Your fractals are the result of a process called viscous fingering: As the paint is squeezed between the plates, the viscous paint spreads out evenly in all directions into the less viscous air layer, creating a stable, disk-shaped boundary.
When the plates are pulled apart, the less viscous air penetrates the more viscous paint, creating an unstable boundary. Small indentations of air grow and become fingers of air. Random indentations in these fingers grow as well. By the time the two plates are separated, the fingers of air have formed intricate branching structures in the paint.
The patterns created with this process often remind people of tree branches or root systems, river deltas, or lightning bolts, all of which are outstanding examples of fractal patterns in nature. A few others are clouds, coastlines, jellyfish tendrils, coral reefs, and blood vessels in the lungs. The photo below shows another example of a natural fractal pattern, the branching of a river. This is typical of the pattern formed when fluid flows from tributaries into a central stream or flows out from a main course into several branches.
All the fractal patterns formed in nature—including the ones you just made—are generated by random processes. As the patterns repeat themselves at different scales, each section of the whole is similar to large- and smaller-scale structures, but are never an exact copy. If you break a floret of cauliflower or broccoli off the larger head, for example, you can see that it's like a miniature version of the larger head, but it's not an exact replica. These repeating but nonidentical patterns are called self-similar.
Even though nature can’t generate a perfect fractal pattern, mathematicians can. The fractals they create are called perfect or mathematical fractals. If you look at a small section of a mathematical fractal, the section will be identical to the whole object. And if you were to select an even smaller piece from the first section and magnify it, this piece would also duplicate the whole. In fact, in this kind of fractal you can’t tell the difference between the whole object and a magnification of any section you select—no matter how small! The image below shows a classic mathematical fractal, the Sierpinski Triangle, or Sierpinski Gasket.