Response from Prof. B.

OK, you bring up quite a few points in this chapter, all of which could be discussed at some length. Let me try to begin a few responses.

I agree that it would be a trick for a Dali horse to fool anyone, unless they wanted to be fooled. I think that people would have found the illusion intriguing, and they would have sought clues to dispel it. You mentioned a few, for example the different sorts of lighting that might give an impression different from one might expect to see in a proper sized horse. But there are times when lighting is more or less uniform so that might not help.

Fuzziness of portions of the horse when other portions are in focus might not be that much of a problem. After all, the horse is big to begin with, so even people with trifocals would be using the tops of their glasses. Having just purchased a set of glasses with middle range capabilities so I can respond to these questions for hours on end without strain either on the eyes or the neck, I can appreciate the fact that things that look good at screen distance don't work for across the street. But once across the street is in focus, I pretty much can see all the things beyond that distance without a refocussing problem. Or am I wrong about that> Is there an optometrist in the class?

If you think that aerial views are weird ways to look at sports I suggest that you aviod interviewing with Goodyear.

The Clifford torus is raising a good deal of interest. Perhaps it would be useful to compare this object with some other figures that one might draw on the three-sphere in four-space. Recall that the ordinary sphere that we see every day is called a two-sphere since every point is identified by two coordinates, latitude and longitude. In the same way the ordinary torus is a two-dimensional surface, whether we build it in three-space to begin with or build it up in four-space and then project it down to three-space so we can see it.

In the same way that the ordinary sphere is covered with circles (= one-dimensional "spheres" in our 3D-centrist language), the hypersphere in four-space is covered with two-spheres, starting with a point ( like the South Pole) and expanding up to the full Equatorial distention, then shrinking back to a point, as in the balloon demonstration of the visit from the hypersphere.

Unfortunately the balloon model is not the one that fits well with the discussion of the Clifford torus. Rather one should think of the way that stereographic projection splays out the entire two-sphere (minus one point) onto the plane. In this model, we start with a point as before, but now the circles just grow and grow without limit, never coming back on themselves, and it is only when the circles have grown to an incomprehensibly immense size that we realize that in fact we have reached the furthest point and the big sphere is actually converging on that point. ( Rudy Rucker has something to say about this, right?) Anyway, the comparable thing in the next dimension up is filling all of three-space with progressively larger and larger spheres. We can think of those spheres as starting at a point at the center of symmetry of a usual torus of revolution, then expanding until it hits th torus at one circle, then two circles for a while and finally one circle as it breaks through and continues out toward infinity. The slice sequence is rather like that of the bagel in Chapter Three.

Does that help at all? You are absolutely right that the Clifford torus divides the Hypersphere into two parts that are absolutely and completely congruent. So the hypersphere topologically is two solid tori pasted together along their boundaries, with the latitudes of one pasted to the longitudes of the other. It's quite mysterious.

With respect to the exercise, Muybridge developed his own photographs so he could get away with anything. If you take your walking-the-plank roll into the local PhotoShop, you will probably be reported to the local authorities. It is actually very instructive to put your head inside a mirrored polyhedron, something even more impressive than a Schlegel view. Nothing about Muhammad Ali resembled "normal" perspective.

The "bended perspective" in your link I find difficult to interpret. But I do like the fractal tetrahedron. Note (or compute) that this object has fractal dimension 2, since doubling its size produces four copies of itself, as in ordinary two-space!