Standard geometric objects can be described by simple rules–every straight line, for example, is just *y* = *ax* + *b*–and they stand in neat relation to each other: Connect two points to make a line, connect four line segments to make a square, connect six squares to make a cube.

These are not the kinds of objects that concern Scott Sheffield. Sheffield, a professor of mathematics at the Massachusetts Institute of Technology, studies shapes that are constructed by random processes. No two of them are ever exactly alike. 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.

Yet in work over the past few years, Sheffield and his frequent collaborator, Jason Miller, a professor at the University of Cambridge, 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.”

In the coming months, Sheffield and Miller will publish the final part of a three-paper series that for the first time provides a comprehensive view of random two-dimensional surfaces — an achievement not unlike the Euclidean mapping of the plane.

“Scott and Jason have been able to implement natural ideas and not be rolled over by technical details,” said Wendelin Werner, a professor at ETH Zurich and winner of the Fields Medal in 2006 for his work in probability theory and statistical physics. “They have been basically able to push for results that looked out of reach using other approaches.”

### A Random Walk on a Quantum String

In standard Euclidean geometry, objects of interest include lines, rays, and smooth curves like circles and parabolas. The coordinate values of the points in these shapes follow clear, ordered patterns that can be described by functions. If you know the value of two points on a line, for instance, you know the values of all other points on the line. The same is true for the values of the points on each of the rays in this first image, which begin at a point and radiate outward.