This Week on Portsmouth Point: Fractals

By Ruthie G, Year 11

‘In the mind’s eye, a fractal is a way of seeing infinity’ - James Gleick

Fractals are some of the strangest and most beautiful shapes. These structures were first studied by the 20th Century mathematician Breniot Bandlebrot, who defined them as ‘rough’ or ‘fractured’ shapes - shapes which get more complicated as you zoom in, so eventually they become infinitely complex. This includes many shapes of nature; the squiggly coastline of the UK, for example. Madlebrot aimed to understand the rough structures of real life, and push back against the idealised curves and lines of conventional mathematics. In this article, we will follow in his footsteps.

The image above is of a classic fractal called Sirpinski’s Triangle, which we’ll use as a ‘test dummy’ throughout this article. If you enlarge any of the shaded regions, you will see a replica of the whole triangular structure. Zoom further, and you will discover more, even smaller replicas. This ‘self-similarity’ gives the Sierpinski Triangle the infinite complexity of a fractal.

One way to go about studying this strange structure is to ask what ‘dimension’ it is. We all understand that a line is 1D, a square is 2D, and a cube is 3D - but it’s challenging to define the dimension of a fractal.

For example, the Sirpinki’s triangle doesn’t seem to be 2D because it doesn’t have an area. Each time you add another repeating triangle structure to the fractal, you cut out ¼ of that area (the middle triangle is removed). Because you are adding infinite triangle structures, going all the way down the structure, eventually you eat away the area of the entire shape by removing so many little chunks.

That said, the Sirpiniski’s triangle definitely isn’t just a line; imagine drawing the fractal’s structure at a desk with pen and paper. The patterns get more and more intricate, and it almost seems to be filling the entire page - it has a certain full-ness which isn’t attributed to 1D shapes like the simple line.

However, these two observations are very vague; they’re based largely on intuition and feeling. What’s more, they seem to be paradoxical. If the Sirpiniski’s Triangle isn’t 1D or 2D, what is it? Because they get more complex as you zoom in, all fractals (so even shapes like the UK’s coastline) have this problem. They don’t all lie between 1D and 2D; however, they all seem to lie between some dimensions.

We can quantify this idea by using the Hausdorff measurement of dimension. The Hausdorff Dimension looks at how an object’s area changes as it scales up, which tells us a lot about its characteristics. If the object’s area scales up in a normal and predictable way, this means it must be a typical shape which fully fills the dimension it is in. If the object’s area scales in an abnormal way, there must be strange gaps or holes in the shape - so it doesn't fully fill the dimension it's in, and is ‘between dimensions’ like our fractals. This idea is analogous to how when you zoom into a normal shape it just reveals uniformity - but when you zoom into a fractal, it just gets more complex and intricate.

To understand the method, let’s start by calculating the Hausdorff dimension of a normal square. If you double the side length of a square, you end up with 4 times the area of the original square (imagine this as a 2x2 grid of four squares - each side is now double the length of the original side, and it has a four square area). If you triple the side length, you end up with 9 times the area, a 3x3 grid.

There’s a pattern here. We got an area of 4 when we doubled the side length, which is equal to 2^2. We got 9 when we tripled it, which is equal to 3^2 Every time you change the length of a 2D shape like a square, the area becomes the amount you scale the length by^2. Because the ‘amount you scale the length by’ (2 and 3 in our cases) is to the power of 2, this means that our square has Hausdorff dimension of 2. In fact, all 2D shapes share this quality - try it for yourself!

In contrast, let’s calculate the Hausdorff dimension of the Sierpinski triangle. To double the side length of a Sierpinski Triangle, we create a new triangle with double the side length. You can think of it like this: you have your original Sierpinski triangle; you add two more sierpinski triangles, one above and one to the side (leaving the one-triangle gap in the centre); and you have an exact copy of your old sierpinski triangle, just with double the side length. It also has three times the area, because you now have three Sierpinski triangles where before you had one.

Now, we must use this data to deduce the Sierpinski Triangle’s Hausdorff Dimension. When we doubled the square, we considered 2^What?= 4. The answer was 2, as 2^2=4, which told us its Hausdroff dimension was 2. Now, we have the equation 2^What?= 3. It isn’t half as easy to solve. We can use logarithms, a mathematical technique to deal with powers, to deduce that 2^1.585....=3. So, the Sierpinski Triangle is a 1.585 dimensional shape.

What does this actually tell us? Remember, the Hausdorff Dimension is about how much of a dimension an object fills. So, the calculation of 1.585 for the Sirpinski triangle fits with our previous hypothesis that it is more ‘hole-y’ than a 2D shape, yet more complex and dense than a line. It’s a bizarre discovery - a fractional dimension - but somehow it feels intuitive. Because of the oddities of fractal nature, a Sirpinkski triangle lies on a different plane to all of the ‘normal’ mathematical shapes.

In fact, all of Mandlebrot’s ‘rough’ shapes share this fractional-dimension property. Using the Hausdorff dimension, a line which draws the coast of Great Britain is 1.21 dimensional. This is, again, pleasingly intuitive. The coastline isn’t just a normal line, because it is so complex, infinitely so at a small scale. On the other hand, it certainly isn’t an area; it’s only made out of a line! It even seems right that the line to draw the coastline should be a little closer to 1-dimensional, and the Sierpinski Triangle should be a little closer to 2-dimensional.

Finally, we’ve reached the end of our quest to make fractals just make sense. Hausdorff’s method has offered us the opportunity to see fractals in terms of the dimensions they reside in and the space they fill. These structures are still incredibly complex - in fact, infinitely so. However, I take comfort in the fact that we’ve managed to quantify and understand these shapes despite their complexity. To me, it’s like we’ve taken a step away from the artificially perfect model of mathematics towards the real, tangible world we live in; the world with a 1.121-dimensional coastline.