Configuration Spaces

Timothy Faulkner

The term configuration space is a broad and flexible way of examining sets of information, and this chapteršs main concern seems to be to present a few examples of the many forms of config. spaces seemly to convey to the reader what can be represented in such a way. Essentially, this is a real application of dimensionality--configuration spaces allow multiple variables to be presented in a clear way in a space. . . of lower dimension.

The first examples concern movements of the body which are highly complex. The charts generated from the goniometer readings convey a lot of information about a seemingly simple movement. Once we attempt to examine config. spaces of the entire body--say the movements of the dancers--a graphical representation would not be very clear. Thus, it is necessary to constrain the space which is being looked at to a single limb, or a single action.

The pendula example is beautiful and illustrates how sets of information can be dealt with together--two circles with a point circling each one independently is not very revealing. . . to be honest the pictures really didnšt reveal anything to me about pendula movement to me either, but it is an elegant example of a four-dimensional config. space.

Although many of the things in this chapter were difficult to understand, what absolutely went completely over my head was the brief introduction of differential structures. . . I understand manifolds, but once the book mentioned the seventh dimension and exotic structures--I felt that what was being discussed was entirely unnecessary--of course, the work of these mathematicians involved config. spaces and higher dimensions, and maybe I didnšt understand--but the topic seems ridiculously too difficult for a book that was necessarily written very simply. How was it determined that there are infinitely many ways of constructing a differential structure in the fourth dimension, and what does that mean in relation to anything that we have learned or will learn in this class?