# B3D Chapter 5

## Lisa Eckstein

**Suggestions**
Get a compass and a straightedge, and draw some figures. I'll bet you haven't used a
compass since high school geometry - I know I haven't. Construct polygons, bisect
angles, go wild!

While you're constructing polygons, think about their duality. Why is it that all
regular polygons are self-dual, while the majority of higher-dimensional regular shapes
are not self-dual? What properties must an object have in order to be self-dual?

**Questions**

Is there any explanation for the existence of 6 regular 4-dimensional polytopes when
there are only 5 regular polyhedra? I would expect the number of regular figures to
decline continuously with higher dimensions, not increase and then decrease again. Why
does such a counter-intuitive phenomenon occur?

*moocow@brown.edu*

Prof. Banchoff's response