How Do You Like Them Apples?

Eulers Identity - 003

Named after the Swiss mathematician Leonhard Euler – Euler’s Identity is often described as an example of deep mathematical beauty.  Some see it almost as proof of the existence of some hidden order of the universe that we can only catch a glimpse of…

This equation is considered amazing because it seems to sets up a relationship between three apparently completely unrelated subjects.

“e” originated from the study of compound interest (and was subsequently found to be the base of the natural logarithms); “π” originated from Geometry and Trigonometry; and i” was “invented” to help with solving equations involving negative numbers.

Compound Interest, Trigonometry, and Imaginary Numbers?…  The question that has intrigued mathematicians and physicists since Euler first published his famous formula in 1748, is “Why in God’s name do these (apparently completely unrelated) areas of mathematics fit together so neatly in this beautiful equation?”

Well here’s the answer – Watch!

[Note: What follows is just a quick jump-thru, if you want the in-depth understanding click here]

e  =  Limit as n →  (1 + 1/n)n  =  2.718281828459045……

For practical purposes, “e” is difficult to pin down is simply a consequence of the fact that “e” is a manufactured quantity, defined using “limits to infinity”, and infinity is an idea not a number.

In reality there is no definitive value of “Transcendental” “e” because we can theoretically infinitely add more and more decimal places to the value of “e” by constantly increasing n all the way to infinite (which doesn’t actually exist).

However although the concept of infinity might be vague, “the concept of “e””, on the other hand, is actually all about very finely-grained precision…

The concept and formula for “e” is neat and very precise because the number of steps (n) must be the exact inverse of the tiny step-size (1/n).  

In order to discuss the preciseness of “e”, it is better, for the purposes of this discussion, to limit the influence of “infinity” and define “e” with very precise numbers.

To do so, we will use both a very large number and a very small number which are exact inverses of each other.  We will use the symbol Ħ for the very large number, and we will use the symbol ħ for the very small number.

Ħ  =  10,000,000,000,000,000,000,000,000,000,000,000  =  10+34

ħ  =  0.000,000,000,000,000,000,000,000,000,000,000,1  =  10-34

Since ħ = 1/Ħ we can now write a precision equation for “e” as

e = ( 1  +  ħ )Ħ  =  2.718281828459045……

Okay, so now away we go…

e = ( 1  +  ħ )Ħ

e1/Ħ = ( 1  +  ħ )

eħ = ( 1  +  ħ )


Set       d  =  eħ  =  ( 1  +  ħ )


 dxĦ  =  (1 + ħ)xĦ  =  (eħ)xĦ  =  (eħĦ)x  ex 


d  =  (1 + ħ)1Ħ  (eħ)1Ħ  =  (eħĦ)1  =  e1  =  2.718281828459…

[Ħ micro-steps of (1+ħ) is equivalent to 1 macro-step of 2.71828]

Okay that was for real numbers, now let’s do complex numbers…

(d)i  =  (ex)i  =  cos(x) +  i sin(x)

(d)i  =  ei  =  cos(1) +  i sin(1)

di  =  eħi  =  cos(ħ) +  i sin(ħ)

di  =  (0.99999999… +  iħ)

euler -4

di(xĦ)   =   ( 1 + iħ )xĦ   =   (eiħ)xĦ   =   (eħĦ)ix   =   eix


di(1Ħ)   =   ( 1 + iħ )1Ħ   =   (eiħ)1Ħ   =   (eħĦ)i1   =   ei1

[Ħ micro-steps of (1+iħ) is equivalent to 1 macro-step of 1 Radian]


di(3.14159…Ħ)  =   ( 1 + iħ )πĦ   =   (eiħ)πĦ   =   (eħĦ)iπ   =   eiπ      

[3.14159…Ħ micro-steps of (1+iħ) is equivalent to 1 macro-step of π Radians]


     di(πĦ)    =    -1.00 + i0.00  

⇒     di(πĦ)  =    -1



        How do you like them apples?

So why is any of this important? 

It is important because despite what is generally believed, Euler Identity is not telling us something special about “rotations”; it is telling us something special about “oscillations”.

[Remember: this was just a quick jump-thru; for a more detailed explanation click here.]

Euler’s identity is a beautiful example of the “quantized mathematics of change”.

