Chapter 5 began by reminding me how fun it can be to play with a compass. In my younger days I used one to find midpoints of line segments and to drop perpendiculars, but I never really used one to construct even such simple shapes as a dodecagon or e
ven a hexagon--it is quite nice to go back to our geometric roots and check out what can be done with the basic tools of a straight edge and a compass. I also think it is a good exercise to try to convince oneself that such things like constructing a reg
ular enneagon or trisecting an angle * cannot * be done with these useful tools.

The section on regularity was filled out nicely by Friday's discussion. I find it fascinating that there are so few regular objects in dimensions higher than 2. It is useful to draw pictures of two dimensional shapes around a point to demonstrate the limits that are placed on folding these up into three dimensional objects. Considering the folding of three dimensional fold-outs is naturally more difficult to visualize, and both Lisa E.'s and Professor Banchoff's 3D models made it clear to me that th ree dimensional models are necessary to show the flexibility of the models themselves--folding begins to seem a lot more like something that would be a viable option given the space in which to work.

**Duals.** How amazing! Why is it that the duals of regular objects are regular themselves? Is the connection of the midpoints of faces of any shape called a 'dual', or does the name assume that a regular polytope was the st
arting shape?

Is the hypercube kite pictured on p.104 flexible? Could it be and still be capable of flight?

I think one of the best exercises for this chapter is to carefully consider each regular polytope and think about how it can exist as well as why others are impossible. Another is to build fold-out models and see what they are like in three dimension s. Are all of them flexible if made from an appropriate material? As for thinking about the duals of regular polytopes, I think it would be nice to construct a way to see several inside each other, one shape alternating with the other. I am not sure ho w to do this physically, but the computer would probably be a wonderful tool for that kind of animation.

Lastly, does anyone remember a toy that was a 'loop' of triangles that was flexible enough to bend in several directions quite easily, but not in every direction--it could fold up into a cube, a triangle with a different colored triangle inside of it, a set of three connected things that looked sort of like two dimensional pictures of cubes, and maybe one or two other things? If so, what was it, what were the shapes, and do they still make them?

Alexis NogeloProf. Banchoff's response A.N. additional comments