Since I had been having vision of the Mandelbrot set as a manifestation of god, I figured I'd better learn a little more about it. And what better place to start than the classic from the man himself? The only problem turned out to be that the book doesn't make it to a discussion of the famous bug until page 188, roughly halfway through, which turned out to be about as far as I could follow it. While Mandelbrot's style is mostly informal and aimed at 'the general reader', his explanations still often suffer from every mathematician's annoying habit of over-compressing things because they're 'obvious'. For me, that was fine with the less complicated material early on in the book (self-similar and scaling 'linear' fractals), and with a little effort, I felt pretty good about my grasp of his examples. But as the material became more complicated (non-scaling, self-inverse and non-linear fractals) more and more obvious things were anything but, and while I was still getting the gist of it, I was mostly just taking his assertions on faith.
Still, it was an enjoyable and mind-expanding journey while it lasted. And I did finally come away with a clear understanding of what saying that something can have a non-integer dimension means. It turns out that there are various ways of defining 'dimension', and the topological definition we are accustomed to is only one of them. Mandelbrot motivates another definition (the Hausdorff Besicovitch dimension) by considering the question: how long is the coast of Britain? The answer is that it depends on what size ruler you use to measure it. A kilometer long ruler appropriate for making a map will give you a smaller length than the roughly meter long ruler you would use if you were trying to walk the complicated edge of every bay and inlet. Similarly, an ant-sized ruler forced to crawl the perimeter of every rock would calculate an even larger length, and so on ... till we discover that the length of a coast seems to be infinite. Needless to say, since we don't usually think that the size of our ruler influences the length of the thing we are trying to measure, this realization poses a problem.
It turns out you can solve this problem and calculate a determinate coast length independently of the size of the ruler, only if you allow the 'dimension' of the coast to be a number between 1 and 2. This works because the coastline length calculation exhibits an approximate empirical relationship where the total length measured by a ruler of length e is proportional to e^1-D. The D in this equation is empirical and varies by coastline, but if we make an abstract model of a coastline using something like the Koch snowflake, we can calculate it exactly. If we choose D=1, we find that the coastline length tends towards infinity, for D=2, it tends to zero. If we pick the correct D in between these we get a well defined total length that no longer depends on which e we chose to use as our ruler size. This is exactly how we intuitively think measurement of length, area, and volume should work -- we break the original shape into smaller pieces, and then we raise the size of each piece to the appropriate power and add them back together again. It's just that the "appropriate power" for a rugged coastline happens to be non-integer.
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