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The Fourth Dimension

Sooner or later everyone hears that time is the fourth dimension. That idea, however, limits the idea of dimensionality. Already in the last century writers realized that there are many situations in which time can be viewed as a fourth dimension, but by no means does it demand any special role as the fourth dimension. When physicists, especially relativity physicists, specify an event by giving three space coordinates and one time coordinate, they are using a four-dimensional configuration space. This space has its own geometry that is not the same as the geometry of four-dimensional Euclidean space, where distance is given by the generalized Pythagorean theorem. In the theory of relativity the distance between two events is given by the expression

(x - x´)2  +  (y - y´)2  +  (z - z´)2  +  (t - t´)2

where time is measured in special units related to the speed of light.

The three-dimensional configuration space of spotlights provides a useful analogy for a four-dimensional space used in molecular modeling. The atoms that make up a molecule can be represented by small spheres of different radii. The description of a particular molecule, like the description of stage lighting, consists of a list of spheres of different sizes in different positions. Each sphere requires three coordinates to specify its center and one coordinate for the radius. Thus the configuration space of atoms is four-dimensional, and a molecule is a collection of such atoms arranged in a particular formation.

Using the language of the configuration space, we can describe a molecule to a computer and ask it to display different views. If we ask the computer to check that two atoms do not intersect, this involves an algebraic condition in four coordinates, namely


(x - x´)2  +  (y - y´)2  +  (z - z´)2  -  (r + r´)2  >  0.

The geometry of this configuration space is much closer to that of relativity theory than it is to ordinary Euclidean four-dimensional geometry. Interestingly it is this sort of question — avoiding intersections — that appears in the science of robotics, using large numbers of coordinates to keep track of objects moving through configuration spaces of high dimension.

Suppose each light on our sample stage possesses a rheostat that can control the current — hence the brightness — of the spot. If we add brightness to the coordinates of the spotlight, then the configuration space will be four-dimensional. If we want to encode the color of each spotlight as well, then the dimensionality jumps again. The specification of color requires three more coordinates representing either hue, saturation, and value or the relative amounts of red, yellow, and blue (for pigments) or red, green, and blue (for lights). So the lighting director will now have seven coordinates for each spotlight: two for floor position, one for radius, one for brightness, and three for color. Thus even a simple example can lead to a configuration space of high dimensionality.

Relativity physics began by considering four-dimensional collections, with three dimensions for space and one for time. Recently modem physics has become much more complicated. Some current models keep track of seven dimensions that act like space and four that act like time, to give an 11-dimensional configuration space. Another important model uses a configuration space with 26 dimensions. In each case the choice of the model depends to some degree on the kinds of mathematics that apply in these dimensions, as an aid to keeping track of the complex interrelationships among events in these high-dimensional spaces.

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.