The potted fractal and the history.

A very easy experiment that you may comfortably do in your kitchen. Let some water rest in a pot or other container for a sufficiently long time at a low temperature. I can't say exactly how much time is necessary, but leaving the pond for a whole day in the refrigerator would suffice. I happened to look at the results after casually leaving some water in a pot for a winter's night in the kitchen. The salt dissolved in water is going to aggregate in an irregular structure that floats on the surface, consisting of an incredible sequence of clusters, which resembles the image that I quoted in the title. That structure has the properties of a fractal, and arises from a process of gradual growth due to the fortuitous meeting of the salt particles dissolved in water and with the floating growing object. As new particles are drifting into contact with a particle already established, a chemical bond is formed and the new molecular grain of salt aggregates to the solid and increases it.

This process is described mathematically as the Diffusion Limited Aggregation (DLA in English - further details and animations are available on these pages of the Hong Kong Polytechnic University), which may well represent the physical process of crystallization far from the equilibrium point. Yes because under conditions of chemical and physical equilibrium there are no interactions between particles: all possible binds would be established, and the water molecules in Brownian motion oppose to the salt aggregation process. The process take-off consists of an accidental event: a pair of particles, which met first, begins to set other grains around them. This event accidentality makes it impossible to generate the same geometry if we were to repeat the experiment a second time: whatever the fortuitous circumstance that led to the formation of the first granular aggregate, this one will characterize the subsequent development and the form of the final structure. It is impossible, in retrospect, to understand why the solid has taken a specific form and not another equally plausible, because the structure will be fully determined by its history.