With apologies to Dr. Struik
I thought I would give some other examples of configuration spaces; these cases are, of course, from fields where I have some not insignificant experience.
Survival ("fitness") of an individual, based on that individual's genetic content, can be expressed as a function--and therefore a function graph. Usually biologists limit their graphs to three dimensions, which means they can only plot two gene loci at a time. (Height expresses the dependent variable--fitness). This method, regardless of dimension, yields an N-surface that has optimal maxima and minima. Most interesting is that the distinction between global maxima and local maxima, a distinction most people learn in the beginning of calculus, is critical for an understanding of biologists' use of this model. Local maxima are another way to express the idea of an "evolutionary dead-end"; a maximally specialized creature that would lose fitness with any small changes but if it could get bumped (through a fairly large mutation) off its local "hill" it might "climb" (move on this surface) to a different local maximum with a higher surface.
The surface is a multi-dimensional function of gene frequency, yielding a gene-space--gene frequencies are most dimensions, F = f(g1,g2,g3) But why should it be limited to discussing the interaction of two genes? Here we have the potential for a configuration space with as many dimensions as there are genetic loci--that is, millions. Something of a difficult space to examine.
Sound space (the possible sounds humans make in language) is described by phoneticians as having several dimensions--most of them have to do with the position of the tongue, but since several other parts of the mouth perform in speech, the space is multidimensional.
Consonants, for example, can be described by a point in a six-dimensional space depending on the configuration of these six parts of the vocal apparatus: the lips, the tongue blade, the tongue back ("dorsum"), the tongue root, the voicebox (larynx), and the soft palate. Each has at least three states: completely obstructing airflow, allowing some air to pass, and allowing completely free passage of air. This yields six dimensions with at least three regions in each dimension = 3^6 = 729 possible cells within the consonant space. Most of them aren't used, but the distinctions of which are and which aren't is largely dependent on the specific language. Each language would have a different graph of frequency, since you could take a speaker's production and graph it as a traveling point in that six-space. The frequently-traveled regions would differ from language to language.
Vowels, on the other hand, can be described almost entirely by a two-dimensional configuration space: the position of the tongue in the mouth ignoring lateral (side-to-side) motions. Much less complicated. If you wanted to make it complicated, you could add a few "coloration" dimensions from the other six: opening the soft palate adds a nasal quality, and pulling back the tongue blade adds an "r"-coloration, but in general the space is defined in a two-space box. In the front of the box are the sounds /i/ and /e/, the back has /o/ and /u/, and /a/ lies in the bottom middle. /u/ and /i/ are high, while /e/, /a/, and /o/ are low. Much simpler than the consonant space.
I don't understand why mapping a pendulum onto a circle needs two dimensions--isn't a circle a one-dimensional object? Or maybe we're talking about something else when we speak of mapping. Creating a new circle and fixing it in space needs lots of dimensions to be specified. We have to define the plane's relationship to the space and then define the circle's position on that plane and the radius. In three-space this can be done by defining the position of three points on the circle, but that requires nine numbers. Seems like a lot.
Jeremy Kahn x6753