Beyond 3-d Chapter 5

Lisa Hicks

Most of the concepts of Chapter Five basically made sense to me. The biggest question that I remember having as I read it was, "What is an antiprism?" A reference is made to it on page 107, but if it was defined in the chapter, I missed it.

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.

Lisa Hicks

Prof. Banchoff's Response