In any complex system we inevitably come across the predominance of some patterns or relationships over others. For example we talk a lot in trading circles about the ‘fractal’ nature of markets or the dominance of certain features such as the Golden ratio or Fibonacci sequence. Now is this just BS for the tea leaf readers or is there something deeper in the significance in certain features of the data?

A gifted mathematician sent me a YouTube presentation the other day about the tendency of all river systems in a depositional environment (such as a floodplain) to adopt very strict rules in the way they meander. In fact this is evident in all rivers across the world. When unimpeded by structural bedrock, they simply do not flow in a straight line towards the river mouth. They all adopt a meandering pattern by virtue of a feature related to one side of the river which is faster flowing eroding into the floodplain while the other side of the river with corresponding slower flow depositing sediment at its edge. This is what causes the meander. The torque associated with the variable flow of the river with its passage on top of the geomorphology. This relationship is a mathematical one and imparts structural rules into how the river flows…..and this is repeated across all river systems in the world.

Now the local natives may have embedded some mystical quality into the meandering feature of the river system…imbuing the river with ‘serpent-like’ qualities but the scientists amongst us would nod their head and say….that’s a very rational explanation.

Now in the same way…we need to be careful about treating concepts such as the Golden ratio…or the Fibonacci sequence as simply ‘woo science’….because with the emergence of a concept of quasi-crystals we start seeing the why in the significance of data relationships such as the Golden Ratio.

Until the emergence of quasicrystals into the scientific lexicon…we only had crystals.

Basically, a quasicrystal is a crystalline structure that breaks the periodicity (meaning it does not have translational symmetry, or the ability to shift the crystal one unit cell without changing the pattern) of a normal crystal for an ordered, yet aperiodic arrangement. This means that quasi-crystalline patterns will fill all available space, but in such a way that the pattern of its atomic arrangement never repeats.

Since their discovery in the 1960’s thousands of examples have been found in nature…..but how and why?

We are all familiar with the properties of a crystal. Most crystals are composed of a three-dimensional arrangement of atoms that repeat in an orderly pattern. Depending on their chemical composition, they have different symmetries. For example, atoms arranged in repeating cubes have fourfold symmetry. Atoms arranged as equilateral triangles have threefold symmetries. But quasicrystals behave differently than other crystals. They have an orderly pattern that includes pentagons, five-fold shapes, but unlike other crystals, the pattern never repeats itself exactly.

Now one of the features of quasicrystals is that there are vastly many more quasicrystal shapes than crystal shapes…..why?

We can create quasicrystals by projecting a crystalline shape in 3D onto a 2D surface. The shadows of the crystal on the 2D surface are not periodic in nature….but they do hold certain relationships when the 3D crystal is rotated. The result is a cross-hatched sequence of shadows on the 2D surface that are not periodic but are definitely orderly patterns that ‘hold’ onto specific properties. The rules of the 3D geometry are transcribed onto the lower dimensional surface. The 2D representations have multi-variate 2D rules that they now have to obey……they have co-dependent relationships (features of co-integration) built into their 2D arrangement.

In other words, the 2D shadows cannot simply be geometrically arranged in ‘any way’ as chaos would suggest….but now must be configured in ‘very specific ways’ geometrically defined by the higher order symmetry. Now nature following the ‘principle of least resistance’ is bound to follow the simplest way to geometrically arrange itself….but some ways are forbidden by the laws of the universe…whereas other geometrical arrangements become self-evident.

Now the natural features of fractals also hold this curious feature. They are the simplest way for a geometric structure to unfold with ‘specific rules applied’ that govern their geometry. Nature always chooses the simplest path and it does this by repetition of these rules again and again and again. They obey the ‘Laws of the greater system’ or are restricted by these Laws in the way they can unfold….but they use these laws to govern what is possible and what is not. As a result, some patterns or relationships start to dominate over others. This is referred to as self-regulation by the lesser system….but the reality is that it is the wider system that is actually regulating what are the possible configurable ‘states’ in the sub-system.

…..but you do not need to step into a higher dimensional geometry to produce these quasi-crystalline shapes or fractal geometries. You just need to project the rules of a global system into the space of a nested system to achieve this. What you are doing here is projecting multi-variate conditions of a global geometry (such as the laws of our universe) into the sub-space of a nested system (such as a living system…….a financial market……a cosmological cluster etc. etc. etc.). What this projection does…is that it gives credence to bias in the underlying system and creates a ‘dominant’ feature in that nested sub-system….which has more ‘weight’ than any other random shape.

The reason that we get order from the chaos is because when dealing with a chaotic subsystem that is contained within a greater system such as our very ordered universe….the rules of the order from the greater system start self-organizing the chaos in the sub-system by a principle of simply ‘biasing the data set’ of the information contained in the sub-system.

When you get a persistent feature arising throughout a complex system such as a certain mathematical ratio or relationship which appears more frequent than a random happenstance…..there is inevitably a ‘real’ reason for it’s occurrence. Not a ‘woo’ reason….but a real physical reason. For example in Euclidean space no matter what size of triangle I geometrically describe, the 3 angles of the triangle always equals 180 degrees. This is not a tea leaf reading event of circumstance…but a real aspect of a particular geometrical property. Because this planet approximates Euclidean space at the small scale, the triangle I draw in the sand at my beach in Australia will have that same relationship that a large triangle drawn in Europe will have….or Jupiter….or any very large planetary system where on the small scale the surface is effectively flat.

You will therefore see this feature with any small scale triangle you observe on this planet. In fact the reason why that triangle at very large scales **does not **demonstrate this relationship, but rather the angles of the triangle drawn across the horizon is greater than 180 degrees….is actually a mathematical proof for the earth being round and not flat.

So don’t be too hasty to draw definitive conclusions regarding recurring features of a system such as a Golden ratio or Fibonacci sequence as there is a very strong likelihood that there is a very ‘sane’ reason for its occurrence that just may be hidden in plain sight.

Is the Golden ratio ‘woo science’…or are we still too naive to really understand it?

Make sure that you are not blind-sided by reductionist logic and can still step back and absorb the big picture. What you will find in doing so and in looking hard enough is that it is all ‘systems within systems’ without end. It is all about processes guys and the rules in which each of these systems play out….not the arbitrary ephemeral patterns that we may call in our naive restricted viewpoint as the ‘things’.

Trade well and prosper

Rich B