Fractals

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How is it that Norway, which is 67th in the list of the world’s countries according to the size of their landmass, has the world’s 7th longest coastline? The answer, as fans of The Hitchhiker’s Guide to the Galaxy will know, is that the Norwegian coast is made up of ‘fiddly bits’. That insight illuminates just one of the fascinating characteristics of the strange shapes known as ‘fractals’.

The definition of a fractal is actually quite simple to state, if not always to understand. Though it’s a mathematical phenomenon, often involving very complex formulae, here we’re simply concerned with fractals which can be drawn in two dimensions. So for present purposes a fractal is a shape made up smaller versions of the same shape, with each of those smaller versions made of even smaller versions – and so on ad infinitum. For our purposes we divide these shapes into two groups, those where the reproduction of the original shape occurs outside the border of the shape, and those where it happens internally.

The fractal rectangle design shown above is of the internal type. We start with a pair of multi-coloured squares, placed alongside each other to produce a rectangle. Each of the two squares is made up of 9 smaller squares, 8 of them coloured, and the middle one empty. In the second line of the design, each of the smaller squares is itself broken up into 9 even smaller squares – again 8  coloured and one empty. In the third line of the design, the even smaller squares are broken up in the same way. The design could go on forever in this way – though it would be impossible to reproduce on paper – and if you could zoom in on a single square, no matter what the size of the magnification, it would look exactly the same.

This ‘internal’ fractal has many strange qualities but two of them stand out. Looking at just a single original square we can see that it is 1/9 ‘empty’. When we replace the 8 coloured squares with small copies of the large square, we take away 1/9 of their area. In the first line, each large square is made up of 9 smaller ones, one empty square and 8 coloured ones. The large square is 8/9 coloured.

Now we break up each of the smaller squares into 3*3 even smaller ones, so we have 81 of the even smaller squares. All the original empty squares remain empty, so we start with 9 of the 81 empty. For each of the eight remaining 3*3 groups, we empty the one in the middle. So that’s another 8 of the 81 empty, making a total of 17 of the 81 empty, and leaving 64 coloured.

For the next stage we break up the even smaller squares further into 3*3 groups. Now have 9*81 or 729 tiny squares, 153 (9*17) of them are empty and remain so. 576 (9*64) of the tiny squares were originally coloured but we empty the middle one of each 3*3 cell, so that’s another 64 empty tiny squares. Now we have 217 empty squares, compared to 512 filled ones.

If we express the ‘filled’ space very roughly in percentages, the original squares were 89% filled, after the first stage they were 79% filled and after the second stage it 70%. The filled area is becoming a smaller and smaller proportion of the whole. And you may have noticed a pattern. Each time we break up the original shape into smaller and smaller squares, the number of squares increased by a factor of 9 and the number of filled squares increases only by a factor of 8. In other words the filled area will be equal to \frac{{{{8}^{n}}}}{{{{9}^{n}}}}, where n is the number of levels we have ‘descended’. As we continue to add new levels the filled area tends towards 0%, without ever completely disappearing.

Just as interesting, and without doing the calculations, the internal borders between filled and empty areas become longer and longer – in fact they tend towards infinite length. As a result our internal fractal has the fascinating quality that if we go to the extreme, the shape has infinitely small filled areas which have infinitely long borders.

The second type of fractal, illustrated by the star, is probably better known. In mathematics it is known as a Koch star. Here the complexity is added outside the border of the original shape. In this case we start with a star and notice that it is made up of an equilateral triangle, with smaller equilateral triangles added to the middle of each side. The smaller triangles have sides 1/3 the length of the biggest triangle. From now on, all we do is take each outward-facing edge and add a smaller triangle, with sides 1/3 the length of the edge. Then when every outward-facing edge had been processed we do the same again, only now the edges are smaller and so are the triangles we are adding. Each time we add a layer of triangles every outward facing edge increases in length by 1/3.

If we were to draw a circle around the original star, touching the six points, no matter how far we continue adding levels to our fractal, it will never extend beyond that circle. In fact the area of the star approaches 8/5 of the area of the first star but math is more complex ad we won’t go into it. But once again, and it is a big but, the length of the border around the star will increase every time we add a layer of complexity, tending towards infinite length around the limited area of the star.

Norway doesn’t quite qualify, but you get the point.

Where mathematics becomes an object of desire.