D and Di - Yellow

This quantization of mathematics allows us to define very precise “coarse” macro step-sizes (both linear and angular) using integer amounts of the quantized micro units d and di.

  • d(xĦ) : defines the coarseness of linear growth and decay
  • di(xĦ) : defines the coarseness of oscillations

d2ipiH 001

[This post is adapted from What is Euler’s Identity?.]

Long Term Economic Growth

This Cartoon is from The Economist -0

“Austerity” or “Quantitative Easing” ?

In nature after an evolutionary collapse, mother-nature will begin to rebuild, and in a normal business cycle so too would an economy.  However in a boom and bust cycle things are not quite so easy!

Disposable Income

Economics is often thought of as a study of the allocation of “scarce” resources.  But that’s just another way of saying that economics is the study of “people making rational decisions”; and unfortunately we now all know that this is not true.

The starting point for all of economics is actually the ability to “produce excess”.  In a time before money, being able to produce more than one’s own immediate needs lead to the emergence of barter.  But with the introduction of money – which acts both, as a means of exchange, and a store of wealth – “excess productivity and production” could lead to the accumulation of “disposable income”; and it is this disposable income that drives an economy in 2 separate ways

  1. Consumption: disposable income drives trade, and thus economic activity.
  2. Investment: disposable income drives innovation, and thus self-organized, economic-complexity.

[Note: Complexity can be defined as “integrated diversity” and thus a Complex Adaptive Economy is a highly diverse & integrated  system.]


In a boom, investments are funded not only from accumulated savings, but usually also with overly-available credit (i.e non-accumulated future income).  In a bust, investments funded out of future income (i.e. debt) continue to incur ongoing interest costs despite the fall in value (or in many cases wipe-out) of the investment.  Not only that, but unfortunately for the borrower (and the economy at large), the principle also still needs to be repaid.

Thus an economic boom funded by excessive credit, eats into disposable income long after the boom has completely disappeared.  And since disposable income is what drives both investment and consumption, it depletion affects not only the growth of complexity in the economy but also the level of economic activity/trade and thereby the size of the economy.

In an economy the only time we really have stability is when nothing much is happening – and nothing much will happen when the driving force of the “Invisible Hand” is severely weakened.

In a weak economy, with very little funding only the very best ideas have a chance of being funded; but with no funding at all, and no consumer spending to boot, not only is new business growth stunted, but many existing businesses will likely decay! [In an economic bust, even healthy business go under; because a large amount of pre-existing disposable income is been redirected towards paying off debt.] 

In a dysfunctional economy, the weight of unproductive debt substantially weakens the driving force of growth and innovation in a normal functional economy.  An economy mired in debt will stagnate!  An economy devoid of disposable income is akin to a planet devoid of sunlight; nothing will grow!

Credit and Excess

Capitalism is perceived as the optimal growth engine of economic development and growth!  Financial Markets are supposed to be the invisible hand that optimally allocates the supply of, and demand for, investment funds in a Complex Adaptive Economy (CAE).  Chaos Theory alerts us however, to a possible vulnerability in the system!

The mathematics of chaos shows us that there is in fact an “Optimal Rate of Investment” in this Driven-Damped System – beyond which the system is vulnerable to “Coarse Synchronicity and Chaos”!

We know that, too much investment can overdrive the system causing the emergence of the so-called “business cycle”; excessive investment however has the effect of converting a functional cycle driven (even if slightly out of sync) by “The Wisdom of Crowds”, finely balancing “Supply and Demand” (of investment funds); into a dysfunctional cycle driven by “The Madness of Mobs” coarsely balancing “Fear and Greed” (of a financial market gamble).  What tips the system from one to the other (i.e. from functional to excessive) is when over-investment becomes “supercharged” by credit-fuelled investment!  Fundamentally, there are 5 levels of Driving Force in a Complex Adaptive Economy

  1. Minimum: Stops the natural decay of a naturally damping system.
  2. Small Excess: Under-Drives the Economy – a lot of good ideas are left unfunded.
  3. Critical Excess: Critically-Drives the Economy – optimally funds all the best ideas.
  4. Large Excess: Over-Drives the Economy into a “Business Cycle” – many poor ideas also get funded.
  5. Excessive Excess: Supercharges the Economy into a “Boom and Bust and Stagnation-Dead-End” – basically any idea will do!

