Reaction:

One thing that struck me right away was this dance *Dimensions*. I was wondering if Prrofessor Strandberg consulted Professor Banchoff to get some ideas or if she came up with her exercises on her own. If she did not talk to him about it in advance I would point to her loss since he obviously has many ideas to share, but due to the nature of the piece I would guess that he was, in fact, a contributor.

The sequence of pendula pictures is beautiful. I cannot imagine how excited Kocak and Bisshopp must have been to see their data so wonderfully displayed! (Too bad Hopf could not enjoy them as well.)

On a personal mathematical note, I would like to say that the brief mention of manifolds did wonders for my mathematical thinking. I am partially horrified to realize that throughout MA113 and MA114 I did not have a good *image* of a manifold--I just tried to work with them using the theorems quite blindly. I hate these loose ends that keep turning up! It is nice to have some of them tied up finally, but I never seem to get a good grasp on them regardless because I do not have everything **at all** fresh in my mind so I am unable to let new information have the impact that it should. Anyway, I am glad to think once again about manifolds and maybe even think about what they are, if only in a small way.

The parallel curves and surfaces are happily quite fresh in my scattered mind, on the other hand, and I am really glad to ponder them again before they dash off somewhere--repetition is the best thing for firming these things up. I particularly like the image of the rays coming straight out from the points of the ellipse. When the additional dimension is added to this image, it is fascinating that the slices reveal the same info. Very nicely pictured.

I have just concluded that visualizing things is everything in my way of thinking about math and too often I am just lacking a good image. This picture book route is one great way to go--the interactive use of the computer is even better. (I realize also that this book is forever indebted to the computer's help.) My biggest regret during my time at Brown is definitely that I did not work harder to picture mathematical things. Here I am, living with all of these amazing tools which our forefathers would have given anything for, and I waste my time being lazy! Shameless! Think of poor Cayley working on his one picture of the form of the focal surfaces for a **single** example.

Now that I have gone off on that tangent, I would like to just add my opinion that I think it is the coolest thing ever that Professor Banchoff has been working on computer imaging throughout its history. It must be extremely rewarding to see how far it has come, knowing that your influence has been so great. I am completely amazed at how lucky I have been to be able to work with him, if even only a little.

Discussion:

7.1 What other motions could be considered similar to that of the pendulum? How would planetary motion compare?

7.2 Think about wave fronts for a variety of two and three dimensional objects, particularly concentrating on singular points. Is it possible to conceive of the wave front emanating from a four dimensional object?

Prof. Banchoff's Response