It is interesting to see the hypercube with labelled coordinate axes. I think the delay of the introduction to this was appropriate, for by this time the reader is quite comfortable with the appearance of this once-unfamiliar representation of the hypercube and is better equipped to understand the image. The generalization of the Pythagorean theorem is similarly timely.
The stella octangula strikes me as extremely cute. Perhaps I am biased by the name (little eight-angled star?), but I also appreciate the generation of the cube's dual by means of the intersection of these simplexes.
As I seem to mention every week, I am really excited to have my attention brought back to material from a course that I took previously--in this case, Complex Analysis. I do keep complex numbers more in the forefront of my mind than I do some of my above mentioned (in previous weeks) topics, and it is great to see these pictures--as it was to see the 'necklace' pictures in Professor Banchoff's art exhibit. My appreciation of thinking about the dimensionality of coordinate systems is enhanced immeasurably by these beautiful complex pictures. It is in cases such as these of the most sublime beauty that I feel computer responses are the most limiting--what can I possibly say in typed words about these images when the images themselves are so vivid and are saying so much? At any rate, I encourage everyone to go back and ponder the pure beauty of these graphs again. The power of complex numbers has always astounded me--I cannot get over them.
Were complex numbers really "invented", or were they discovered? The title imaginary certainly did nothing for their image as legitimate--so sad that in order to gain respect they were forced to demonstrate their usefulness. I guess the positive side is that they certainly proved themselves--showed those nay-sayers where it's at!
Topics for Discussion:
8.1 One problem with my own high school mathematics education was that although complex numbers were introduced several times, no development of their behavior was ever done. We also never went into why mathematicians study them--they were seen as just another thing to perform operations on, and those operations were the end of their treatment. I am interested in how this could be changed. I have several ideas. One is to demonstrate in Calculus classes how nicely some functions which are messy when limited to the real number system can be differentiated and such when one takes advantage of complex numbers' behavior. Another is to perhaps expand on the relationships among sin, cos, exp, pi and i more directly--ie. rather than just stating cis(t) = cos(t) + isin(t), delve into the other ways to come at that equation by thinking about e^i(pi) as a point on the unit circle and the resulting identities for sin and cos. The geometric examples in this chapter provide a third means for filling out the treatment of complex numbers in an accessible manner.
I would be interested to hear about others' experiences with complex numbers in high school--were they developed at all and if so, how?
Prof. Banchoff's Response.