## Response from Prof. B.

The definition of regular polyhedron is very restricted, and if you relax any of the conditions, there are a number of beautiful and intriguing (but not regular) figures that you can make, some by following the sorts of procedures you suggest in your Chapter 5 comments. In particular, but removing cubelets it is possible to find figures which satisfy most of the conditions for a regular polyhedron, without being convex. If you are willing to go to higher dimensions, you can find some really nice examples, but there are also quite a few in three-space. Why not try to make some models to illustrate some of the figures you have in mind and we can see in what sense they are or are not regular in the restricted definition, and then we can see whether or not you have all the examples that satisfy certain conditions (that is the way a lot of papers get written). In the library we might have a book by Bonnie Stewart (a man, despite the name) who wrote "Adventures among the Toroids".

Removing more and more of these cubelets with smaller and smaller sizes leads to a three-dimensional fractal known as a "Menger sponge". I forget it's fractal dimension, but you can probably figure it out.

The method you describe for rolling a figure out onto a plane could run into overlapping difficulties, I agree, but in most cases you should be able to find a "development" that avoids these problems, at least for regular or semi-regular figures. I have a research paper around here somewhere that makes precise what "most" means in the previous sentence. What about higher dimensions? Can you use the inkprint method to get a development of a regular polytope into three-space?