Response from Prof. B.

That 52nd edition of B3D is so much better than all the rest that I don't see why anyone would have bought one of the first 51. Agreed it would be very nice indeed to have some animations to let people explore the effect of changing their positions for viewing works of art with a particular three-point perspective built in. Or seeing Eadweard stride across that plank over and over, courtesy of some standard in-betweening technology. Or moving these various polyhedral skeletons continuously from their position under a bright light into their shadows below, even in the icosahedral and dodecahedral case. I agree that it is easier to imagine pulling a pentagon out than a triangle, and probably easier to pull out the 120-cell than the 600-cell. It is still possible to see the 16-cell though and worth the effort. I think that the 24-cell is best of all, and I'll have to remember to bring in a nice model sent by a friend in California, a very persuasive rendition.

Ah, yes, holograms. Won't they be wonderful when they finally work? Theoretically we should be able to produce an on-line holographic experience that would enable us to explore four-dimensional virtual space, but not yet, at least not yet the last time I asked.

The three-sphere is a little more complicated than just a double cap over the Clifford torus. You have to fill the torus surface in in two different ways, by capping off not just two of the circles but by capping simultaneously a whole family of circles that cover the torus. There are several ways to do this, and perhaps the easiest (but not the most symmetrical) is to think of a collection of vertical circles around a standard torus filled in by discs to give a solid torus. Then the family of horizontal circles should also be filled in by disjoint discs, and that is a bit harder to accomplish. The smallest horizontal circle gets filled in by a horizontal disc all right, but then the next discs bulge up or down so that by the time you get to the top horizontal circle, it is the boundary of a disc that is virtually hemispheric. As we go further, the discs balloon further and further out so that ultimately we get arbitrarily close to the region _outside_ the biggest horizontal circle, thought of as a disc that is so large that it goes through "the point at infinity".

It is quite mysterious, but a bit more plausible when we consider a lower-dimensional case (as usual). The whole construction I have been sketching is a collection of surfaces of revolution of something resembling the lines of force in a magnetic field between two discs in the plane. One of these lines of force starts at the horizontal position, goes all the way out to infinity, and comes back to the other side, always remaining horizontal. With a bit of effort, this can be seen as the projection of something happening on a sphere in three-space that is then projected into the plane. That helps the analogy quite a bit when we try to go back to four-space, doesn't it? Or does it get lost? Let me know if this is a useful way of looking at things.