Here's yet another type of configuration space, set up by a simple story. For the school sculpture show two students want to decorate the back wall of the hall with a pattern of plastic strings. They decide to stretch them from the left-hand edge of the wall down to the floor. By trial and error the week before the show, they come up with a pleasing design, using more than twenty strings. They can't leave them up until the show so they have to find a way of recording the positions so they can put them up again later. How many numbers do they need to specify the position of each string? What is the dimensionality of the collection of strings?
It is easy to see that the dimensionality of this configuration space is two: it takes, just two marks to locate a given string, one along the floor and one up the left edge of the wall, and each of these locations can be specified by a single number. The pair of numbers (4,3), for example, could represent the string that goes from the point four feet over on the floor to the point three feet up on the wall edge (Figure 34). The collection of pairs, one pair for each string, tells the positions of all strings. It is even possible to record these ordered pairs in a specific sequence so the students will know which order to follow when they replace them.
In a way this coding is like the old game of "connect the dots" where a polygon is determined by a sequence of ordered pairs, so by connecting the dots in order we draw the polygon. In our sculpture story the basic elements are not points but segments: by forming the sequence of string segments, we re-create the wall sculpture.
Figure 34. A configuration space of two dimensions can represent the positions of strings that run from the floor to locations on the left edge of the wall.
If we increase the dimensionality of the configuration space, we can allow the bottom of the string to be placed anywhere on the floor, with the top still somewhere on the left edge of the wall. We still need one number for the height, but now the record will have to include two numbers for the floor coordinates. The collection of segments would then be three-dimensional, yielding greater possibilities of more interesting sculptures.
By allowing the strings to start anywhere on the vertical wall and end up anywhere on the floor, we would have a realization of a four-dimensional system. Simple algebra would then enable one to predict, for example, whether or not two strings are going to intersect. When we are laying strings along a wall, it is commonplace for them to intersect. Such intersections are rare if we are in a three-dimensional collection and rarer still for the four-dimensional system of segments in space. It is also interesting to look for configurations of segments that correspond to familiar configurations in ordinary space. What collection of segments in a two-dimensional configuration space corresponds to a line joining two points? What segments in a three-dimensional collection correspond to a coordinate plane in three-space? Questions such as these can yield striking and unpredictable visual effects in the string sculpture.
The dimensionality of a configuration space becomes especially important when we consider dynamic problems. When a point is moving on a line, we can describe its state at any given time by giving two numbers, one for its position and a second for its velocity. The state space is therefore two-dimensional, and a point moving according to a given physical law, like a ball bobbing up and down on a spring, will describe a curve in that state space. Similarly a point moving in a circle, like a swinging pendulum, will have a two-dimensional state space giving its angular position and angular velocity.
The state space of a point moving in a plane will be four-dimensional, with two points for location and another two for velocity. Scientists analyzing the motion of a satellite have to work in a six-dimensional state space, with three coordinates for position and three for velocity. The laws of physics will restrict the actual states of a system to some lower-dimensional space. Indeed, scientists devote a good deal of effort to analyzing the shapes of these spaces. For example, the motion of two pendulums corresponds to a curve on a torus in four-dimensional space. The study of such high-dimensional dynamical systems is an extremely important subject in modern applied mathematics.
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.