Tuesday, November 16, 2010

Fractal: Growth and Development

Update 2010-12-05
Growth and Development

Take a long look at the fractals above and ask yourself: are they developing? are they growing? There appears (to me) to be no obvious, or distinguishing differences between (fractal) growth, and (fractal) development.  When describing fractals, the terms growth and or development are often used  interchangeably. As if to be a law, the fractal fact is that the two are inextricably linked - as the fractal grows, the fractal develops.
The fractal demonstrates Development: this is to do with the increase in complexity of a fractal as it iterates towards fractal equilibrium; it is a qualitative measure of fullness, completeness.
The fractal (also) demonstrates Growth, and may be seen as an increase in either the area, or number of triangles, or even the perimeter of the snowflake - which is apparently infinite.

To analyse growth (more), we need to go back and look at one of my early diagrams I 'developed' (above).
The red Total Area curve (TA) actually shows the GROWTH in the area of the Koch Snowflake: it rises quickly at the early stages and then at a slower rate as the 'snowflake' or fractal gets closer to equilibrium. There appears to be limits to growth.
The problem with this approach of measuring growth is that traditionally we do not think of Area (or utility) as a measure of growth, rather we use change in quantity  - traditionally measured on the x axis.

Update May 2017
This is by far my best idea; I have written it up in a working paper at my academia.edu 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

Friday, November 5, 2010

Price Inflation - fractal

Inflation - a fractal explanation          

Update: 29th 11, 2012
If the fractal demonstrates the likes of price and quantity, supply and demand - as I have shown in my previous entries - then it should also be able to demonstrate inflation.
This entry has been a real challenge for me, but has been very existing : my thoughts maybe different, but they at least can't be over looked.

Just what is inflation?
Of course we have the textbook answer, a general increase in the average price level - over time, but to be fractal, it must be a universal definition, and go beyond prices of goods.
A fractal explanation will have to demonstrate the following - I am convinced that the general fractal does.
  1. the increase in the descriptive value placed upon an 'unchanging' object 
  2. there must be devaluation of the measure.
  3. there must be a notion of equilibrium, where one gets used to the new nominal change or size
For inflation to be fractal, inflation has to be a universal concept: I am interested to learn that hotels now come with 6, 7, and possibly 8 stars to rate their quality.  Have hotels really changed so much to deserve more numbers than the ‘5 star’. Of course they have, but maybe the system is open to some distortion - 5 should be the ultimate. As the 'stars' increase or inflate above the original ‘5 star’, that 5 star loses value.  Other examples exist:  academic grade inflation - where the school leaving qualification creeps up and up, from a Degree to a Masters to where you need a PhD to work behind a bar.

We know, from experience, and from neo-classical economic's, that price inflation fits the above 'rules': (nominal) price's rise for the same object;  consumers get used to the new price's - there is new equilibrium; and finally there is a devaluation of money itself, when relating to its past value.

I also consider changes in models of car (for example) without any inovation or evolution, just the colour of style, as inflation - design inflation. The new car devalues the old just because it is has something different (no practicle) - even if they are basically the same car.

A fractal explanation for inflation
When we look at the  Koch Snowflake development (below), what is it that inflates - the area,, the number of 'triangles', or the perimeter - or nothing?

Inflation and perimeter

I have become convinced that it is the perimeter, the reason being that the area of the Koch is finite, while the perimeter is infinite. Also note that area and perimeter are connected through the base length of the triangle side too.

It came to me after thinking about what the perimeter length of an island, or the length of a piece of string. Fractal theory says both are infinite, and that the result depends on the actual length of the measuring tool. The shorter the tool, the larger the perimeter length: all this while at the same time the shape – the object – remains the same, or at equilibrium. This measuring problem is well described from around 2 minutes into the following:

youtube clip: BBC, How long is a piece of string?

 P= s.l
The perimeter (P) is calculated by multiplying number of sides (s) by length of a side (l).  In the below diagram, as the fractal iterates, and the shape develops towards equilibrium, at iteration 7 (plus or minus 2), the perimeter of the snowflake rises exponentially to infinity. This is due to relationship between the diminishing length of each side, and the exponentially rising quantity of sides.

This to me is also analogise to price inflation and the concept of value in economies: (Domestic) Production (GDP), where the measuring tool is units of currency or money.
Measuring an economies GDP should be the same principle as measuring an island, or the Koch snowflake's perimeter, only the measuring tool is money, or currency in this context.

To go any further we need to clear what money is? By definition, money should be - with the help of the circular flow of income model: Y=P=E - a liquid means of exchange, derived from income (Y), which comes from production (P), and will in turn be what exspended or spent (E).
It follows that if the Product of the Koch snowflake is the sides, then sides are the money in the 'money' economy.
As we grow (domonstrated as a fractal zoom) the sides get smaller and smaller, while the perimeter gets larger and larger.

the Koch fractal zoom - growth
If we were to stop the zoom at any time, we see the same 'snowflake' like shape as earlier in the zoom, that is to say, in our analogy, the money looks the same, only the the numbers have changed. We see an equilibrium shape.
With that, I conclude that inflation is natural, and that growth, development and now inflation are inextricably linked through the mechanics of the fractal.
However, it begs the question, what about printed or fiat money? Does it distort the reality? Is there such a thing as unnatural inflation?
It is my thinking that fiat money is another fractal of its own. If we print (more) money - than what our resources call for - it follows that we will shorten measuring tool more than what is natural, distorting the reality with an even longer perimeter, or larger (misleading) GDP.

I am still of minds that there is more to this: there could be something in the quantity of 'triangles' to explore.