I really liked this chapter as well, especially the part at the end about the rotating torus in four space. Watching the hypercube video a couple extra times really helped understand a lot of this chapter. I did, however, get kind of confused about the stereographic projections.

One thought I had was on perspectives for the cube and hypercube. It would seem to me that the diagram of the cube looking straight at it - on the top left of page 114 - is inconsistent with the Shlegel diagram on 117. At most , one of @these is correct. The edges that connect the front and back faces are of different lengths, and thus the proportions of the cubes are different. However, proportions of a cube are quite demanding, in that all of the edges must be of equal length. So, one of the cubes is not really a cube, but a figure with two square faces and four rectangular faces (if there is a name for such a figure I do not know it). There must be only one way to properly depict a cube when drawing on a plane.

The same restrictions must apply then for a hypercube. My question here deals with the size of the inner cube. In three space, if the smaller square is very small in proportion to the larger one, we say that the figure must be very LONG. In four space if the inner cube is very small, what adjective would one use for the hypercube?DSET|ÿÿ(H:ÿÿÿÿÿÿXœ„ÔˆÃÆl ˆÔX6ÿÿ*ÃÆlDSET|ÿÿ(HÈtlPÿÿÿÿÿÿXœˆlˆÔïRl6ÿÿ*ÃÆlFNTMTÿÿH HelveticaCUTSDSUM& Performa UserHDNIETBL8FNTM}CUTSÙDSUMáHDNIETBLÿþýüûúùøðñòóôõö÷