Understanding Fractional Dimensions
Measuring those infinitely crinkly things called fractals
Fractals started to enter the public consciousness in the early 1980s thanks to Benoit Mandelbrot’s landmark book The Fractal Geometry of Nature. In this book, Mandelbrot argues that many natural structures — clouds, coastlines, trees, rivers, blood vessels, lightning — possess a type of geometry that resists description using simple shapes like lines, planes, and spheres.
Those were the early days of affordable, mass-produced computers with graphics capabilities, and the times were ripe for exploring these fascinating mathematical objects. I spent many of my undergraduate days in the late 1980s writing Turbo Pascal programs on an IBM PC to render images of Mandelbrot sets, Julia sets, Sierpiński triangles, dragon curves, and other mind-bending imaginary structures.
As a budding computer programmer, it fascinated me that very simple formulas could produce images of frightening and sublime beauty.
Beauty Beneath the Surface
At the time, I was so entranced by the raw beauty of fractals that I didn’t spend much time learning more about their underlying mathematics or their implications to our understanding of the natural world. I was content to tinker with small algorithms that created a huge variety of amazing images.
Only later did I start to read about the mathematical theory behind fractals. Much of it was, and still is, far beyond my understanding. Indeed, there are many unsolved problems in this field that have stumped the brightest mathematical minds. (One example: what is the exact surface area of the Mandelbrot set?)
But here I would like to share one fascinating fractal concept that I could understand, one that is approachable by anyone with high school level mathematics: fractional dimensions. This is the surprising idea that mathematical and physical objects can be described using a number of dimensions that is not an integer like 1, 2, 3, etc.
Not only will we establish that fractional dimensions exist, we will learn how to calculate their numeric values for a certain category of fractals called Koch snowflakes, whose dimensionality lies between 1 and 2. That’s right, we can prove that such objects are more than one-dimensional but less than two-dimensional! Even better, we can calculate exactly what that number of dimensions is.
The Koch snowflake starts with a straight line segment like this:
We proceed to build the Koch snowflake in a series of steps. In the first step, we split the line segment into three equal parts and remove the middle section. The missing middle section is replaced with two line segments having the same length as the missing section and oriented to form an equilateral triangle with that missing section. Here is the result after the first step:
Now we have 4 line segments that are each the same length as the 3 sections we split the original line segment into. Therefore, the total linear length of the curve is 4/3 as long. This will be important later.
We continue by repeating this process of splitting each of the four shorter line segments into three parts and replacing the middle segment with two segments to form a triangular “bump.” So the second iteration looks like this:
Note that this third object is 4/3 times 4/3, or 16/9, as large as the first object. Continue repeating this process to form a fourth object:
… and then a fifth object:
Each time, the total length increases by a factor of 4/3. Anyone who has studied investing and the effects of compound interest is starting to get excited now. At least they would be, if this abstract structure were priced based on how long all the line segments are. This thing is growing in length exponentially! It’s like you bought a stock that goes up in value 33% year after year without fail. Even if you started with a small investment, pretty soon Jeff Bezos will be seething with envy.
To Infinity and Beyond
But we still don’t have a fractal yet. For the object to be truly fractal in nature, we have to repeat the procedure an infinite number of times. In the real world this is not possible because we human beings can’t do anything an infinite number of times, and real material objects are made of finite atoms, not an infinitely divisible substance.
But mathematicians don’t let things like that bother them. In the ideal world of our imagination, let’s pretend it is possible. After all, there are many times in traditional geometry where we create idealized simplifications of reality: we can imagine a perfectly straight line segment with zero width and infinite length, circles that are flawlessly round, and so on. Such perfection is nowhere to be found in real life. Mathematics is useful and beautiful even though we know it doesn’t exactly represent the real world.
If we take a leap of imagination and declare that we can repeat the procedure outlined above an infinite number of times, we end up with the Koch snowflake, an object that resembles this:
Now we have an interesting situation. This object has an infinite length, because it grew exponentially an infinite number of times. So there is no way to describe the snowflake as a one-dimensional object.
On the other hand, it has zero surface area. It lies completely inside a finite region of two-dimensional space. With a small enough pair of scissors, you can imagine snipping away more and more of the paper it is printed on, leaving less and less paper each time, without limit. With enough time and precision, you can make the remaining paper weigh as close to zero as you want without damaging the snowflake.
It seems like the Koch snowflake is too “large” to be considered one-dimensional, yet too “small” to be two-dimensional. That might make sense. But does it really? Is this just sloppy thinking? To find out, we need to establish exactly what dimensions are. It turns out there is more than one way to think about dimensions.
Dimensions by Counting Coordinates
Intuitively we know that some abstract shapes like discs and squares are two-dimensional, while others like spheres and cubes are three-dimensional. Numbers of dimensions are typically determined by counting how many perpendicular directions are required to specify a point inside an object. If you can move up or down, left or right, and forward or backward inside an object, and you can report a distance number for each of those movements, that object has three dimensions because there are three different perpendicular ways you can move.
