Wednesday, December 29, 2010

1.8 Sustainability and the Fractal

Sustainability and the fractal:
update: 2011-01-04
This entry follows on from my fractal growth and development entries - published earlier.
There is never one snowflake alike, but there are snowflakes.

The fractal offers insights and helps us understand growth and development, change and evolution, then it should also help us understand sustainability. It should clarify what sustainability is. Is it real? Is it possible? Is it an illusion, or is it a delusion?

Fractals, by definition are patterns that show: 'same' but 'different', or regular irregularity - at all scales.
Fractals do support sustainability in one way; but not in another or the way 'we' currently associate sustainability with, the notion of keeping the environment or the economy today without compromising future generations. It maybe that the notion of sustainability is a (mathematically) non-sense. Here's why.
Fractals and sustainability analysis
To see why sustainability is a false statement - and doesn't hold -  at  least in the way it is commonly used - we need to split the above definition to the 'same' and the 'different'.

The 'same' or the 'regular' part of the fractal definition suggests that patterns, rules, and knowledge all repeat, at all scales: this part is sustainable or constant, it will happen, it does happen. This feature of fractals is explained by strange attractors found in the study of chaos and fractals; the repeating of a rule.
An example of the 'same' component - what I term lines of fractility - is exposed within the study of biology with 'evolutionary convergence' or 'analogous structures' - as shown in example diagram below. 'Same' pattern - a wing,  but 'different' forms (Bat, Bird, and Insect). The line of fractility is clearly flight by wing, and this is a repeating pattern, repeating at all scales. Today, we could add to this (at least) the human developed aircraft wing . 
repeating evolutionary patterns: same (function) but different (form)

The 'different' or 'irregular' part of the definition alludes to change, roughness or variety, it alludes to evolution: this component part of the fractal definition is not a sustainable, or fixed notion. This is where sustainability is a nonsense.
Fractals show us how there will never be a repeat of anything again:  there isn't, or has never been, another snowflake the same; there will also never be another 'me', or another 'you'; there will never be another repeated moment. Things come and go, for a complexity of reasons.
As objects (species) evolve, so to will they become extinct, whether we like it, or not; but rules or functions the 'same') will not (not to suggest they can't, else I will be breaking the laws of fractility).
In the wing diagram example; we see that there are many different variates of wing, at least Bat, Bird and Insect; but there is more to this than just that: the 'different' suggests a superposition all the wings that have ever been, and will ever be. Patterns repeat, there will again be winged flight,  it is a part of 'life'.

It is with this I suggest that the fractal shows us that the political notion of sustainability is a nonsense. The thought that humans, for whatever (moral) reason, can hold the likes of life, development, or an economy constant forever, when 'scientific' observation and knowledge shows us evidence only to the contrary, is a nonsense.

As an after thought: it is as if this part of the definition (different)  is a flow concept - the coming and going of something: the other 'same' or 'function' part maybe a stock, or standing (wave) concept.

Monday, December 13, 2010

On fractals and statistics

Just what's on my mind today:
What is the connection between the Mandelbrot set and the bell shaped normal distribution curve, or any distribution for that matter?

This is something I have been thinking about for sometime. I am surprised that fractals are not used to describe patterns.
It came to me today while on my bike to work: Fractals are an object thing, and describe the object through all scales; normal distribution or statistics need a parameter to function.
example :Stars are fractal, and will not distribute without a parameter : when we add say star size, star colour or distance, we get a distribution.
So I believe there is a very close relationship between the two - what is interesting is that distribution patterns are very  fractal, absolutely universal.

It is a goal of mine to understand this more: for there is more to it.

Thursday, December 9, 2010

The (fractal) God Illusion - the feeling of being watched.

The (fractal) God Illusion:
This applies to the Koch Curve zoom and links to my early blog on Inflation.

The following video inspired me for this insight, but the insight actually came to me while waiting in a doctors surgery - funny enough.
This is a great video on fractals and the Mandelbrot set : at 4:20 min Arthur C Clark explains the infinite size of the Mandelbrot set.

Two people stand at the edge of the fractal ( the Koch Snowflake), pairing into it - as if it were a tunnel or a computer screen.
What if one of the people (the walker) were able to walk out into the zoom, while the other stayed out and watched (the viewer).
For the walker, it would be like walking into a tunnel, and the viewer would see him or her get smaller and smaller as they walk in.
Now, what if the walker were to stop, turn, and look behind. What would they see?
A tunnel - with the viewer at the entrance, very small, and watching?
They would see infinity: they would see the infinite eye of the viewer - who is actually only looking into what they both know and see is a simple (triangle making Koch Snowflake) fractal.

