My Response to Chapter 7

I was quite taken with the examples of configuration spaces in the book. In particular, I was fond of the example of stage lighting. Below is a little applet I cooked up to fool around with while thinking about the example of stage lighting, and here is the source code. If you are using a Java capable browser (such as Netscape 2.0 or Sun's HotJava) you should be able to use it as well. Otherwise, you'll just see a pair of horizontal bars.

My applet was to go here, but for unknown reasons, the macintosh web server can't handle java. You can look at the applet here instead, on David Akers' linux box.

I found this example enormously intriguing. Although the book treats it in fairly practical terms, I immediately had the impulse to treat these circles as though they were an alternate representation of a three dimensional set of points. That is to say, given a set of ordered triples, one can think geometrically about the set of ordered triples as representing a set of points in a three space or as representing a set of circles in the plane. What sorts of synthetic relationships do we find between the points in space and the circles in the plane?

I was particularly wondering this about those sets of ordered triples that happen to be the domains of easily expressed functions in two variables. We have to be very careful about how we set up the "graphs" of such functions in the set of 2-circles. If the function is continuous, the set of circles representing its domain will be an amorphous smear of circles intersecting one another, in which one is hard pressed to distinguish one circle from another.

I don't see a particularly satisfactory way out of this. Any continuous function over the real numbers would have a circle graph covering the entire plane. To see why consider this: if the function is expressed as r = f(x,y), and its domain is the real numbers, every point on our plane would be the center of a circle of a certain radius. Although perhaps (as I haven't thought thoroughly enough about this to say for certain) there are certain functions that would leave "holes" in such a graph, but for most of the simple cases I can think of, the graph would (if all drawn in the same color) effectively paint the plane.

Nevertheless, when playing with the applet above, I noticed that if one only takes a systematic, finite subset of the circles to graph (eg, only those variable pairs that are integral and lie in a certain interval), one gets some rather intriguing looking results. As an example, take r=x^2-y. Below are some exemplary values of x and y.

 x | y | r 
 0   0   0
 1   0   1
 2   0   4
 3   0   9
 0   1   -1
 1   1   0
 2   1   3
 3   1   8
 0   2   -2
 1   2   -1
 2   2   2
 3   2   7
 0   3   -3
 1   3   -2
 2   3   1
 3   3   6

When one plots these circles, they are quite interesting. There are obvious generalizations-- for instance, the farther to the right a circle is (ie, the greater the value of x) the larger its radius. It is obvious why this should be so. There is something of a problem when it comes to the radii, some of which are negative. How are these represented on the stage? The only obvious reasonable alternatives are to say that they don't, because there's no such thing as a positive distance, or to say that these circles are identical to the circles having radii equal to the absolute value of the negative radii. In the first case, we clearly have a problem with translating from the set of points of three-space to the circles, since some are simply not represented. In the second case, we have a problem translating from the set of circles to the set of points in three space, since each circle represents is the image of two points.

Oh well, so it isn't as promising as I'd hoped. But I think the reasons that this doesn't work are interesting enough to justify the effort.