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Fractal Dimensions

How long is the coast of Britain? The answer, surprisingly, depends on the size of your ruler. If you measure with a big stick, you will only pick up the rough features, but if you measure with a smaller one, your route will be longer. In 1967, Benoît B. Mandelbrot wrote a paper called How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. In it, he contends that, contrary to popular opinion, “curves of dimension greater than one are [not] an invention of mathematicians.” Instead, Mandelbrot says, many real-world curves are actually statistically self-similar: in some sense, the small parts can be said to be scaled-down versions of the whole. And we can put a measure on the “degree of complication”, a measure that has many things in common with the notion of a dimension.

You are no doubt familiar with fractals, those beautiful self-similar patterns the study of which Mandelbrot is credited with founding. You may have heard that they have fractional dimension, a term that seems prima facie nonsensical. If we think of dimension as, say, the number of coordinates needed to specify a point in a figure—one on a line, two on a plane, three in a volume—it makes no sense. You can’t have 1.58 coordinates. But as Mandelbrot explains in his paper, and as explained excellently here (9-min video) or here (text), it follows directly from a different, but equally intuitive conception of what a “dimension” is.

Consider a line segment. Suppose you want to double its size. How many copies of the original do you get? Straightforwardly, you get two. But if you consider doubling both sides of a square, you get 2*2 = 4 copies of the original square. And if you double each side of a cube, you get 2*2*2 = 8 copies of the original cube. If you triple the sides, you get 27 copies. In fact, we can quite simply make an equation relating the dimension D, the enlargement factor e and the number of copies, c. It looks like this: c = eD. Using logarithms, we can transform this into the following: D = loge(C)=log(c)/log(e).

You can easily verify this for the cube, the square and the line. Tripling the sides in the cube, we get D = log(33) / log(3) = 3log(3)/log(3) = 3.

Now consider a simple fractal like the Sierpinski triangle. Each iteration of the fractal is made by removing the middle third of each triangle, creating three identical copies. Each one is half as big as the previous iteration. When we enlarge one piece, we get three copies of the original. Plugging this into the equation, D = log(c)/log(e) = log(3)/log(2) ≈ 1.585. So the fractional dimension of the Sierpinski triangle is almost 1.6; it is neither 1, like a line, nor 2, like an ordinary figure in the plane.

You can find a list of fractals by fractional dimension on Wikipedia. Some fractals do actually have integer dimension; many do not.

The quadratic cross, for instance, has a dimension of 1.49.

What does this have to do with science? Well, as Mandelbrot suspected, fractals and fractional-dimension curves are very useful tools for describing a number of real-world phenomena, not limited to pretty pictures and coastlines. Everything from earthquakes to heartbeats to ice crystals has been described as a fractal phenomenon.