Response from Prof. B.

Your initial apologetic dedication reminded me to call Dirk Struik, and it is quite possible that he will be coming to my gallery talk on the 14th. Thanks for reminding me.

I'm glad to see your collection of personal configuration spaces. That is such an obvious exercise to suggest to anyone reading this chapter that I hope everyone in the class shared your reaction. Once you take a number of courses at Brown, you should have a whole range of such examples to share.

Your "fitness graph" is a very good one and I hope you can give me some references so I can lead people to some interesting visualizations. The situation here is quite close to the setup used by economists trying to maximize things like profit rather than another kind of fitness. Pareto optimality is a way of trying to increase profitability by a method of steepest ascent, moving up the scale as fast as possible, but that can lead to relative maxima that miss out on the true bonanzas, equivalent in form as well as spirit to the evolutionary dead ends you mention. You are quite right that the potential dimension of this genetic configuration space is huge, but as in other similar situations, we can still get some information by restricting to a low number of variables and using our old partial derivative techniques.

The consonant and vowel space examples are very nice, especially since each has its own preferred dimension. Although the tongue position moves continuously, it looks as if the configuration space is considered to be discrete, again a standard way of reducing a more complicated situation to something more manageable. Restricting the number of variables in the consonant space might be possible in some languages and not in others? I wonder if in some sense an unfamiliar language might go off in a different direction in this six-dimensional space, so that a click language or one that uses tones in essential ways would lie in some four-dimensional subspace that intersected some other language space in a two-dimensional subspace. There are all sorts of statistical packages that try to analyze phenomena like this, using techniques of masking and slicing and projection to isolate populations that share similar characteristics, and then to apply regression analysis to lower the dimensions still further if the samples all happen to lie close to a hyperplane, or some other linear or well-understood subspace.

You are quite right that the positions of the two pendulums depend only on two variables, the angles. The state space on the other hand does involve not just position but velocity, so the real reason that the state space is four dimensional is that each pendulum requires two variables for its position and velocity. These can be given by an angle and by a signed number that determines the velocity vector lying in the tangent line to the circe.

To define a circle in space requires fewer than nine numbers, as you suspect. To give a sphere requires four variables, one for the radius and three for the center, and then you can determine an equator on this sphere by specifying its north pole. That requires two more dimensions, the latitude and longitude coordinates of the pole, so the total is six. It's still a lot, but better than nine.

Another way of seeing this is to start with your description of a circle in terms of three triples of numbers. Of course this is redundant, since we can determine the same circle by other triples. In fact we can move one of the points in a one-dimensional way and define the same circle, so that lowers the dimensionality by one. Altogether you lower from nine to six in this way, but of course you have to be careful when you are moving the points around to see that no two of them coincide during the process. In general mathematicians have to be careful about such special positions when identifying different ways of describing a configuration space. That's part of the job.