Al-righty, Chapter 5. This chapter yearns from the start to be accompanied by three-dimensional models. I really feel like I'm beginning to "lose it" with respect to this visualization thing. I need to palpate these models, to rotate them and see them fitting within and around each other. And I'm a construction paper clutz. What to do?
For starters, I liked the models of the 5 regular polyhedra and the illustrations of their duals. But I really wanted to see more angles, especially for the higher ones. There's a point at which mere transparency is confusing. Thank goodness for the process of analogy--I feel like it gives me a tenuous grasp on the concepts, at best. Still, it's better than nothing. I'm terribly frustrated with my inability to visualize the four-dimensional things, which I thoroughly feel to be a failure of my puny three-dimensional brain rather than this well-intentioned two-dimensional text. I keep asking the hopeless question of A Square, only in my case, it's "how can you fold that OUTWARDS, but not UPWARDS? (Or downwards, or sidewards...) And far be it from me to comprehend a five-dimensional polytope. My visuospatial cortex is imploding.
I am intrigued by the idea that once you pass the fourth dimension, there will only be three regular n-dimensional polytopes. It seems odd, somehow.
Now, the 600-cell is a piece of work. I have a lot of admiration for the mathematicians who cooked up this one. Do you arrive at this sort of concept via induction, superior visualization skills, or ephiphany? Or all of the above? I'll tell you, one thing that popped into my head while looking at this creation was, of all things, the child's toy--yes, you guessed it--the Spirograph. Did you all have one of these things? Prof. Banchoff, did you give one to your grand-daughter? Apparently it dates back to the 60's and can be used to draw curves called trochoids. Now, I realize that the 600-cell couldn't quite be created with a Spirograph (it's not a trochoid), but I had a sudden image of 3-and-4-dimensional Spirographs and thought that there must be useful geometric principles at work that could apply here. I can't remember the exact method of establishing a link from this page (I still haven't got the hang of this html thing fully, y'know), but I did find a bunch of Spirograph sites on the net for all of you yearning for nostalgia or virtual playthings. Here are three addresses (if someone reminds me how to fix up a link, perhaps I'll do so):
I haven't tried them yet, but I will do so as soon as I get this thing posted to MA8 tonight.
Now, on to the fold-outs. I've always thought it curious that a cube looks like a cross when unfolded. Divine revelation for those of the Christian faith? Perhaps, till you look at some of the pentagonal stars and things you can construct with other geometrical toying-about. At any rate, this must be what occurred to Dalí in The Crucifixion (Corpus Hypercubicus). Too bad he didn't create a three-dimensional projection of a four-Christ. Now that would be art. The frustrating thing about the model we looked at in class today, as well as the pictures in this book, is that I kept wanting to grab the cubes and fold them OUT or SIDEWAYS or whatever that nameless, incomprehensible direction is until I could see the fully extended hypercube in all of its splendor. And I can never do that. Very frustrating. Still, it's intriguing, and I just might make my own model yet.
One more thing that I've been thinking about and finally (reminded by all this folding) decided to bring up was the "hexaflexagon". It's a simple enough fold-out model, made from only one flat sheet of paper, and scored along the triangular boundaries till it's folded into a hexagon (it's made from a strip of conjoined equilateral triangles). There is some overlap on some parts of the hexagon, and a little creative maneuvering turns the thing inside out. By applying different colors to the faces of the hexaflexagon, one can create a puzzle reminiscent (sort of) of the Rubik's Cube. You have to align the faces so that they are all of the same color, or so that they form some design or other. Hmm, now that I think, it is possibly folded out of two strips of paper. At any rate, I have a model of one that I could make, currently lying around somewhere. Is there relevance to all this folding stuff? Are people interested in seeing it?
Lastly, I feel negligent because I haven't yet turned in my fiction story. I had this idea for a story that I started working on early and kept writing and it just kept going and I decided it was stupid and wasn't going anywhere. I have started another, however, and hope to drop it in the slot sometime this week. If anyone is motivated, check it out.