Response from Prof. B.

Certainly the stella octangula is cute. Your comment about seeing the dual of the cube as the intersection of the two makes me wonder about intersections of other figures. What other interesting configurations can be obtained by intersecting two or more regular figures of appropriate sizes? We should get some nice semi-regular figures in this way, right?

I agree that the geometry of complex numbers is neglected in high school, or at least that was my experience. I did run into de Moivre's formula at one point, giving the rule for the power of a complex number (r cos(t) + r sin(t)i)^n = r^n (cos(nt) + isin(nt)), but I didn't understand the geometric significance of it. In fact my senior project in high school was devoted to an analysis of the function f(t) = (-16)^t, which would have been much, much easier to deal with if I had rewritten it as (16)^t e^(it). Then I could have graphed the spiral (t, (16)^t cos(t), (16)^t sin(t)) directly with no fuss. But then I would not have had the opportunity of discovering the form for myself and I would not have won the regional science fair and I would not have gone to the national science fair in Oklahoma so I would not have had a chance to stop off at Notre Dame where I decided to go and study mathematics, so I probably would have ended up studying structural linguistics at Georgetown and I would have been analyzing run-on sentence structure rather than higher-dimensional geometry. So perhaps it's just as well that my teachers didn't do it right the first time.

I do believe that at one level the complex numbers were discovered but at another level, they had to be invented, at least their representations had to be invented. You are right that the term imaginary set the PR aspect of the subject back a few notches. It helped me when I realized that it was not necessary to rely on mysterious symbols, but rather that complex multiplication was just a way of operating on pairs of real numbers. That algebraic approach then tied in very nicely with the Argand diagram, representing complex numbers as points in the plane, and that leads to representing quaternions as pairs of complex numbers, in our four-dimensional space.

I will be interested to see if other members in this class have comments on this general subject.