realm of visualization is asked of the Chapter 7 reader. The notion of using four-dimensional representations to most clearly illustrate a complex relationship is a good one, yet it requires a viewer who is able to interpret and synthesize, to some extent, views of four-dimensional objects. The description on page 137is precise, yet takes a good deal of concentration to follow. And (gripe, gripe) I think that the author was trying to stuff so much meaning into a given sentence that sometimes they were a bit overloaded (sorry, Prof. B!.)...The sentence I blinked at for a few minutes, also on page 137, is "The film The Hypershpere: Foliation and Projections, made together with KoÁak, Bisshopp, and computer science graduate students David Laidlaw and David Margolis, presents a visualization of all possible orbits of synchronized pendulums, named Hopf circles after the Swiss mathemitician Heinz Hopf, who studied their properties in the 1930s." Yipes!! Count the commas!

On the positive side, I enjoyed the especially creative examples of interdisciplinary applications of configuration spaces in Chapter 7. From goniometers to hunter-gatherer circles to spotlights...something for everyone. I confess that one of the best parts of this book is reading about something mathematical and learning little tidbits about fields with which I am unfamiliar, to boot.

I was also glad to finally get a good solid answer to a question which plagued me since before the beginning of this class: namely, "Is time the fourth dimension? If not, why not?" The fact that there are multitudinous fourth dimensions is really nicely laid out here, I think, for the first time. It becomes obvious that in the realm of configuration spaces, even a "space' kind of dimension is not necessarily a spatial thing.

The end of the chapter really seemed to me a good example of lots of information crammed into a small space. Singularities and catastrophes sound interesting, yet the concepts elude me. (Apparently, they also elude my differential geometry-taking math major boyfriend. It would be cool if I could explain a mathematical concept to him, for once!) I can appreciate the attempts at simpler explanations of these phenomena, but I think I'd like something even more condensed. Or maybe I should just go back and read the end of the chapter a few more times!

--Michelle

Prof. Banchoff's response.