Dimensions are a topic in mathematics that everyone has some experience with. From movies and video games to the 3D printer, we talk about dimensions all the time. Everyone has an idea of what the word "dimension" means. While it may seem simple, the study of dimension in mathematics is full of interesting mind blowing facts! I mentioned this to one of my students this summer and he responded with:
"A drawing on a piece of paper is in 2 dimensions, we live in 3 dimensions, and 4 dimensions is some kind of mind trip that we can't picture. What's so interesting about that?"
"Well, yes trying to picture 4 dimensions and beyond is hard, but you don't have to go up to those dimensions to have interesting things happen." The student scoffed and turned in his chair.
I continued, "you just mentioned dimensions which are whole numbers, but have you ever thought about what a world in dimension one half might look like? Or what about dimension log(3)/log(2)?" I had just lectured on logarithms so this got his attention. This particular student has picked up on the fact that when I say something is interesting or a "good problem" he should take note since there's a good chance he'll see it on a quiz later.
Like a good fractal geometer I took to the white board to start drawing some pictures. Consider the following 4 sets: a collection of points, a line segment of length 2, a square whose sides have length 2, and a cube whose sides have length 2.
Now lets think about how we measure how big sets are. This is also a brilliant subject in math, but I'll only give you the basics here. One way to measure a set is to count how many points it has. This is how we measure sets in dimension 0. We'll write this as a function c (for counting) which takes as an input a set and spits out the number of points in that set. For example,
We can also measure how big a set is by looking at its length, area, and volume. These are how we measure the sizes of sets in dimensions 1, 2, and 3, respectively. We'll write these as l for length, a for area, and v for volume.
At this point my astute student says "wait a minute what does it mean to take the length of a square or the area of a cube?" Trying my best to control my enthusiasm so as to not frighten the poor boy away, I responded "THAT is a fantastic question! You see the fact that you've never been asked to find the length of a square or the area of a cube has to do with the dimension of the square and the dimension of the cube".
Let's take as examples the 4 sets from before and let's find their size by counting the number of points in the set and by finding the sets length, area, and volume. Remember the sides of the square and cube are of length 2.
Now let me explain a few of these calculations. First note that the length, area, and volume of the set of points is 0. Intuitively this is because a set of points just isn't big enough to have length, much less area or volume. You just can't "run" along a set of points like you can "run" along a line. A similar idea explains all of the other zeros. For example, a square isn't big enough to have volume since it has no depth. If you think of volume as how much water the shape can hold, a square has no depth so can hold 0 water whereas a cube can hold some water so its volume is a positive real number.
The values of infinity come from measuring a set that's to "big" for the measure we're using. For example, counting the number of points in a line gives a value of infinity since a line segment contains infinitely many point. A similar idea explains the other values of infinity.
These calculations show how important it is to use the correct measure on each set. A value of infinity or zero doesn't really tell us much about how big the set is; however, if we measure the set in just the right way we get a meaningful number. For example, using the measure v on the cube told us that the cube has volume 8 and using the measure a on the square told us that the square has area 4.
"What does this have to do with dimensions?"
A lot! It turns out that finding the right way to measure a set will also find what dimension the set lives best in. Our set of points has dimension zero because its measure in dimension zero, counting, gives a number, 6, which is not 0 or infinity. Similarly, the dimension of our line segment is one since its measure in dimension one (length) is the number 2 and not 0 or infinity. The same can be done for our square and cube.
"Things can get a lot more complicated and I'm leaving out a few things but this is how we describe dimension as you know it", I said to my student. "Now let me give you an example that throws a wrench into all of this".
Lets talk about the Cantor set. I could say a whole lot about the Cantor set. It's one of the most interesting things in the world to me so maybe I'll write a whole post on it later. For now I'll just tell you some facts about the Cantor set and you'll just have to believe me or try to justify them yourself.
We build the Cantor set in the following way. Start with a line segment of length 1, cut it into 3 equal parts, and remove the middle third. What we're left with are 2 line segments each of length 1/3. Now let's do it again. Take each of the 2 line segments from the previous step, divide each into 3 equal parts, and remove the middle third of each segment. Now what we're left with are 4 line segments each of length 1/9. Doing this again gives 8 line segments of length 1/27 and so on. Continue to do this over and over infinitely many times. The set of points that are not removed is called the Cantor set.
To convince you that this set isn't empty (that we didn't remove the entire line of length 1) notice that the point 1/3 is in the Cantor set. In fact, the end points to all the intervals resulting from our construction are in the Cantor set, so 1/9, 2/9, 1/3, 2/3, 7/9, 8/9, and 1 are some of the numbers in the Cantor set. It turns out there are infinitely many numbers in the Cantor set, and in fact there are as many numbers in the Cantor set as there are real numbers!
Another interesting fact is that the length of the Cantor set is zero. This can be shown by looking at the sum of the length of the segments that were removed in our construction. At each step, say step n, we removed 2n-1 segments of length 3-n. Then from the original segment of length 1 we removed pieces whose lengths sum to
This means that the remaining set, the Cantor set, has length 0.
So what do we have then?
But we just learned that the dimension of a set has to do with where the set can be measured in a way that gives something other than 0 or infinity!
So here is my big tada, my shining moment, my mind blowing result! The Cantor set is such a strange set that it doesn't really belong in 0, 1, 2 or any whole numbered dimension. The Cantor set belongs in a dimension which is not a whole number at all. It belongs in dimension log(3)/log(2)? And the reason it's dimension is log(3)/log(2) has to do with how we constructed it by cutting into pieces of length 1/3, giving us the 3 in log(3)/log(2), and getting 2 segments after removing the middle third, giving the 2 in log(3)/log(2)!
I've said all of this to my student and failed to contain my enthusiasm. I look at him anxiously awaiting to see if I've lost and frightened him or if he's understood and is at least a little excited. He places the five fingers of his right hand on his temple and then moves his hand back while expanding his fingers into an open palm. The universal signal for mind blown. I turn around, drop my whiteboard marker like a mic, and walk away.
"My job here is done."