Beechey in Fractals

What are the odds that this tiny rock, which I picked up at random from the gravel at Beechey Island, would happen to be shaped almost exactly like Beechey Island? Common sense would seem to suggest that it's incredibly unlikely, but -- as a friend to whom I showed this rock remarked, seen in terms of fractal geometry, the odds of such a thing are actually far closer to one than to zero. Fractal geometry often exhibits this property of "self-similarity," and there is even a mathematical formula for it. Like a glass hologram shattered into bits, each shard of a fractal form contains within it the same patterning forces and tendencies that shaped its parent. Of course, my Beechey rock has not been subject to erosion, ice-scour, tides, or other larger-scale forces; it's a bit more angular and rough. It appears to be a sort of igneous rock, and is pocked with little round holes that look like the craters on some irregularly-shaped moon. Nevertheless, the signature of nature lies within it; what I had picked up as a casual souvenir turns out to have a connection with its source that is anything but accidental.

These patterns were first noted by the mathematician Benoit Mandelbrot, after whom the 'Mandelbrot set' is named. His 1983 book, The Fractal Geometry of Nature, shows the prevalence of such forms in all manner of realms, ranging in scale from microscopic particles of dust to entire galaxies in space. His ideas were at first received with considerable skepticism, but have since been widely accepted and better understood; their combination of widespread applicability and mathematical elegance sets them as a remarkable leap in the understanding of the world around us.

In passing, I'd like to note that this is the 100th post on this blog since I began it in February of 2009 -- I'm still finding new things to write about, and I look forward to many more posts and responses to come. Since the start, we've had 34,833 pageviews, from all around the world -- over 4,000 from Canada, 3,000 from Australia, 1,800 from Poland, 1,500 from Germany, more than 1,000 from Ireland, the Netherlands, Itay, and Hungary, and even 409 from India!

My thanks to everyone who has read and posted on the site.