Comments on Beyond 3d

Laura Gigi Lee

Response to Chapter 4:

Regular Polytopes and Fold-outs:

The concepts of duality and self-duality, in particular are very interesting. I have a few questions about this chapter in general...

The concept of fold-outs is interesting in respect to Flatland. The idea that a Flatlander would be able to create the fold-out of a 3-dimensional figure but be unable to assemble it is compelling. I wonder what needs to be perceived for us to be able to assemble the fold-outs of four-dimensional and higher dimensional figures.

I understand why a 3-space square casts a shadow of a square, but why does a hypercube do the same to us in 3-d? Shouldn't a light from the fourth dimension cast a hypercube shadow that appears to us in the 3rd dimension as a "shadow cube?"

Prof. Banchoff's Response