A couple of weeks ago, I ended up trying to explain my current line of research to one of my wife’s friends. It turns out that it is not that simple to describe. To begin consider this deceptively simple little question: what is a three dimensional figure?
Most people can probably give examples of three dimensional figures based on experience in elementary or secondary lessons in geometry. When pressed, I imagine that many could also give some kind of description of a three dimensional figure. For instance, we might state that a one dimensional object has only length, while a two dimensional object has length and width, and a three dimensional object has length, width, and height.
Hence we might define dimensions as follows:
The dimension of an object is the number of directions required in order to measure it.
There is a nice intuition here, and there are some interesting relationships between objects of different dimensions under this interpretation, but there are also some problems.
From a mathematical standpoint, the biggest problem is that the definition is vague. What do we mean by “direction” and “measure?” How do we determine the minimum? For instance, is the surface of a sphere two dimensional or three dimensional? We could find points on a sphere by measuring forward-backward, left-right, and up-down; or we could measure using some notion of latitude and longitude. We could justify calling the surface of a sphere either two dimensional or three dimensional.
Another problem, almost paradoxically, is the specificity of this definition. Mathematicians like to be able to generalize ideas to talk about different kinds of objects. In modern mathematics, all of the shapes shown in the first figure are simply collections of points. They can be described by various branches of mathematics, including set theory and topology. Unfortunately, the definition above really only makes sense in the context of Euclidean geometry. Because of this specificity, our initial definition turns out not to be terribly useful.
So what is a mathematician to do?
First and foremost, the mathematician needs to discard the idea that a good definition should refer to some real object or collection of objects. The goal is to create a definition that captures the intuitive idea from geometry, but that can be applied in a meaningful way to more abstract objects.
That being said, the mathematician would like his or her definition to be consistent with the intuition. If he or she writes a definition that would make spheres one dimensional or points two dimensional, it probably isn’t a very good definition.
One early formalization of the concept of dimension is very similar to that shown above. Based on the intuition that we are, in some sense, attempting to use dimensions to measure objects, we could define dimension as the number of coordinates needed to identify the points of an object.
The dimension of a space (or subset of a space) is the minimum number of real valued coordinates necessary to uniquely identify points in that space (or subset of that space).
There are a few terms in that definition that may not be obvious to the uninitiated. We will not attempt to define these terms with great formality, but the basic sketch is as follows:
A space is simply a collection of points. A line is a collection of points, as is a plane. Hence a line or plane is a kind of space. However, spaces need not be the familiar kind of geometric spaces with which we are familiar. Any collection of points (or objects which could be represented as points) can be a space. We might consider a cow to be a kind of point, in which case the collection of cows in a field is a space.
A subset is a collection of points that are all contained in a larger space. For instance, a line segment is a subset of a line, and a square is a subset of a plane. We do need to be a little careful about how we discuss the original space, however. For instance, a line segment is also a subset of a plane. Subsets can be empty, or they could contain the entire space (that is, any space is a subset of itself). Also, subsets need not be exclusive. In the case of cows on a pasture, brown cows might form one subset, while cows with calves my form another subset. Note that a cow could be both brown and have calves, hence the subsets are not exclusive of each other.
The next bit is a little complicated. By analogy, think of the number line. Each point on the number line can be uniquely identified by a single real number. Given any point on the number line, I can identify that point with a real number. Positive numbers indicate numbers to the right of 0, while negative numbers are to the left of 0. The magnitude, or size, of the number indicates a distance from 0.
By contrast, we cannot identify every point of a plane (or square) with only one coordinate. The formal argument is a little complicated, but the basic idea is that there is no function that can take every point on the number line and map it uniquely and reversibly to a point on the plane. This means that a plane must be more than one dimensional. In fact, it is two dimensional, as each point can be uniquely identified with two real numbers—say an x-coordinate and a y-coordinate.
This definition is pretty good. It matches our intuition about geometrical objects pretty well, and it is a much more rigorous definition than our first attempt. Moreover, it generalized very nicely. Once we understand how coordinate systems work, which we can figure out by working with two and three dimensional spaces that can be easily drawn or visualized, we can extend this knowledge into higher dimensions. It may not be possible to draw or visualize a four dimensional cube, but we can describe it mathematically using a system of coordinates.
In fact, in the study of economics and optimization theory, it is not uncommon to work in hundreds or thousands of dimensions. For instance, if you were the owner of a grocery store, you might be interested in how much of each item to stock. Each item represents one variable in a very large system of equations, and the amount of the item to stock is the value of the variable. Each variable represents a single dimension, hence making stocking decisions can be done using very high dimensional spaces.
Unfortunately, this definition ran into problems in the early 20th century. Mathematicians like Hilbert and Peano showed that it was possible to bend a line segment in such a way that it completely covered a square. It became possible (more or less) to identify each point in the square with a single coordinate.
Additionally, there are kinds of pathological figures that don’t behave in the expected way. There are things that look like lines, but which behave like planes. Are these objects one dimensional or two dimensional?
And finally, this brings me to my current area of research. Mathematicians such as Lebesgue and Hausdorff came up with other ways of defining dimension. Some of these approaches are really quite nifty, though they are beyond the scope of the current discussion. Rather, for the moment, it is sufficient to say that I am studying the nature of dimension, and the results that follow from defining dimension in a particular way. This is a very abstract study, and has very little direct bearing upon the real world, or even the abstract geometrical figures that we are familiar with.
To return to the original question, the nature of a three dimensional figure depends upon how we define the concept of dimension. Cubes are certainly good examples, but sets of solutions to systems of equations in three variables might also be considered three dimensional, as well as certain fractals. There is a kind of abstraction in evidence here, and traditional notions of what a dimension is must be discarded in order to make meaningful progress.