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.

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.

Monday, October 18, 2010

Fractal Dimension, (Economic's) Elasticity and Complexity

Fractal Dimension and Elasticity - is there a connection?
I have found - after analysing the Koch Snowflake fractal with standard economic analysis - that there maybe a connection between Economics 'Elasticity coefficent' (which is a universal measure of the change in one variable to a change in another) - and the fractal dimension. That is to say, Price Elasticity of Demand (PED) is possibly the same as , or at least related to, the fractal dimension (D) - a measure of complexity. This may help explain the logarithmic nature of the demand curve.
Here's my reasoning:
Below you can also listen to a video blog of me on this topic too.

 Recap: Marginal Analysis of the fractal

In my last blog I established that the blue (MA) curve  in Fig. 1 -showing the change in area in the 'Koch Snowflake' after an other iteration- is mimicking or demonstrating something central to traditional marginal analysis and utility theory: that it maybe the origin of the consumer demand curve as we know it.   The next question begs - what's the MA's elasticity?

Sunday, October 17, 2010

The Passing of Benoit Mandelbrot

Last night I was camping out with my wife  in the forests in Sweden and slept under the stars and this very wonderful scotch pine above.
Apart from the romance of the setting, I went to sleep thinking over fractal explanations of (mostly) inflation of money, and it's connection to economic growth, development, and sustainability.
They are really on my mind at the moment and I will write on my findings in the future in the near future.

I heard this morning that Benoit Mandelbrot - the father of Fractals has died.
I don't know what to think: I thought I might get to meet him someday.
He curtainly hadn't made it into the mainstream: when I ask, hardly anyone knew of him or his fractals - outside maths of course.
Yet he to me made what I believe the greatest discovery in human history (and I mean it) cause I am sure everything is fractral - evolution and all.  We just don't really know it yet how to use it.
His name and input will only grow now - in  the very same way (funny enough) as the fractals he founded.

If I (and I am sure many other thinkers) have anything to do with it - we will put his fractals up front.

Wednesday, October 13, 2010

Fractals - Marginal Analysis of the Koch Snowflake

Marginalism, and Marginal Analysis : derived from the fractal (?)

For some time now I have pondered on the fractal, particularly the development the Koch Snowflake fractal, (seen below), I have noticed how it uncannily demonstrates many of the key features of economic theory.
Could it be that the economy - as we experience it - is a fractal? Is it that the fractal can be understood as an economy?
I am now confident, after doing the following analysis of the (Koch Snowflake) fractal that fractal geometry is the foundation of neoclassical economics and our reality. 
The following is the first in a series of insights I have found within the fractal:

The marginal utility (or benefit) of additional triangle area - after each iteration

To show that the fractal demonstrates marginal utility (or benefit) I substituted area for utility, and then analysed the change in area over the development of the Koch Snowflake fractal - from a triangle (iteration 1 below), to the complete shape of the snowflake at iteration 4 (below). Shape is usually forms at and around 7 plus or minus 2 iterations: it is at iteration 4 (below) because of its size and relative thickness of line.

Towards  (Fractal) Equilibrium

Koch Snowflake development: iteration 1 to 3
Once a rule -  in this case new triangles - is set in action, and allowed to iterate (repeat) step by step, it will go on to eventually form a snowflake.  After iteration 4, the shape will set at what I term fractal equilibrium: no more iterations will change the shape after this iteration - assuming a static viewers perspective, ie no zooming.

Koch Snowflake development animation: iteration 1 to 6

If one were to continue the iterations beyond the point of fractal equilibrium (iteration 4 in the figure above and iteration 6 in the animation above) the computer generating the pattern would soon crash: the effort (cost) would multiply while the extra benefit to the viewer - in terms of the shape – would fall close to, but never equaling, zero.
Take the time to look at this webpage where you can experience the fill working of the Koch Curve first hand.

Marginal Analysis: Method.
I analysed the growth of the Koch Snowflake using marginal analysis (a method found in any basic economic textbook) which looks at (in this case) the extra 'area' added - iteration by iteration.

