Now we're talking...

Brilliant discussion of this feature here...

http://scienceforums.com/topic/13548-do-a-spiral-galaxy-and-a-hurricane-share-a-similar-formation-mechanism/

Virtually every aspect of fractal geometry and type of dynamical system can be expressed by variations upon the simple quadratic iterator: X = X**2 + c which expresses the particular type of feedback being examined, Phi can be expressed by a related but more archetypal variation to derive the Fibonacci series: X[n+1] = X[n] + X[n-1] which incrementally gravitates towards a particular ratio which possesses unique qualities. Numerically, it can be derived from the relation: (1 + sqrt(5))/2. For example, if one diminishes Phi by Unity you derive its reciprocal. Additionally, Phi is the unique ratio that fulfils: 1/Ø + 1/Ø**2 = 1 in other words, Phi is also the only possible geometric and arithmetic, expansion and partitioning of One.This leads us to the other cardinal feature of Phi. There is only one proportional division of One possible using two terms, with the third being One itself. From Euclid's ELEMENTS Book Five, Theorem Three (Alexandria, 3rd century B.C.):"A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."The Golden Mean then, is an archetypal fractal in that it preserves its relationship with itself (its inherent similarities under scaling are conformal symmetries - with topological consequences, that are invariant about themselves), in the most mathematically robust, economical but also elegant, way. It is analogia exemplified.As we shall see, this reciprocal, squaring behaviour about One, or Unity, as it is more properly termed, is far from being mathematically trivial.All feedback loops deterministically involve the passage of time. The quadratic iterator is derived from Newton's differential calculus, and from a period when nature was seen as a mechanistic and time-reversible automaton. Recent science demonstrates that in fact it consists of both the above and irreversible processes, known as the entropy barrier, or the arrow of time. The Golden Mean can also be seen as mathematically (because of the above) the simplest and most stable way of communing or mediating between the two, as we shall see.Nigel Reading
Quote
You may assume.But it is never safe
Quote
According to the Nobel laureate Ilya Prigogine, these far from equilibrium dissipative systems locally minimise their entropy production by being open to their environments --- they export it in fact, back into their environments, whilst importing low entropy. Globally, overall entropy increase is nevertheless preserved, with the important caveat that the dissipative system concerned often experiences a transient increase (or optimisation) of its own complexity, or internal sophistication, before it eventually subsides back into the flux.This is known as the region of alternatively, Emergence, Maximum Complexity, Self-organised Criticality, Autopoiesis, or the Edge of Chaos. (Nascent science debates nomenclature routinely - and appropriately, in this case, the crucial point being that they are all different terms for essentially the same phenomena.)Lifeforms, ecosystems, global climate, plate tectonics, celestial mechanics, human economies, history and societies, even consciousness itself - all manifest this feedback-led, reflexive behaviour; they maximise their adaptive capacities by entering this region of (maximum) complexity on the edge of Chaos, whenever they are pushed far from their equilibrium states, thereby incrementally increasing their internal complexity, between occasional catastrophes.Remarkably, this transition zone is mathematically occupied by The Golden Mean. This ratio acts as an optimised probability operator, (a differential equation like an oscillating binary switch), whenever we observe the quasi-periodic evolution of a dynamical system. It appears in fact, to be the optimal, energy-minimising route to the region of maximum algorithmic complexity, and to be a basin of attraction for the edge of Chaos.

[credit Nigel Reading]