You probably know fractals as cool-looking mathy shapes. Maybe you’ve also heard that fractals can have non-integer dimensions. This is extremely weird. I hope to give you an idea of what fractals are and help you to understand why we say that fractals’ dimensions can be non-integer.
Prerequisites: Not being scared of logarithms.
Originally Written: January 2017.
Confidence Level: Established math since the early 1900s.
Zooming In
Consider a circle. What do you see when you zoom in?
If we zoom in far enough, eventually it looks like a straight line. Boring.
Most other geometric shapes – triangles, squares, ellipses, parabolas, etc. – look like straight lines once you zoom in far enough. If you zoom in on most things, you will pretty quickly get to something that looks boring.
I will de fine ‘fractal’ in opposition to this. A fractal is a shape which does not become boring, regardless of how far you zoom in on it.
Let’s consider an example: the Koch curve. As you zoom in on the Koch curve, it doesn’t eventually look like a straight line. Instead, the shape repeats itself. When a shape repeats itself as you zoom in on it, the shape is called self-similar.
Another fractal is the Mandelbrot set. Although the Mandelbrot set is not exactly self-similar, it certainly does not become boring when you zoom. Go watch some of these videos on Youtube !
Dimension
Most people’s understanding of dimension is entirely intuitive. A line or curve is obviously one-dimensional. A circle or square is obviously two-dimensional. Spheres and cubes are obviously three-dimensional. The surface of a sphere or cube is two-dimensional.
How could we mathematically determine what the dimension of an object is?
Let’s consider how the size of the object changes when you change its radius or the length of one of its sides.
- The length of a line segment (1-dimensional) is: $ \ell = \ell . $
- The length of the circumference of a circle (1-dimensional) is: $ C = 2 \pi r . $
- The area of a square (2-dimensional) is: $ A = \ell^2 . $
- The area of a circle (2-dimensional) is: $ A = \pi r^2 . $
- The area of a sphere’s surface (2-dimensional) is: $ A = 4 \pi r^2 . $
- The area of a cube’s surface (2-dimensional) is: $ A = 6 \ell^2 . $
- The volume of a sphere (3-dimensional) is: $ V = \frac{4}{3} \pi r^3 . $
- The volume of a cube (3-dimensional) is: $ V = \ell^3 . $
Look at the exponent of the radius or length for each of these examples. The exponent is the same as the dimension of the object.
Let’s extrapolate a general law from this. The size of any object of dimension $ d $ is related to a length of that object according to: $$ S = c \ \ell^d , $$ where $ c $ is a constant which depends on the object you are considering (circle vs square vs triangle). The length $ \ell $ could be the side of the object or it could be the radius.
Solving for $ d $ gives: $$ d = \frac{\log(S/c)}{\log(\ell)} . $$
In particular, if you increase the length by a factor of $ 2 $, the size of the object will increase by a factor of $ 2^d $ . If you increase the length by a factor of $ 3 $, the size of the object will increase by a factor of $ 3^d $ . If $ \ell_i $ is the initial length, $ \ell_f $ is the final length, $ S_i $ is the initial size, and $ S_f$ is the final size, then the dimension of the object is: $$ d = \frac{\log(S_f/S_i)}{\log(\ell_f/\ell_i)} . $$
To demystify this formula, let’s look at some simple examples. What happens when you triple the length of a line, a square, and a cube ?
Dimension of A Few Fractals
Let’s see how this definition works for a few fractals.
Koch Curve
We will begin with the Koch curve of Figure 2. To construct the Koch curve, start with a straight line segment. Take the middle third of the line segment and use it to draw an equilateral triangle. Remove the original line, leaving the other two lines of the equilateral triangle. At this stage, the partially-made Koch curve has four line segments, each of which is a third the length of the original line. Repeat the process for each of the four line segments. After you do this, the partially-made Koch curve has sixteen segments, each of which is a ninth the length of the original line. Repeat this process for each of the sixteen line segment. The Koch curve is the end result after you’ve repeated the process in finitely many times.
At each stage, the length of the partially-made Koch curve gets longer. The final Koch curve has in finite length, even though there is a finite straight-line distance between its start and end.
The Koch curve doesn’t get boring at short length scales because, as you zoom in, you can smaller and smaller triangular structures formed at later and later stages of the Koch curve’s creation.
So what is the dimension of the Koch curve? Since it’s a curve, it would appear to be one-dimensional. However, it crams in finite length into a finite distance, so that throws off your intuition some. When your intuition is suspect, it is a good idea to go back to the definition of dimension that we established earlier. What happens when you increase the side length of the Koch curve?
