# Cerebration on Chapter VI, Beyond the Third Dimension

I find it pretty hard to believe that Dali's horse would fool anyone. It's a wonderful idea, but there is more to human depth perception than father objects being smaller than closer objects. For one thing, the horse's butt would basically be in a different time zone than the nostrils. Wouldn't it look a little fishy if his head was lit by the sun, but it was dark where the sun don't shine? Also, our accuity decreases with distance. The horse's rump would probably not be very clear to us. Moreover, in general, thnigs farther away are higher in our visual field, why then did Dali put the horse's rear end so high? Maybe it would work on a computer... that would be a neat final project concept.

The boxing ring picture looks fake. Maybe because it's a view that we never see, or because it's so "perfect". It has a very surreal quality. I imagine it would be pretty weird to watch a boxing match from this position.

Is the Clifford torus the same as a 3-torus. It would seem not. It looks like it is formed the same way that a flat torus is formed in 3-space. The only difference would be that we do not have to distort the squares. If it is not the same as a 3-torus, how is it similar, and how is it different? Is is basically like the equator of a sphere?

All of the points on the clifford torus are points on a hypersphere. If we were to actually draw this hypersphere over the image of the Clifford torus on page 127, I would say that it would have the image of a sphere in the hole of the doughnut, a sphere circumscribing the doughnut, and lots of lines connecting corresponding points. Since the light source on the sphere was basically a line shone onto a plane, we would have to put a light on the hypersphere which was a plane projected into space. This makes sense looking at the illustrations. I realize that it would be very complex, but at some point I would like to see images of the rotating clifford torus with the hypersphere drawn in. It would be nice to see what is happeneing to the hupersphere and the light source while all of this is happening. This could even be a separate diagram (to avoid cluttering), and it could be left to the reader to merge the two in his mind.

Question: After reading about this whole Clifford Torus thing, it would seem that the inner surface of a torus is topologically equivalent to the outer surface. Is this true?

Exercises are getting more and more difficult to create. For this chapter, some things that I think might be fun/worth doing are:

1. Take pictures of yourself walking naked on a plank. put them in sequence on a turntable, and reinvent animation.

2. construct large polyhedra so that if you press your face against one of the edges, you get a schlegel diagram.

3. draw the muhammed ali boxing ring from a more "normal" perspective.

p.s. A while back I talked about fractals with 3-dimensional objects. a good example of a fractal tetrahedron can be found on this page.