Aggregation is a process in which the particles stick together to build characteristic structures. Such aggregation can often be identified by complex, disorder appearances, but it is not entirely random in the global sense.
For many years scientists have been trying to use different words to describe the shape of such aggregates - grainy, seaweedy, tangled, wiggly, tortuous, to name but a few. However, Mandelbrot, a mathematician called such shapes as fractals (In Latin, fractus means to break to create
irregular fragments). In nature, aggregates often show fractal shape. For example, the aggregation of soot and dusts. Beside particle aggregation processes, fractals also occur in the motion of air bubbles in oil, random patterns of coastlines and even tree and plant growth.
One typical characteristic of factals is that they always look similar at any scales although the detailed underlying structures are formed by random processes. For example, the photo below shows a tree branches. Look at a small group of branches (as highlighted). When it is magnified, the details may look different but
the overall shape look similar to that of the whole tree.
This characteristic feature is called self-similarity. This self-similarity is valid only in average. The new object obtained after enlarging part of the original object is statistically equivalent to it. Because of the properties of self-similarity fractal objects are said to be scale invariant. Scale invariance is a 'symmetry' of fractals.
Just like round objects are symmetric under rotations, fractals are symmetric under dilation, or changes of scale.