Clearly credit is the lifeblood of business and trade in the economy.  Credit however has no place in the funding of asset price speculation.  Excessive credit fuelled speculation is simply a build-up of instability in markets; an accident waiting to happen!


Implicit to the idea of the “Wisdom of Crowds” is that no one player in the market might know all the available information, but the market as a whole does.  This apparent “rational wisdom” however, relies heavily on the “Law of Large Numbers” (LLN)!

The LLN is a statistical concept that deals with the idea that statistical accuracy is related to the size of the sample.  Most people would recognize the central thesis: if you toss a coin four times, you won’t necessarily get a 50/50 split of heads and tails: indeed, you could get 4 tails, suggesting (wrongly) that the coin will always land on tails. But if you toss a coin a trillion times, you will get something close to a 50/50 split between heads and tails.

The tendency in financial markets towards “herding behavior” weakens the apparent wisdom of the crowd by reducing the large number of independent players in the market into smaller collective groups, and in so doing, effectively engineers a Reverse of the Law of Large Numbers (RLLN)…

Macro-Economic Behaviour -1

Debt and Inequality

The RLLN is the reason why financial markets don’t always do what they are supposed to do.  Financial markets are a Complex Adaptive System and thus inherent vulnerability to the emergence of herding and the RLLN.  If we want these markets to operate in the most efficient fashion, we need to guard against any amplification of the RLLN; we need better control over the amount of credit in the system; both before and after a market bubble.

Vulnerability to the RLLN leads to the sub-optimal allocation of resources.  The misallocation of savings and credit has 4 main effects:

  1. In the short term, it burns though savings wasting a valuable economic resource.
  2. In the long run, it starves good ideas of investment.
  3. It creates inequality, because bad investments always suck the poor in last, allowing the rich to get out…
  4. It creates suffocating debt, and with that debt, it sucks the driving force out of the economic system…

Complexity Theory shows us that the “optimal” Complex Adaptive Economy (CAE) self-organizes from the bottom up.  Chaos shows us that there is an optimal rate of investment in a CAE – beyond which financial markets are vulnerable to “coarse synchronicity and chaos”.

Moreover the mathematics of chaos tells us that

In the short term artificially pumped up asset markets, can have a trickle-down effect, but ultimately in the long term, the primary effect is to increase inequality!

The Economic Engine

There has been a long running argument in economics as to whether an economy should be left alone or needs supervision.  In a way both sides are right!

The economy is like a car, it has a battery, but we never use the battery when the car is running – the battery’s sole purpose is to kick-start the car into ignition.  Using excessive credit in a booming economy is always going to flood the engine; lack of credit when the economy is stalled however is like trying to start the car without the ignition.

The takeaway message is that in any Driven-Damped Complex Adaptive System, positive feedback is always required in order for the system to be able to work its way away from the gravitational pull of natural decay.  In the push and pull of a Complex Adaptive Economy, a collapse in positive feedback can collapse the complex equilibrium which can mean that the “Business Cycle” itself effectively disappears.

Excessive Consumer Credit can overdrives an economy, but Excessive Consumer Debt can kill the drive altogether!

We should treat the economy as we would a car – we should leave well alone when the engine is running and the battery is charging; but if the engine is not running, then we have no option but to use the ignition…


But if there is but one single take-home message about understanding the behavior of the Driven-Damped Complex Adaptive Economy for the current economic climate, it is that: a CAE naturally self-organizes “from the bottom up”, driven by “disposable income”, and consequently

“Austerity” severely weakens the driving force of economic growth, and, in such a weaken environment, “Quantitative Easing” will simply widen the inequality gap!

So in answer to the implied headline question, “what will really drive long-term economic growth; austerity or quantitative easing?” – Neither of the above!

[This post is adapted from Incompressible Chaos in Financial Markets.]

Coarse Synchronicity

Lorenz Attractor

Chaos & Complexity are related; both are forms of “Coarse Damping”.  While complexity is a form of coarse damping in “structure”, Chaos is a form of coarse damping in “Time”.

Chaos is Coarse Synchronicity.


