Sunday, January 30, 2011

Fractal: Equilibrium Perfect Knowledge and Output

Fractal Equilibrium:
This entry follows on from 'production of the fractal'.
Here I am suggesting that equilibrium - in any sense - is a fundamental of the fractal.

Koch Snowflake development: source, Wikipedia

The above animation shows the development of the fractal, at iteration 5 or 6 fractal equilibrium is reached - where the shape (of the snowflake) is made; or where benefit production is equal to cost of production.

Fig. 2b below, shows a closeup analysis of the fractal equilibrium, at least from a static point of view.  MC  intersects, or is equal to MA, at iteration 5 where the Area is equal to 1, due to the reciprocal of 1 itself being equal 1.

Equilibrium - Perfect Knowledge*
Any iteration less than fractal equilibrium will result in an imperfect (snowflake) shape or incomplete knowledge or information.
Any iteration point greater fractal equilibrium will result with little added gain in information or shape -  and it will come at a great cost.
Perfect knowledge, or the perfect shape, is gained where MA =MC.  This, it taken directly from  economics theory, resembles where allocative efficiency is maximized - at the equilibrium point in market theory. Competition gives the best shape or knowledge, opposed to a monopolist that would produce or output somewhere left of equilibrium.
Notice Total knowledge will never be gained - the cost is too high.
*I am aware that the fractal demonstrates infinity, and thus shows the infinity of knowledge - of the rule. To say perfect information may well be a relative statement, said in context, or in a  paradigm.

A theory of knowing?
Development of the little man is my attempt at understanding how we know: check out my theory of mind
Equilibrium Output (quantity)
Of course, iteration number is not the output or quantity ( as in classical economics). So, what then is it?
It came to me that it is the number of triangles - that are added after each iteration.
In the below diagrams below (fig.3 and 3b), shown in orange is the increase in the quantity of triangles, after each iteration. I have recently learned that this curve is exponential.

 Equilibrium quantity or output at fractal equilibrium is  ..... triangles or point 'a', on the diagram.
Update May 2017
This is by far my best idea; I have written it up in a working paper at my and vixra, and named it:  Quantum Mechanics, Information and Knowledge, all Aspects of Fractal Geometry and Revealed in an Understanding of Marginal Economics.
I shall post the Abstract, followed by the original post, followed by the paper. I hope to have some collaborate and review my work in time.
Fractal geometry is found universally and is said to be one of the best descriptions of our reality – from clouds and trees, to market price behaviour. As a fractal structure emerges – the repeating of a simple rule – it appears to share direct properties familiar to classical economics, including production, consumption, and equilibrium. This paper was an investigation into whether the mathematical principles behind ‘the market’ – known as marginalism – is an aspect or manifestation of a fractal geometry or attractor. Total and marginal areas (assumed to stand for utility) and the cost of production were graphed as the fractal grew and compared to a classical interpretation of diminishing marginal utility theory, and the market supply and demand. PED and PES was also calculated and analysed with respect to (iteration) time and decay.  It was found the fractal attractor demonstrates properties and best models classical economic theory and from this it was deduced the market is a fractal attractor phenomenon where all properties are inextricably linked. The fractal, at equilibrium, appears to be a convergent – zeta function – series, able to be described by Fourier analysis, and involves Pi, i, e, 0, and 1 (of Euler’s identity) in one model. It also demonstrated growth, development, evolution and Say’s Law – production before consumption. Insights from the fractal on knowledge and knowing are also revealed, with implications on the question of what exactly is ‘science’ – and what is ‘art’? A connect between reality and quantum mechanics was identified. It was concluded marginal, classical economics is an aspect of a fractal geometry.  
Marginal, Fractal, Elasticity, Utility, Cost, Production, Price, Growth and Development, Say’s Law

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