Tuesday, July 16, 2013

Music Mathematics Part II. Timeless Music Theory: Intuitive Fractal British Coastline

Read Part I first!

The repetition at multiple scales we just constructed with Sinatra's musical phrase occurs throughout nature – many examples are found in Benoit Mandelbrot’s book, The Fractal Geometry of Nature.

The best known example is the problem of how to measure the length of the British coastline. There is a straightforward approach, which is like using a stopwatch to measure the duration of music: you buy four very long rulers, you and three friends stand at the four corners of Britain, and voila, you add the 4 lengths together, and calculate say 2,000 kilometers.

But rulers don’t account for the fact that the British coastline is craggy at different scales, ranging from a promontory seen from a space satellite to tiny pools in the sand seen with a magnifying glass. If you measure the coast with a 200 kilometer ruler, you need to place the ruler in 12 straight lines that approximate this irregular coastline end to end, and calculate the coastline is 2400 kilometers.

Or with a 50 km ruler, we place it end to end 68 times, and come up with a length of 3400 kilometers instead of 2400.

The reason the coastline length is so hard to measure is that a straight line measures one dimension, length alone, but not the pattern of the coastline, which is better described in at least two dimensions, length and width.

The problem  is similar to the question of how to measure the circumference of a circle: it’s hard to measure the edge of a circle with a ruler, but you can measure it’s diameter with a ruler in one dimension, and then use a trick: multiply the diameter by the number pi. This  yields a precise measurement of the circumference in one dimension, even though the circumference itself has a two dimensional shape with length and width. If we measured the surface area of a ball instead of a circle, we can use the same kind of trick, measuring the radius with a ruler, squaring it, and multiplying it by 4 times pi. The ball is in three dimensions, but the surface area in two dimensions.

Mandelbrot’s insight was that the coastline doesn’t expand infinitely, but converges on a length. He introduced a math trick, which is to add a little bit of an extra dimension, called Hausdorff or fractal dimensions: the description of the length of the British coast turns out to be accurate if you estimate it using a about 1 and one quarter dimensions. So closer to a line than an area, but getting towards an area. Countries with straighter boundaries like Germany have a dimension of about 1.15.

While imagining fractional dimensions is frustrating, you can imagine that the repeated smaller and smaller rulers placed along the coastline fill up more and more of the country’s area. If the smaller eddies and promontories continued forever, they would eventually cover the entire country. Then Britain would be all coastline, and either all land or all sea, in which case our linear coastline would be completely two dimensional by occupying all of the length and width. Britain would either be all land or all sea, or some kind of muck between: since it isn't, the length converges somewhere.

In a couple of more entries on this blog, you'll learn to make music that is analogus: without a time dimension, and neither music nor not-music, but some kind of muck that is in between.

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