Ah, the aura of a university in united anxiety, the gleam of dormitories encased in ice, the aroma of freshly photocopied midterms... yep, it's that time of year again. This time, March 8th marks the Mid of the Second Term--a time for reflection. Most of this reflection is done through time, but while we're at it, we may as well reflect our selves over lines, palnes, and spaces.

"Yeah, yeah... but what exactly is THE FOURTH DIMENSION??"
That was my battle cry in the beginning of the semester. I guess
I was looking for a one line definition buried somewhere in __Beyond
the Third Dimension__. Since that time, I've learned that dimensionality
is (in a sense) an acquired taste. Dimensionality can only be
ingested in small, individually wrapped quantities--try to bite
off the whole thing at once and you'll choke.

So, now it's time to masticate on some of the stuff I bit off earlier in the semester... here goes:

Prof. Banchoff's response

>What about the dimensionality of juggling, by the way? --Prof. Banchoff

In assigning dimensionality to something, it is often useful to find the smallest set of numbers needed to describe a certain phenomenon. With juggling, lets say we start out with several constants (number of balls, acceleration due to gravity, number of hands). With these constants, an infinite amount of patterns can be created. We can alter the order in which the hands throw. We can alter the hand that each hand throws to. We can change how many balls each hand throws/catches at a time. We can alter the direction (up, down) of the throw. We can change the position of the throw/catch. Three-dimensional juggling is pretty complex. I imagine it would be only slightly less complex in Flatland. If gravity pulled Southward, A Square could juggle by pushing circles Northward. Actually, now that I think about it, How would a ball (circle) be different from a round house? What prevents a round house from being thrown up (Northward) like a ball? How are houses stationary? What's holding them there? Can balls exist in Flatland? How? Anyway, I have a nifty program called jugglepro which condenses juggling into a simple matrix. A cascade pattern (the most typical form of juggling) can be represented in a 2 by 2 matrix with this program. I haven't been able to do too much with the program, but if anybody out there would be interested in figuring it out with me, let me know! I think it would be a fun project.

>I like your suggestions of exercises, trying fractal constructions with other figures as starting points, or generalizing to higher dimensions. Have you >tried some of these things yourself? That's the best way to generate really good, and doable, problems. --Prof B.

Actually, I've tried a couple with semi-interesting results. As I am scannerless, I will not be posting them. One algorithm I was using was to start with some sort of polygon and just start connecting midpoints. What I found when I started with a trapezoidal-like figure was that I got 4 sets of similar triangles on the periphery, and a trapezoid in the middle with each step. Even more interesting was when I started out with a concave pentagon. This polygon has four angles less than 180 degrees, and one angle greater than 180 degrees. I'll have to do a more precise copy to verify this, but it seems that as I continue to connect midpoints, I (of course) get a pentagon each time); the amazing thing is that the pentagons are getting more and more regular! I started out with the most irregula rpentagon I could think of, and after only six iterations, I have a pentagon that looks very, very regular. This even works when I start with pentagons that have intersecting edges. I will definitely work on this some more and try to scan some of it onto the WEB.

Speaking of reflections, it seems to me that there is only one way to reflect something. We can reflect a linelander over a point. We can reflect a Flatlander over a point or a line--with essentially the same result (would a reflection of A Square over a point be a reflection over a line through that point, or would such a reflection not exist?). Can humans be reflected over a line? I would think not. We can obviously reflect over planes (mirrors). What about higher dimensions. Would A Square's reflection be any different over a plane? I would think not. Following this analogy, I would guess that a human's reflection over an n-space would be no different from his reflection in the mirror.... any comments? What does this mean?

Prof. Banchoff's response

Dan.