Fine vs. Coarse

Imagine you are using a camera with a large telephoto lens and you are trying to focus in on a subject in the distance.  As you focus, you adjust the lens back and forth, clockwise and anticlockwise.  Now imagine the lens does not move smoothly clockwise and anticlockwise, but instead has a limited number of preset positions (say 5 to the left, 1 midway, and 5 to the right).  If this were the case you would find it impossible to truly focus on your subject (unless the focus length just happened to be exactly at one of the 11 preset positions), the best you could do is jump back and forth between the 2 closest positions.

Furthermore the degree of focus depends on the step-size between the 2 closest positions; the larger the step size, the less focused the shot.  In this focusing set-up, it is the number of preset positions available that determines the step-size between each individual position; and the smaller the number of pre-set positions, the larger the step-size becomes.

Imagine the volume control on your home music system.  A smooth continuous dial effectively offers an infinite number of positions up to maximum volume, and so the step-size is in effect zero.   A dial with 11 preset levels (from 0 to 10) is obviously coarser in terms of step-size but nonetheless still usable to achieve the type of volume you want.  It becomes more difficult if the dial’s presets are just “Off”, “Low”, “Medium”, and “High”.  And you have no real control, if the dial basically allows you only “Off” or “On”; your step-size is now a single jump from zero to maximum volume…

In both these systems the step-size determines the coarseness of calibration.  This concept of degrees of fine or coarse calibration can be applied to the behavior of all systems, of any type.  Imagine if your personality only had 2 modes of expression, super nice or super angry, I suspect the vast majority of people would think your behavior a bit unpredictable.

The Butterfly Effect 002

The Butterfly Effect

The common perception about chaos is that it is all about unpredictability.  This unpredictability of chaos is popularly known as “The Butterfly Effect”, a sort of exaggerated version of the domino effect; a small change over here can cause a large change over there sometime down the line.  The scientific phraseology is that systems which exhibit chaos are “Sensitive Dependent on Initial Conditions” (SDIC).

The whole initial conditions thing relates to the classical physics idea that if we know all the forces currently acting on a system then the only remaining thing that we need to know in order to predict the future behavior of the system, is the exact state of the system as it is right now.  If a system is overly sensitive to these “initial conditions” it makes prediction impossible because we need to know these “initial conditions” with an immeasurable degree of exactness.

This explanation of chaos however is somewhat misleading.  While not untrue, this explanation places too much emphasis on unpredictability and fails to capture the reason for the extreme sensitivity.  SDIC is the result of internal interactions being too “coarse” to negotiate stability, which leads instead to the emergence of tipping points. 

The unpredictability of chaos might be better described as “Sensitive Dependence to Emergent Tipping Points”

Emergence of Tipping Points

Now imagine we have or we design some sort of self-stabilizing or self-balancing system.  Such a system would require the ability to find its way to the exact point of balance; this point being the “true-equilibrium”.

Many such self-stabilizing systems exist in nature; most of which self-stabilize in the most obvious way; that is they start off coarse-tuning and subsequently fine-tune their way to equilibrium.  If the system however is unable, or becomes unable, to reduce coarse-tuning to fine-tuning, it will be unable to micro-adjust and hone-in on the single true-equilibrium.

When the step-size of the internal adjustments remains too coarse, or become too coarse, it causes constant overshooting of the true-equilibrium, which means that the unobtainable equilibrium has now become a point about which the system oscillates.  [Think of the classic so-called business cycle of economic growth and recession]. 

At first glance it might appear that all such oscillations look the same, but actually they come in two different flavors.  The emergence of an unobtainable equilibrium means that there are now two different path trajectories that the system can take.  The oscillation can be a “tick-tock” oscillation or a “tock-tick” oscillation.  Depending on which side of the equilibrium the system evolves to, determines which path trajectory the system will take.  So although the system cannot find true-equilibrium, it nevertheless still exists, only now it behaves as a tipping point.


Many Evolutionary Paths

The inability to obtain equilibrium means, the behavior of the system has become sensitive to the emergence of a tipping point.  Given any two different sets of initial conditions we now find that their future evolutions are either in phase with each other, or out of phase with each other.  The coarseness of the step-size has thus caused two different (but complimentary) evolutionary paths to emerge…

But it doesn’t stop there!  The coarseness of the step-size not only determines the ability to self-synchronize to equilibrium, it also can determine whether the system is even able to synchronize the two legs of the back and forth oscillation.

