Response from Prof. B.

Is there somewhere where it is possible to see examples of the kinds of photographs your friend is doing? I think that the idea of using photography to do surrealistic effects is intriguing, and at the same time disturbing. For a long time, people were accustomed to think of a photograph as a faithful representation of reality, as if a photograph could never lie. That wasn't true even in the beginning of newspaper photography, where heads of some characters were pasted over others to accompany news stories about hard-to-photograph individuals. Such practices were decried as unethical very quickly so that that sort of doctoring is usually only done by parodists these days.

But now we know how easy, or relatively easy, it is to do wonders by digital modification of images, in "Forrest Gump" for example, or in "Rising Sun". It does get a bit disconcerting.

Dali worked with photographs in setting up some of his more impressive perspective effects, especially those that had to deal with stereoscopic oil painting. He used a stereo pair of photographs to capture the realistic position of a central figure, then played with different ways of distorting the perspective in other elements of the composition, sometimes with striking effect.

You are quite right that the hypercube is not rotating about a line, but rather about a plane, in the movie that I showed. The rotations we are most familiar with in ordinary space are those where we fix one axis and rotate the other two, using formulas from trigonometry to find the new coordinates. By combining these "elementary" rotations, we get more complicated ones, but there is a theorem that says that anything we get in this way is still a rotation about a single fixed axis somewhere.

In the case of four dimensions, the corresponding fundamental building block for rotations is obtained when we hold two coordinates fixed and rotate in the other two. In that sense the fixed set is a plane, about which we can say we are rotating. But if we follow one of these rotations by a rotation about a different plane, it is possible that the combination does not have any fixed axis at all. It is a rather surprising phenomenon at first, since we are used to orienting ourselves by looking for the fixed axis in something that we know is a rotation. It takes a while to get used to these things. Maybe right after the break we can set up a video showing, or even put a tape on reserve over in the media lab so people can view it at their leisure.

There is a box in the basement of the math building filled with early versions of our films and outtakes as well. Whether or not any of them are in shape to be projected is another question. I agree that sometimes it is quite useful to see the preliminary versions of things, but I'm not sure how easy it is in this case. Want to try?