Patterns exist everywhere, in space and in time. Some can be easily explained mathematically while others are too chaotic to make sense of.
Fractal geometry is the branch of study that tries to find sense in this chaos.
But what could possibly be the relation between fractal geometry and self organization? Well, it seems that self organization follows fractals laws and results in systems with a fractal pattern.
The aim of any system is, generally, to continuously be in a state of stability.
This stability is constantly being tested by both internal and external forces. So when a complex system (and non linear) becomes stressed, it becomes unstable. The system then tries to make changes to reestablish its stability. These changes can either be first order (linear, predictable, moderate, and incremental) or second order (chaotic and drastic). (Sandra Bloom, 2000). Generally in simple cases, these changes are either a division of the system or integration with another system. More often than not pattern of this integration or division is constant across the whole system both in space and time.
When a drastic change (2nd order) occurs to bring back stability, we call it Chaos. But Chaos always has a higher order but it comes on its own, spontaneously and unpredictably. It is self organized. (http://www.fractalwisdom.com/#paradigm). The order implicit in Chaos is unpredictable on a case by case basis, but still reliable and workable on the long run.
Quoting physicist Paul Davies (from his book The Cosmic Blueprint (1988)):
There is no detailed blueprint, only a set of laws with an inbuilt facility for making interesting things happen. The universe is free to create itself as it goes along. The general pattern of development is “predestined”, but the details are not. Thus, the existence of intelligent life at some stage is inevitable; it is, so to speak, written into the laws of nature. But man as such is far from preordained.
In this video Dr. Ron Eglash explains his research on the fractal patterns he noticed in villages across the African continent.
Some good references about fractal applications and self organization: