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Configuration Spaces

The coordinate descriptions that are so useful in giving locations and direction in familiar spaces of one, two, and three dimensions work equally well for phenomena whose specification requires more than three numbers. Exploratory data analysis, a statistical technique for dealing with these representations, is one of the most important applications of dimensions in current research. The ability to visualize and interpret multidimensional data sets may be one of the best gifts we can present our students in this modem age.

Some of the most useful and interesting examples of higher-dimension — phenomena occur as configuration spaces — collections of geometric objects representing certain structures or motions in the natural world. The most familiar spaces are the one-dimensional collection of points on a line, the two-dimensional collection of points in a plane, and the three-dimensional collection of points in space. But we can also consider the collection of lines in the plane, the collection of planes in space, the collection of all possible circles in a plane, or the collection of spheres in space. We illustrate this process by presenting several examples of phenomena that lead to higher-dimensional configuration spaces. Consider the following (slightly unrealistic) situation: The lighting director of our local theater has to arrange a set of lights over the stage so as to illuminate certain parts of the floor at certain times. Sometimes the size of a spot is supposed to change during the course of a performance. Sometimes one colored circle is supposed to be contained in another. How can she keep track of all the circles of light and then design lighting directions so that an assistant can carry them out?

In this particular theater the lights all have the same form. A single bulb is suspended from a wire hanging down from the ceiling, and conical shade directs the light out in a beam that meets the floor in a disc of light. The sides of the shade come down at a 45° angle, so the radius of the disc is equal to the height of the bulb above the floor (Figure 33). This makes it easy for the director to specify the location of any light, since she can indicate the position of the center of the disc using the same coordinates that the director of the play uses to give instructions. That uses two coordinates, but the lighting director needs another number to represent the radius of the disc. She could, as an alternative, specify the height of the bulb above the floor, since in this idealized situation these two numbers are the same. Hence any particular disc can be represented by three coordinates, the first two being the location of the center and the third giving the radius (or, in our special case, the height).

Figure 33. A spotlight with a shade set at a 45 degree angle will illuminate a spot on the floor of a stage whose radius equals the height of the light above the stage.

In this way we see that the collection of discs in the plane is three-dimensional; this collection is an example of a configuration space, each disc representing one element in the configuration of spotlights. To exploit the three-dimensionality as a bookkeeping device, the director can record the position of each light by giving three coordinates: for example, (6,8,5) refers to the light with center at the (6,8) position on the floor and a radius (or height) of 5.

To call this a space indicates something more than convenience of recording. It is a signal that the arithmetic of the coordinates reflects properties of the geometry of lights. For example, a spotlight with coordinates (6,8,5) stays on the stage, while the light (6,4,5) shines off the front of the stage. It is easy to determine a rule to tell when a light stays away from the front rim of the stage, namely that the second coordinate be larger than the third.

More complex problems facing the lighting director can also be solved by referring to the coordinates. For example, when will one spot be entirely separate from another? In words, this happens when the distance between the points in the plane given by the first two coordinates is greater than the sum of the third coordinates. In symbols, the condition is expressed by

(x - x´)2  +  (y - y´)2 > r + r´

In this configuration space the three coordinates do not play the same sorts of roles; so even though the geometry of the configuration space is three-dimensional, it treats the last coordinate differently from the first two. It is not identical to the usual geometry of ordinary three-space, where the Pythagorean theorem treats all coordinates the same way. An important aspect of configuration spaces are the special symmetries they possess.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.