Dimensions as Coordinates

Common to almost all of these ideas of dimensions are lists of numbers, coordinates that specify some quantity associated with an object or a phenomenon. For example, the driver in the Baltimore tunnel can record her position by noting the tenths of a mile from the starting point and the number of feet from the tunnel wall. By far the most familiar example of such coordinates are the length, width, and height of a rectangular box. These three numbers completely specify the shape of a box. Once we know these numbers, we can construct the box, or picture it in our minds even before it is constructed. We can use the familiar concept of the coordinates of a box to help visualize patterns of data in different dimensions — one, two, three, and eventually four and more.

In many applications of mathematics, from the economics of health care to the mapping of distant galaxies, the information we have to process consists of many different measurements for each observation. Making sense out of such complicated collections of data is one of the greatest challenges to social scientists or physical scientists, and it is in this area that the experience of mathematicians concerned with visualizing higher dimensions can be of most help. All observational scientists rely on their ability to recognize trends and patterns, to identify regularities that lead to predictable behavior. Our visual sense is our most powerful faculty for discerning such patterns. One of the most effective ways of bringing our visual faculty into play is to interpret the sequence of measurements for each observation as a point in a space of the appropriate dimension.

One number for each individual is sufficient to record the heights in a family, and all the coordinates can be entered on the same number line, for example the doorjamb at the kitchen entrance. To record both the height and width of the armspan for each family member, we could mark the two quantities on two different number lines, but we get much more information by displaying the data on a two-dimensional surface, a kitchen wall next to the doorjamb. Each family member determines one smallest-fitting rectangle, with the width marking the armspan. The definite advantage of a two-dimensional display is that the single point on the wall indicates both quantities, height and armspan, for each individual. A pair of measurements has led to a two-dimensional quantity. We are better able to see relationships between the quantities when they are displayed on the same two dimensional diagram.

As an example of such a relationship, for most adults the smallest rectangle that records height and armspan is nearly square. Once we observe this trend, we can reduce the dimension of the system. We do not have to record the armspan since we can deduce it once we know the height. This simple example lies at the heart of the modern subject known as exploratory data analysis.

To record foot size as well as height and armspan for each family member, we determine the smallest rectangular box that contains that person. An upper corner of the box gives a three-dimensional quantity, the record of all three numbers at the same time.

The power of these familiar one-, two-, and three-dimensional frameworks becomes evident when we use them to record observations that have little to do with height or width or length. The three numbers height, weight, and age can be recorded and visualized on the same three-dimensional framework that we used to keep track of the spatial coordinates height, armspan, and foot length.

We have ready-made ways of visualizing data of dimensionality one, two, or three. Marks on a number line, points on a piece of graph paper, points in space are all available to us as a means of picturing a set of coordinates. But what if we were concerned with more than three measurements, say height, armspan, weight, and age? We would have four measurements for each family member, and where would we be able to record them, and how would we visualize the records we create?

Once we see the data laid out on a familiar framework in two or three dimensions, we can identify relationships that are simply invisible if all we can see is a long list of measurements. The coordinate structure forms the backdrop against which we organize our observations and achieve our insights. In order to use visualization techniques effectively in more complicated situations, we need to become familiar with frameworks of even greater dimensionality.

A thread that runs through all considerations of dimensions is the attempt to use insight obtained in one dimension to understand the next. We use this process automatically as we walk around an object or a structure, accumulating sequences of two-dimensional visual images on our retinas from which we infer properties of the three-dimensional objects causing the images. Thinking about different dimensions can make us much more conscious of what it means to see an object, not just as a sequence of images but rather as a form, an ideal object in the mind. We can then begin to turn this imaging faculty to the study of objects that require even more exploration before we can understand them, objects that cannot be built in ordinary space.
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