The Minkowski–Bouligand dimension of a clay brick

In the early 1990's, fractals and chaos were hot. In 1987, James Gleick had published "Chaos: Making a New Science", popularizing non-linear dynamics. Hydrologists played an important role in the development of fractal theory. Hurst had discovered that sequences of dry and wet years for the Nile showed very long memory effects. Instead of the chance of a dry year following a dry year being 50%, Hurst found that there were surprisingly many long series of dry or wet years. Seven fat years, seven lean years, as it is noted in Genesis. Scott Tyler found fractals in soils ("Fractal processes in soil water retention"). At Cornell, where we were at the time, David Turcotte described "Fractals in geology and geophysics". A few years later, Ignacio Rodríguez-Iturbe and Andrea Rinaldo would publish "Fractal River Basins: Chance and Self-Organization". In short, fractals were exciting scientific gold.

A fractal is not just an obscure mathematical object but something that can actually be found everywhere in nature. Early on, a paper was published in Nature with the title "Fractal viscous fingering in clay slurries" by Van Damme, Obrecht, Levitz, Gatineau, and Laroche. They "only" did an experiment on a fractal embedded in 2D; we should be able to do one better and find the fractal dimension of the surface of cracking clay embedded in 3D. So out we went, collected some clay, mixed it with water in a cement mixer, siliconed together a shallow "aquarium", and poured in the slurry. To observe the cracking of the drying slurry, a video camera was mounted above the experiment, looking down and taking time-lapse images. To access the views from the sides, mirrors were installed at 45 degrees at each of the four sides. Lights made sure the camera captured high quality images. The whole set-up was enclosed in a frame with dark cloth to ensure that lighting was always the same.  We already had some box-counting code ready to calculate the fractal dimension of the surface, called the Minkowski–Bouligand dimension. One variable needed some extra attention, namely the boundary between the clay slurry and the glass sides. If the clay would cling to the sides, it would be difficult to understand the effects that this boundary condition had on the outcome of the experiment. Moreover, the cracks may not have become visible in the mirrors when the sides were covered with mud. So, instead, it was decided to make the sides hydrophobic with some mineral oil. This ensured that when the clay would start to shrink, it would come loose from the sides. Now, all we had to do was wait. It took only a week or so before the consolidated slurry started to shrink and to come loose from the sides. After that, the clay continued shrink for many weeks. This is how we learned that the fractal dimension of a shrinking brick of clay is (very close) to 3.0.