## Response from Prof. B.

Your discussion of the dimensionality of motion is right on target. The number of dimensions necessary to describe a precise placement of even one arm is substantial. After all, we have more than one finger on each hand and more than one joint on the non-thumbs. It is also true that these joints do not have full range of motion, and that elbow position does limit hand movements. This became especially clear to me two years ago when I taught this course with my left arm in a cast for the first seven weeks, then four weeks of physical therapy before I could begin to rotate my left hand into typing position, something I had taken for granted.

Your comments on joint hierarchy made me think of an epistle of St. Paul where he compares members of a congregation to members of a human body, which had better be cooperative if they want to achieve anything. The recent magazine story about Siamese twins with separate control over the two legs raises all sorts of other questions about degrees of freedom.

There are definite Moire patterns in the wave-front pictures, and Prof. Shapiro and I were consciously playing on that when we did that video "Fronts & Centers". The music was also making patterns of reinforced beats, to complement the visual illusion. The experience of multiple perspective resolutions of a single image is something difficult to explain, and I found it interesting that people have such varying impressions even when the object is moving.

Any configuration of non-intersecting lines will generate a hyperbolic paraboloid, as long as the lines are not parallel either. This can be "verified" by experimenting with elastic strings attached to wands, which then can be twisted to produce a variety of shapes that are all hyperbolic paraboloids, again as long as the lines of the wands neither intersect nor are parallel. This can be shown formally by observing that we can get from the pair of lines in the picture on page 143 to any other pair of non-intersecting non-parallel lines by an "affine transformation" that takes lines to lines and necessarily takes hyperbolid paraboloids to hyperbolic paraboloids. This last fact follows since the affine transformations are just linear transformations followed by a translation, and composing linear functions with quadratic functions just leads to other quadratic functions. There is a limited set of shapes that can come from such quadratic expressions, and you can't get an ellipsoid or a hyperboloid of one sheet from such a composition. Another hyperbolic paraboloid is just about the only thing that you can expect. Have you had enough matrix algebra to put that argument into a formal proof?

It was partially in response to your question that I spoke Wednesday about the dance that Julie Strandberg choreographed. There certainly was a rather free adaptation of the story line, but several elements of the original book show through clearly. The critics didn't have too much to say about the performances, and even some of the audiences were a bit silent, rapt in thought I suppose. In general people liked it, especially since it seemed to "humanize" mathematics.