In Mandelbrot's classic text, the second chapter is titled: How long is the coast of Britain? I will describe the sandy beach in the two-dimensional context of a map. Thus, the ocean and the land are mostly two- dimensional. Before fractal geometry, the map showed the boundary between the ocean and the land as a smooth curve: a one-dimensional coast. But now, thanks to Mandelbrot, we may zoom in on the coast, and see that it has very small islands, even pebbles, in a densely packed structure. Zooming in again, we see grains of sand on the beach, and in the ocean close to the beach. All this is the coast: it has a fractal dimension. Land penetrates into the ocean in a frothy structure of sand, ocean penetrates into the land in a frothy structure of water in the wet sand. Not only is the coast a fractal, with a dimension more than one but less than two, but it is a fractal region: the coastal zone. The ocean and land are not divided by the coast in a binary fashion: they interpenetrate in a fractal geometry. The fractals of chaos theory (attractors, separatrices, and bifurcations) are all of the sandy beach variety.
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Thanks for the notes too. I never really studied the subject academically but do know that endless fractal possibilities exist. Your photo is an elegant example.
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