Wheels within wheels

“Clouds are not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight line.”
~ Benoit Mandelbrot

 

Of course, there’s more to mathematics than numbers. Mathematics is a way of modeling reality, and there are a range of approaches to that modeling. One of these is to simply interrogate some aspect of reality, and attempt to devise a representation that models the result.

The French mathematician Benoit Mandelbrot took on the formidable challenge of modeling the physical contours of things in the objective realm. He pursued a method of modelling the shapes of clouds and mountains and coastlines, which were not immediately apparent as obeying any coherent mathematical principle. In 1967, he published a paper that asked the innocuous question “How Long is the Coast of Britain?” – it was to prove revolutionary, and would give rise to the concept of fractals. This concept is very important to my own parasimplistic worldview, but of course it has far more important implications than that.

Coastlines, it turns out, are tricky things to measure. If one attempted to measure the coastline in units of, say, 10-meter lengths – approximating the actual contours of the coastline – one would find a shorter result than if one used 1-meter lengths. The 1-meter lengths would give a better approximation, and would be longer as a result. In fact, as Mandelbrot’s paper illustrated, the shorter the unit of measurement becomes, the longer the overall measurement becomes. In the limit of an infinitesimally small unit of measure, the coastline of Britain becomes infinite.

This might seem at first blush to be absurd, but Mandelbrot expanded on this result to show its consistency with a whole family of known mathematical relationships that exhibit the property of self-similarity – that is, the curve viewed at a large scale resembles the same curve at successively smaller scales. With a self-similar curve modeling variable x against variable y, the appearance of the curve between, say, 1 and 2 will be the same as the appearance of the curve between 1.0 and 1.1, or between 1.00 and 1.01, or at any smaller scale of measure. Such curves are said to have a Hausdorff dimension between 1 and 2 – the upper bound, curves of Hausdorff dimension 2, are known as Peano curves and have the property on successive iteration of completely filling the space over which they are measured, after the fashion of a ‘Greek key’ motif. Moreover, Mandelbrot listed several examples of naturally occuring self-similar relations – famously including the leaf fronds of ferns.

Mandelbrot was, like most mathematicians, building on the work of predecessors (including in this case Lewis Fry Richardson, who had tackled the coastline paradox himself and posited a mathematical law governing coastlines that foreshadowed Mandelbrot’s result that coastlines were self-similar). The focus of his 1967 paper, which was to form the basis of his work until his death of pancreatic cancer in 2010, was actually a modern application of a very ancient paradox proposed by the Greek philosopher Zeno of Elea – the dichotomy paradox, famously elaborated in his paradigm of a race between Achilles and the tortoise. Zeno’s paradoxes give us a useful structure for considering parasimplicity and being-in-time (what Heidegger refers to as Sein-in-der-Welt), and we will be returning to them.