Chapter 7 Excogitation

The most interesting part of chapter 7 (for me) is the section on "Orbits of Dynamical Systems". I don't think it's very clear--after reading it for a while, I still have some questions. Is there anything particularly special about the two tori in the first picture on page 136? Do they show some "critical point", or is the purple torus show "pendulum A short, pendulum B long" and vice versa? My guess is that those two tori are in the pictures just for reference. Correct me if I'm wrong, but the motions of the two pendula only need one torus for their description. This would make sense because a torus is a circle of circles (R1 cross R1). If left to pendulate for all eternity, would the set of two pendula hit every point on their respective torus? It would seem so, but I'm not sure about that.

Is there a movie to go along with these Hopf circles. It would be helpful to see the two pendula at the top of the screen, and the path on the torus that their position takes on a torus. It would also help to see an animated version of the slide show on pages 136-7. Would this animation look like the flat torus in the 3-sphere rotating in four dimensional space of Prof. B.'s project space.

I took Gould's intro archaeology class and read his book that the book store wouldn't buy back from me. It's a pity he didn't mention dimensionality at all. He did talk a helluva lot about those dang Australian aborigines. But I digress...

Why [(x-x')^2 + (y-y')^2 + (z-z')^2 - (t-t')^2]^.5 ???

Regarding the space of lines: this reminds me a lot of the surface created by a rotating, moving juggling club. For physics last semester my group found the equations for the paths of the two ends of the pins. We then tested these experimentally. They were pretty much correct. The question is: how would I relate this information to another mathematician? How do I write the information in the "standard form" of segment space? Are two parametric equations enough? Also, are there computer programs that would plot the surfaces created in segment space?

I was particularly surprised that there was no mention of wave fronts created by 3-dimensional surfaces. I must say that I am a bit confused as to why the waves from a circle pass through the center of the circle. It seems like all of the forces should cancel eachother out. Do they continue because all of the waves are at the same point in their sinusoidal cycle when they hit the center? Does this make their interference constructive? Does this matter?

Are the involutes of these curvesthe same as parallel curves?

Possible exercises... hmmm... how about this: Take two springs with different spring constants. Hang them form a platform. Hang equal weights (equal so the dimensionality doesn't get out of control.). Plot (or calculate) their positions with respect to time when set into SHM. The result can be plottted on a line of lines (plane). Depending on the spring constants, the grpahs will lie in different portions of the plane. The starting positions also affect the graph...

Another exercise could involve predicting the parallel curves of a certain shape and testing the prediction using wire, water, a video camera, and a slow motion VCR (or a computer program that is provided on the book's CD-ROM).I don't think it's terribly hard to write a program that allows the user to draw a curve using sketchpad or other and computes its parallel curves. If the program is not available, we could always have the reader buils a stadium and fill the rim with lasers.

By the way, do we have "Dimensions" (the Brown Dance production) on tape? I'd like to see it.

Prof. Banchoff's Response
Dan