We could begin the construction process of the Koch curve using a short initial line. Both the short line and the resulting Koch curve are shown in red in Figure 6. Alternately, we could begin the construction process of the Koch curve using an initial line that is three times as long as before. This line is shown in black. If we triple the length of the initial line, then the resulting Koch curve is four times as big. In Figure 6, this is apparent because the larger Koch curve is built up of four copies of the smaller Koch curve, shown in four different colors.
The dimension of the Koch curve is thus: $$ d = \frac{\log{4}}{\log{3}} \approx 1.262 . $$
Sierpinski Carpet
Another well-known exactly self-similar fractal is the Sierpinski carpet.
To construct the Sierpinski carpet, begin with a square. Divide the square into nine pieces, like a tic-tac-toe board. Remove the middle piece. You are left with eight smaller squares, each of which is a ninth the size of the original square. Divide each of these smaller squares into ninths and remove the middle piece. You are left with sixty-four smaller squares, each of which is one eighty-first the size of the original square. The Sierpinski carpet is the end result after you’ve repeated the process infinitely many times.
At each stage, the area of the partially-made Sierpinski carpet gets smaller. The final Sierpinski carpet has zero area.
The Sierpinski carpet doesn’t get boring at short length scales because, as you zoom in, you can find smaller and smaller squares removed at later and later stages of the Sierpinski carpet’s construction.
What is the dimension of the Sierpinski carpet?
Since it was created from a square, you might think that it is two dimensional. However, it has zero area, so that throws your intuition off . Let’s use our definition of dimension again.
We could begin the construction process of the Sierpinski carpet using a square with short side length. That square’s side length is shown in red in Figure 7. The resulting Sierpinski carpet would only be one corner of the Sierpinski carpet – corner three, in this example. Alternatively, we could begin the construction process using a square with side length three times as long as before. That square’s side length is shown in black. If we triple the side length of the initial square, then we get a larger version of the Sierpinski carpet built of eight smaller pieces. Each smaller piece is identical to the Sierpinski carpet built from the square with the smaller side length. When we triple the side length of the initial square, we get a Sierpinski carpet which is eight times as big.
The dimension of the Sierpinski carpet is thus: $$ d = \frac{\log{8}}{\log{3}} \approx 1.893 . $$
Others
You can now use this same procedure to figure out the dimension of any other fractal built by repeating a simple rule on a smaller and smaller scale.
In the previous examples, we tripled the length of one side. That is not necessary. For other fractals, it might be better to double the length of the side or to increase it by a factor of four. Break the fractal into smaller pieces, see how long each piece is, and count how many pieces there are.
Try it on the Sierpinski triangle !
It is more difficult to measure the dimension of fractals which are not exactly self-similar. All of the techniques that can be used are based on the same principle. When you increase the side length of the fractal or, equivalently, decrease the length of your measuring stick, how much bigger does the fractal become?
Intuition?
We have shown that some object have dimension which is not an integer. This is an affront to many people’s intuition. How can we reconcile these results with the way we usually think about dimension?
We typically think that one-dimensional objects have only length, two-dimensional objects have length and width, and three-dimensional objects have length, width, and thickness. This way of thinking works well as long as it is obvious if the object has or doesn’t have each of these properties.
The Sierpinski carpet starts out as something that has length and width. To create the carpet, we punch infinitely many holes in it until it no longer covers any area. Whether or not something completely covered in hole has width is less obvious, so it is reasonable to think that the Sierpinski carpet’s dimension is less than two. The final lattice left at the end of the construction looks more substantial than any familiar one-dimensional object. The Sierpinski carpet’s dimension is more than one. We end up with a result somewhere in between, approximately $ 1.893 $ .
The Koch curve starts out with only length. To create the curve, we bend it infinitely many time, until the length becomes infinite. Although the curve doesn’t cover any area, all of the infinitely small bends give it something kind of like width. It is more substantial than any familiar one-dimensional object, but still not wide enough to be two-dimensional. The resulting dimension is in between, approximately $ 1.262 $ .
There is no significant difference between fractals which are made by folding a curve infinitely many times and fractals which are made by punching infinitely many holes in a two-dimensional object. Both end up having dimension between one and two. Some, like the Sierpinski carpet, can be formed either way, although one construction method is easier than the other. There are even some infinitely-folded curves which fill in a non-zero area. These `space-filling curves’ have dimension two.
To estimate the dimension of a fractal object, look at how close it looks to an object that you are familiar with. Fractals which look more substantial, like the Sierpinski carpet, have higher dimension than objects which look less substantial, like the Koch curve, even though both objects have dimension between one and two.
Fractals defy your typical notions of the length, width, and thickness. We have to resort to a mathematical definition to figure out their dimension. Often, the result is not an integer.