__2004 – Complexity__

- Complexity and Evolution seems to occur in defiance of the Second Law of Thermodynamics (SLOT), but the SLOT is actually really only a
*“probabilistic”*law of nature and consequently it relies heavily on the strength of the*“Law of Large Numbers”*(LLN). - Thus a stable thermal equilibrium relies on the LLN; and the stronger the influence of the LLN (i.e. the more
*independent*elements in the system), the more stable will be the equilibrium. - Adaptive Systems however exhibit
*inter-dependence*and this leads to a Reverse of the Law of Large Numbers (RLLN)) - The RLLN (weakens the LLN and) pulls system ways from equilibrium (where ultimately the system can potentially symmetry-break).
- Complex System exhibit an interplay of LLN and RLLN, an interplay of entropy and reverse entropy.

[Click here for ** “The Reverse Law and Emergent Dynamics”**]

__Nov 2013 – Chaos__

- Chaotic Behavior in Logistic Map is the result of the
*dual*role of driving parameter (R). R not only determines the driving force, it also determines theof the incremental step from one iteration to the next.*“coarseness”*

[Click here for ** “Coarse Synchronicity in the Logistic Map”**]

__Jan 2014 – Coarseness__

- The commonality between Chaos and Complexity is
. Coarseness restricts a system’s ability to find equilibrium. Both chaos and complexity are forms of*“Coarseness”*. Chaos is coarse damping in time (coarse synchronicity), complexity is coarse damping in structure (coarse entropy).**“coarse damping”**

[Click here for ** “What is Coarse Damping?”**]

__Jan 2014 – Natural Damping & Coarse Equilibrium__

- In the log map, for low values of R (i.e. low driving force, & finely-grained step-size) if you start the system from anywhere other than equilibrium, it will spontaneously restore to equilibrium.
- Similarly in a thermal systems of very many independent parts, if we start the system from anywhere other than equilibrium, or pull the system away from equilibrium, the system will restore to equilibrium.
- What sort of equilibrium these system restores to, is determined by the system’s ability to fine-tune to equilibrium

__June 2014 – Natural Restoring Force__

- Both Chaos and Complexity are Damped-Driven System, both are systems where coarseness in the system is resisting the pull of a
. Restoring forces are indicative of both oscillating bodies and elastic materials.**“natural restoring force”**

__July 2014 – Restoring Force in Chaos & Complexity__

- Chaotic behavior emerges because of a coarse restoring force in oscillating bodies
- Complex structures emerge because of coarseness in the restoring/elastic force of entropy; in other words,
*complexity is a deformation of an elastic material*.

__Oct 2014 – Weaken Restoring Force__

- Elasticity reflects a systems restoring force. Most elastic systems have a
*“elastic limit”*where the restoring force fails. - Both Chaos and Complexity occurs as a result of a weaken restoring force.
- Chaos occurs when the coarseness of the incremental step-size weakens the system ability to find or restore equilibrium.
- Complexity occurs when a adaptive system drives against the
*“entropic limit”*, and cause some form of*“elastic/entropic deformation”*.

[Click here for ** “The Emergence of Complexity”**]

[Click here for ** “The Evolution of Complexity”**]

__Dec 2014 – Emergence__

- The emergence of both Chaos + Complexity results from the interplay of coarse-tuning and fine-tuning (i.e. the interplay of coarseness and smoothness).
- So if coarse damping brings about emergence, maybe everything is
*“coarse emergent”*including space and time.*Maybe space and time are also coarse?…*

__Jan 2015 – Time__

- The ultimate in smoothness is continuous Space-Time. The transcendental number
*“e”*, defined using limits to infinity, is usually used to represent continuous time. - But
*“e”*started life as the base of a logarithm, and the basic idea behind logarithm is*very small steps applied a large number of times.* *“e”*is the*“natural logarithm”*because the step-size is exact inverse of the number of steps, and defined effectively as an infinitely small step-size applied an infinite number of time. So*“e”*represent continuous time; but what if time where*not continuous?*

[Click here for ** “What is e?”**]

[Click here for ** “What is Euler’s Identity?”**]

__Feb 2015 – Euler’s Formula__

- If N did not go to infinity in the definition of
*“e”,*then in Euler’s Formula we would find a*quantum angular step.*

[Click here for ** “How Do You Like Them Apples?”**]

__Mar 2015 – Is there a smallest angular step?__

- What if a smallest angular step could be defined in terms of the maximum angular displacement (π) on the x axis.
- So if (-1)
^{1}defines the maximum angular step, then (-1)^{x}defines every other angular step, and so then all we need to do is ask the question*“what value of “x” would we need in order to be the smaller possible step-size?*Well let’s try h-bar…

__Apr 2015 – “i”__

- All rotations are oscillations, and all oscillations are cycles. (i) is a quarter-rotation, which means (i) is a quarter-cycle, and a quarter cycle of oscillation travels the same distance as the maximum displacement from equilibrium (i.e. the
of the oscillation).**Amplitude** - (i)
^{2}looks very similar to amplitude squared (A)^{2 }in the formula for simple harmonic motion… - (-1) to the power of h-bar produces an angular step that looks remarkable like h/2

[Click here for ** “What is Euler’s Secret Identity?”**]

__Jun 2015 – “Cycle Amplitude”__

- (-1)
^{x}is an*“angular frequency” of electromagnetic oscillation* - (i) is the
*“cycle amplitude”*of electromagnetic oscillation