A frost pattern on a window doesn’t spread evenly: it branches out, seizes voids, advances in jerks. Much like that, the growth of many shapes in nature can be described by a mathematical scheme invented a century ago. Here, a differential equation works like a recipe: you feed in a signal, a driving function, and the recipe tells you how the shape will change. The classic recipe demanded the signal be continuous, like slowly turning a knob. But life is full of sharp jolts.

Mathematicians Eveliina Peltola and Anne Schrøder allowed the signal to “jump” and discovered that each jump adds not a point but a whole chunk to the shape. Sometimes these are bubbles — open regions that suddenly turn up inside, like air pockets in a frost pattern. Other times, they’re dense clusters of boundary points, firmly stuck to the edge. This picture is very important for the Schramm–Loewner evolution (SLE), a branch of mathematics that studies random fractal curves. These curves resemble coastlines or lightning bolts, and their behavior is governed by the laws of probability. When jumps occur in such stochastic processes, the very bubbles and clusters described in the paper appear. This provides a key to numerical simulation of rare events — for instance, sudden ruptures in materials.

Now the theory is ready to describe growth in jerks, and the frost lace on your window gains a coherent meaning. Each pattern is a chronicle of temperature and humidity jumps, frozen into the structure of the ice. And mathematics is learning to read that chronicle to better understand the nonlinear effects around us — from the growth of biofilms to the spread of cracks.