When size doesn't matter: our multifractal world

Shaun Lovejoy

Common sense says that if we could voyage through scale, zooming in to smaller and smaller scales, seeing smaller and smaller structures, we would constantly discover new and different worlds. The prevailing scientific views of many complex systems such as landscape topography, the weather or climate, simply take this common sense on faith, believing a priori that at every factor of ten or so in scale, qualitatively different mechanisms, different laws are at work.

Unfortunately, common sense is not a substitute for systematic scientific analysis. Over the last twenty years many analyses have shown that on the contrary many natural systems have quantitative characteristics called "scaling exponents" which - over sometimes huge ranges - are independent of size, they are "scale invariant". However, when we zoom into the systems, they are found to change subtly with scale - becoming for example more and more "squashed" or rotated. In response, scientists generalized the notion of scale to include the squashing and rotation. This means that the system can still be considered to be scale invariant even if it must be squashed and/or rotating after zooming into it. The resulting scale invariance is a symmetry principle (akin to mirro symmetries) which is far richer and far more applicable than the classical scale invariance which requires strict similarity before and after zooming.

Giving examples mostly from the geosphere (clouds, wind, precipitation, climate, mountains) as well as copious synthetic images, we show that this generalized scale invariance can hold over huge ranges giving rise to fractal structures, multifractals fields.