If we were able to manually adjust the coarseness of the internal adjustment we would find that as we increase the coarseness of the step-size we make it increasing difficult for the system to synchronize its own internal self-balancing.  The inability to synchronize a two-step balancing leads to a four-step balancing; the inability to synchronize a four-step balancing leads to an eight-step balancing; and so on to infinity…

This process of continual bifurcation is known as the “period doubling route to chaos”.  Each bifurcation results in doubling the number of internal tipping points, which ultimately causes an infinite number of different evolutionary paths to emerge…


[Many mathematical models exhibit this coarse behavior, the best known of which is the logistic map of population growth.  So for those of you more mathematically inclined, click here for a description/analysis of The Behavior of the Logistic Map.]

Emergence of Desynchronized Diversity


Emergence and Unpredictability

The road to unpredictable chaos may be a gradual breakdown of synchronicity in behavior – leading to an infinite number of different evolutionary paths – but what emerges from this breakdown however is still an equilibrium of sorts.  Each evolutionary path is merely a different version of a previously hidden but now emergent “coarse equilibrium”.

This emergent structure – which results from ever increasing coarse synchronicity – is known in chaos as an “attractor”.  In actuality true-equilibrium is itself also an attractor, just not a very interesting attractor.  True-equilibrium is a “point attractor” which always pulls the system to the same place, regardless of where it starts from.  An emergent attractor however has a diversity of evolutionary paths that it is capable of pulling a system to; and the more emergent the attractor the greater the diversity of the evolutionary paths.

Chaotic Behavior of “Chaos” has what is known as a “strange attractor” – because of its infinite number of evolutionary paths.  This infinite number of paths however is not what makes the system unpredictable.  What makes the system unpredictable is the sheer density of emergent tipping points that system encounters; each one of which alters the behavior of the system, and any one of which can significantly impact its future evolution.  Consequently the more tipping points there are, the stranger the attractor, and the more unpredictable the system’s behavior becomes.

Coarse Deformation

Chaos Theory has revealed the existence of hidden attractors.  Attractors act on a system like a gravitational restraining force or an elastic restoring force.  What chaos theory shows us is that the type of restraining force (of equilibrium attractor) that acts on a system ultimately is determined by the degree of “coarseness of internal interactions”.

Finely-balanced/short-range interactions lead to highly-symmetric uninteresting conforming behavior; coarsely-balanced/long-range interactions on the other hand lead to an interesting mix of creative non-conforming asymmetric behavior.  It is as if coarse damping deforms the true-equilibrium behavior in a manner analogous to how excessive stress causes the deformation of elastic materials.  It is as if increasing the coarseness of step-size causes the emergence of higher dimensional attractors and the associated diversity of surprising patterns of behavior.

Strange Attractor-4


So Chaos, it turns out, is actually a form of Coarse Damping to Equilibrium. 

Chaos is the Coarse Synchronization of coarse competing forces resulting in a coarsely synchronized equilibrium…

So why is this important? It is important because Chaos is showing us that there is more to equilibrium than a finely-tuned, highly symmetric, but fundamentally bland and featureless uniformity.

Practically everything in the universe is both some form of system in itself, and part of a larger system in some form or other; and equilibriums are central to systems.  Chaos is the study of what can emerge from coarse system equilibrium.

Chaos Theory is the mathematical study of coarse equilibrium behavior.  The fascinating thing about chaos is that it is behavior we are accustomed to seeing, not only in everyday life but, in the universe at large. This is because

There is a universality in the behavior of naturally damped systems when coarsely driven; they all have the same universal mathematical methodology in investigating where else the system can go.

Chaos appears to be the universe’s search algorithm; a mathematically driven way of non-randomly exploring infinite possibilities, and higher dimensionality.

Too long has Chaos focused on the idea of SDIC. Chaos is more than unpredictability. What’s interesting about studying Chaos, is not the unpredictability behavior, but the surprising and highly creative emergent behavior that can result from the coarse synchronicity of internal dynamics.

Chaos Theory is the study of the  Incompressible Mathematics that drives The Surfacing of Incompressible Diversity.