Catastrophic Power Law and Everyday Life A recent massive Toronto propane explosion, and the city's pitiful response has inspired me to resurrect my Claptrap article and beef it up for a Knol.
Power Law is the fundamental force behind most large-scale disasters, such as earthquakes, and industrial accidents. It is a consequence of the fractal, self-similar ordering of most natural systems, and human societies. Disasters are a consequence of Power Law coming against the natural linear thinking of the human brain.
Fractal Ordering and Nature
Take a look at most natural scenes, and you will find them pretty. A rugged coastline, a cliff face, a beautiful tree, and a sunflower. All of them appeal to our sense of proportion. These are all fractal patterns, and are created from the application of simple growth rules, applied over and over again. When this concept first came out, I studied it intensely, and could never get over the complicated math that accompanied it. I dismissed it as merely a way to make pretty pictures. Over the years, I have become to realize that it is one of the most fundamental forces of nature, and that the concept is very simple. Take an everyday tree. You can derive some very simple rules for making that tree, simply by specifying the degree of branching (two, three, etc), the length of the base twigs, and then state that a new sprout is simply another tree growing off the shaft. The tree can become immensely complex! The 'roughness' of the tree is a consequence of the branching rules.
Geology is my love, and here's where fractals totally rule! Take this pretty picture, for example.
Fractals have some important attributes that affect us in everyday life. Look at the pretty clouds, nice and fluffy. You can zoom in on the clouds, and still see the same patterns as in the larger clouds. Most clouds have the same fractal roughness, but limiting dimensions are important here! There is no practical limit on the small side, since you could zoom in forever, and it would be fractal, until you hit the scale of condensation droplets. But on the large side, there are varying limits. On a nice day, the fluffy clouds are limited to a certain size, before the fractal pattern breaks down. In Ontario this summer, we have had rough fractal clouds with no limit on the large side. That means you constantly experience all types of weather on the same day! The cloud patterns are perfectly fractal on the satellite and radar. The sunny breaks and the rain follow Power Law. Lots of little ones, and then some super big thunderstorms or sunny periods. Very pretty cloud formations, very depressing weather!
Human societies love fractals. We organize ourselves in families, villages, regions, countries, etc. Most companies also organize in self-similar hierarchies. Every smaller group resembles the larger group as a whole.
Power Law can be a truly horrible thing.