Geological structures

Typical landscapes on Earth are to a first approximation formed by regions of crust being uplifted through tectonic activity, then being sculpted by progressive erosion (and redeposition of sediment) associated with the flow of water. (Visually very different special cases include volcanos, impact craters and wind-sculpted deserts.) Eventually erosion and deposition will in effect completely smooth out a landscape. But at intermediate times one will see all sorts of potentially dramatic gullies that reflect the pattern of drainage, and the formation of a whole tree of streams and rivers. (Such trees have been studied since at least the early 1900s, with typical examples of concepts being Horton stream order, equal to Depth for trees given as Mathematica expressions.) If one imagines a uniform slope with discrete streams of water going randomly in each direction at the top, and then merging whenever they meet, one immediately gets a simple tree structure a little like in the pictures at the top of page 359. (More complicated models based for example on aggregation, percolation and energy minimization have been proposed in recent years—and perhaps because most random spanning trees are similar, they tend to give similar results.) As emphasized by Benoit Mandelbrot in the 1970s and 1980s, topography and contour lines (notably coastlines) seem to show apparently random structure on a wide range of scales—with definite power laws being measured in quite a few cases. And presumably at some level this is the result of the nested patterns in which erosion occurs. (An unrelated effect is that as a result of the dynamics of flow in it, even a single river on a featureless landscape will typically tend to increase the curvature of its meanders, until they break off and form oxbow lakes.)