In the 1980s, two ecologists, Jim Brown at the University of New Mexico and Brian Maurer at Brigham Young University, coined the term macroecology, which gave a name and intellectual home to researchers searching for emergent patterns in nature. Frustrated by the small scale of many ecological studies, macroecologists were looking for patterns and theories that could allow them to describe nature broadly in time and space. ... Brown and Maurer had been influenced heavily by regularities in many ecological phenomena. One of these, called the species-area curve, was discovered back in the 19th century, and formalized in 1921. That curve emerged when naturalists counted the number of species (of plants, insects, mammals, and so on) found in plots laid out in backyards, savannahs, and forests. They discovered that the number of species increased with the area of the plot, as expected. But as the plot size kept increasing, the rate of increase in the number of species began to plateau. Even more remarkable, the same basic species-area curve was found regardless of the species or habitat. To put it mathematically, the curve followed a power law, in which the change in species number increased proportionally to the square root of the square root of the area. ... Power laws are common in science, and are the defining feature of universality in physics. They describe the strength of magnets as temperature increases, earthquake frequency versus size, and city productivity as a function of population.

Consider the most familiar random shape, the random walk, which shows up everywhere from the movement of financial asset prices to the path of particles in quantum physics. These walks are described as random because no knowledge of the path up to a given point can allow you to predict where it will go next. ... Beyond the one-dimensional random walk, there are many other kinds of random shapes. There are varieties of random paths, random two-dimensional surfaces, random growth models that approximate, for example, the way a lichen spreads on a rock. All of these shapes emerge naturally in the physical world, yet until recently they’ve existed beyond the boundaries of rigorous mathematical thought. Given a large collection of random paths or random two-dimensional shapes, mathematicians would have been at a loss to say much about what these random objects shared in common. ... have shown that these random shapes can be categorized into various classes, that these classes have distinct properties of their own, and that some kinds of random objects have surprisingly clear connections with other kinds of random objects. Their work forms the beginning of a unified theory of geometric randomness. ... “You take the most natural objects — trees, paths, surfaces — and you show they’re all related to each other,” Sheffield said. “And once you have these relationships, you can prove all sorts of new theorems you couldn’t prove before.” ... incoherent is not the same as incomprehensible. ... In practical terms, the results by Sheffield and Miller can be used to describe the random growth of real phenomena like snowflakes, mineral deposits, and dendrites in caves, but only when that growth takes place in the imagined world of random surfaces.