The pictures show very different objects, yet their images look rather similar in many ways. Not really, because a line is one-dimensional, whereas a coastline is a random fractal that has a dimension whose value is between 1 and 2. The growth of these structures can be modeled by a process called aggregation , in which random walkers dance around the growing structure and stick to it when they touch it. The resulting jagged pattern is called an aggregate . Depending on the details of the growth process, some aggregates are natural fractals, others are not.
Dancing means to stagger around randomly.
We can use our understanding of random walks to mimic the process of electrochemical deposition. Another word for mimic is model . We model the aggregation process using our knowledge of a random walker who staggers around and sticks to a growing structure. Compare this model to your picture of the motion of ions in the electrodeposition experiment. This applet explores this model.