Comments on Beyond 3d

Laura Gigi Lee

Response to Chapter 6:

After watching the most recent 'film' on the hypercube, it was interesting to read about the mechanism of changing perspective through "exploratory motion" in regards to the hypercube and torus. Changing the viewpoint seems to be the most effective way of seeing an object's properties, especially if the object is rotated around a particular axis. However, it is very difficult to visualize moving around a hypercube in such a way--- I understand he procedure of viewing the hypercube in continuous motion, 'animating the hypercube,' but it is complicated trying to see the projection of the hypercube as a polyhedral torus. I don't understand how the Clifford torus can resemble both the hypersphere and polyhedral torus in the hypercube. As the number of subdivisions increases, and the polyhedral torus begins to resemble the Clifford torus, things start to get jumbled up. How do these relate to each other?

Stereographic projection as a process makes sense to me, the pictures on p. 125 all make sense, but I don't understand the projections from four space. Shouldn't the shadows have some kind of shape and depth? Can shadows have depth?

Prof. Banchoff's Response