I didn't really understand the illustrations that accompany the text about sending a kit for building a square to Lineland. To me, it seems that it would be necessary to go into the second dimension even to get the overlapping collapsed square.
I don't see why there cannot be more than three regular polytopes in n-space. I think most of my problem with this stems from the fact that we are no longer dealing with something that I can visualize. I can picture three squares around a point or three cubes around a line, but I can't picture three hypercubes around a plane, or three 5-cubes around a cube.
If the edges of the 600-cell would all be straight in four-space, why does the projection appear as a circle?
This isn't really related to polygons, but I'll ask anyway: A circle is regular in two-space, and a sphere is regular in three-space. Will the higher-dimensional analogue of a circle always be regular? It seems that it should be, but I'm not really su re, again because I'm unable to visualize it.
I really liked the polygonal progression on page 86. I had never paid attention to the fact that, if a square and a pentagon have sides of the same length, the pentagon looks a lot bigger. Kind of obvious, but you don't really notice it until you see them sharing a side. A three-dimensional version of this progression would be really cool.
This chapter, like many of the others, begs for constructions. If you build it, they will come.
Prof. Banchoff's Response