This is a handy way to understand dimensions, but it’s not the only way. In the case of fractals, counting coordinates simply will not suffice. Coordinate counting leads to a dead end when you consider the movement of a point confined to the Koch snowflake. At first it seems like one dimension works fine, because no matter where you place a point on the snowflake, it can move in only two directions: toward one end of the snowflake or the other, “right” or “left.”
The problem occurs when you ask how far the point has moved. How many meters did it travel along the snowflake? No matter how tiny a movement your point shifts along this object, the total distance traveled is infinite. Even though an outside observer might barely detect a microscopic shift in position, the distance the point had to move along the infinitely crinkly curve is unmeasurable. There is no way to assign a single number expressed in units like feet or meters to describe the motion of a point on this fractal.
And as we noticed above, although the Koch snowflake is infinitely long, it has zero area. We can describe the location of any point on the snowflake using two coordinates, but it seems excessive to call it two-dimensional because the snowflake has no surface area at all.
Dimensions by Measurement Scaling
Because we are stuck, let’s backtrack and consider a different way to define dimensions. Suppose your job is to tile a square floor that is 10 feet long by 10 feet wide. You are given square tiles that are one foot long on each side. Clearly, you are going to need 100 tiles. But what happens instead if each side of the floor is 3 times as long: 30 feet by 30 feet? Now you will need 900 tiles.
Although the linear size of the floor increased by a factor of 3, the number of tiles went up by 9. In general, if the linear size of the floor goes up by a factor of n, the number of tiles has to increase by a factor of n². We know that floors are two-dimensional, and that number 2 in the formula n² gives us a clue.
Let’s try a new working definition of dimensions. We’ll say that an object is 2-dimensional if increasing its linear size by n causes it to get larger by a factor of n², that it is 3-dimensional if it gets larger by n³, and so on.
We want to apply this idea to the Koch snowflake. That means we would like to triple the linear extent of the snowflake and then, somehow, measure the change in its “size.” But there’s the rub: the notion of “size” is slippery for an object like this. How do we proceed?
The solution is to reconsider the square tiles we used above. Why are we able to use tiles to measure the area of two-dimensional floors? It is because tiles are themselves two-dimensional: they have a finite surface area, and we can count how many of them are placed onto the floor to cover it.
How Do We Tile a Snowflake?
To measure Koch snowflakes, we need something that has the same number of dimensions as a Koch snowflake. This is a little frustrating because we don’t know what that number of dimensions is. In fact, that’s what we’re trying to figure out! It’s like a dog chasing its tail.
But if we did have some kind of object smaller than the snowflake and with the same number of dimensions, we could count how many of those smaller objects it takes to “tile” the snowflake.
It turns out there is a perfect tool for this job. In fact, it’s right under our noses: another Koch snowflake, only smaller. We can make a copy of a small part of the snowflake and use it to measure the whole snowflake, just like we use small square tiles to measure a large floor. This works because each section of the snowflake is exactly the same shape as the whole snowflake, a common property of fractals called self-similarity.
In this diagram we see that the snowflake has four major sections as labeled by Roman numerals. Each of these sections is identical in shape to the whole snowflake. If we think of the sections as tiles, it takes 4 small tiles to cover the entire snowflake, and therefore the entire snowflake is 4 times as large as each of the sections that comprise it.
At the same time, we see that each of these sections has 1/3 the linear size as the whole snowflake. For example, section IV is exactly 1/3 as wide at its base as the entire snowflake.
So tripling the linear size of one of the sections results in a snowflake that takes four of the original size sections to tile it. To find the number of dimensions D of this fractal, we need to find a value of D such that
If we try to guess values of D, we see that D=1 is too small, because 3¹=3. Also, D=2 is too large, because 3²=9. Now we are getting somewhere! We have a solid basis for claiming that the number of dimensions D is between 1 and 2. But what is the exact value of D?
Solving the Puzzle
(Don’t worry about this part if you are rusty on the math; it’s not necessary to the overall concept. The important part is that it is possible to arrive at an exact value for D.)
To calculate the value of D, we need to take the logarithm of both sides of the equation and use a little algebra to isolate D to one side of the equal sign:
The result is that the Koch snowflake is about 1.26-dimensional. This makes sense because the snowflake looks closer to 1-dimensional than 2-dimensional; it appears more line-like than plane-like.
Using the tiling idea to calculate the dimensionality of a fractal results in a number D called the Hausdorff dimension. No, this isn’t the name of some obscure German space music band. It is a well-established formalism introduced by the mathematician Felix Hausdorff in 1918. The same type of reasoning allows mathematicians to establish the Hausdorff dimension value for a wide variety of fractals.
I hope this mental journey has opened a new door for you by expanding your concept of what dimensions are. The idea that there is such a thing as a fractional number of dimensions is fascinating. This concept has practical value too. It shows up in economics, biology, geography, and many other real-world disciplines.
Beyond the specific concept of Hausdorff dimension, I propose that breaking the “integer barrier,” establishing that dimensions can be fractional, is an illuminating exercise for one’s creativity. It fosters a spirit of questioning assumptions and expanding one’s mind to reach beyond them. What other ways do seemingly obvious assumptions limit your thinking? What other wonders await you when you notice and challenge the “obvious?”