Working from internet sources I constructed a table showing the area progression of the curve - iteration after iteration:

Taking s as the side length, the original triangle area is . The side length of each successive small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is proportional to the square of its side length, the area of each triangle added in the nth step is 1/9th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are added as in the previous iteration; because the first iteration adds 3 triangles, the nth iteration will add triangles. Combining these two formulae gives the iteration formula:

Area (A) of each triangle (as in column 2) = (s^2 √(3))/4

Results Table

The Fractal Total and Marginal Area Curves
Fig 1. shows the total area (TA) and marginal area (MA) at each iteration.
Notice, from the start, that Fig.1 resembles any marginal utility diagram found in any elementary economic text books.

Fig 1. is derived from the fractal - not from any (fictitious) made up numbers, as in text books. All fractals - whether computer generated or not - will chart as Fig 1.

Diminishing Marginal Utility.

fig 4-1
Figure 4-1 above shows how the fractal analysis is similar if not the same as a typical economics text book diagram: marginal utility reference.
The MA curve is a 'Power law' function, and that it demonstrates Pareto distribution - the 80/20 rule.
As the total area of the Koch Snowflake increases, after each iteration, the marginal or extra area (also) diminishes.
The fractal is the perfect demonstration of diminishing marginal utility; it could be interpreted that diminishing marginal utility is a perfect demonstration of the (universal) fractal - that both are the same thing?
Here are some internet references on marginal utility:

A window into reality:
My fractal-marginal analysis diagram is a snap-shot, taken somewhere (anywhere) in the set: this insight in turn has opened a window of enormous insight into reality, and has led me to the world quantum mechanics. The fractal (above) shows no scale - the numbers assigned are purely arbitrary and serve merely to show the relationship between the total area, 160, and the first triangle area 100. In fact triangles range in size from small, (but never reaching 0), to the infinitely large.
From this early insight (in this blog) I have gone on to unlock the quantum features of the fractal and have recently suggested a direct link between the demand curve and the de Broglie wave function.

negative marginal utility misattributed
Next: Marginal cost, Fractal production.

Saturday, October 9, 2010

The creation of new public goods: the internet (or information age) paradox

First published:October 9, 2010

Update:March 22, 2015.
Blair D. Macdonald.

Based on the economic model used to classify 'goods' in an economy; private goods such as found in the entertainment/media industries, or any item on the internet subject to file sharing or digital copying in any form including – at the extreme – the human genome, solid object 3D printing, and even money in the form of bit-coins, are being slowly repositioned from what are termed 'private' or 'club'/'congestion' goods, to the extreme opposite, public goods. The ‘free rider problem’ of Public Goods has become the ‘free copy problem’.  Public Goods failure in the market, and are therefore provided by Government.  Is this the destiny of internet goods?

Key words: Public Goods, Internet, Copyright, Market Failure

1        Introduction

One of the great questions of our time is what is to be the fate of the entertainment and media industries - newspapers, movies, music production, TV and the like - all of which since the advent of the internet, have been scrambling for new models that pay? Will they survive in this age of Internet and file sharing? I have found something that suggests why it is so difficult for them.
It came to me while teaching the market failure economics and analysing examples of  ‘public’ and ‘private’ goods: I was looking at the (market) differences between 'Cable TV' - which is known as a ‘club good' or 'natural monopoly', – and 'free to air TV' – which is known as a ‘public good:  I began to see other examples or repetition’s of the subject at different scales and contexts, and after doing so, noticed something interesting, if not astounding, newspapers and the like are now – at least in their digital form – public goods.

1.1      Public Goods

The following is a typical textbook definition of a Public Good.
 Public goods are goods that are both non excludable and non-rival. Consumers cannot be prevented from consuming them once they are provided and additional consumers do not reduce the amount left for other people, e.g. national defence. Once a country is defended all of its inhabitants benefit automatically. Many public goods such as lighthouses could in theory be provided by the market mechanism but are not; these are called 'quasi public goods', rather than 'pure public goods '. Public goods suffer from the free rider problem. If asked whether they would pay for them, households would lie and say no because, once provided, they could benefit for free anyway. Because no one is willing to pay for these goods (because they hope someone else will) they will not be provided in a free market. Therefore, the Government must provide them. (Note: for a 'private good' if one unit is consumed by one person it cannot be consumed by another. (This is not the case for public goods.) [1]

1.2      Public Goods and the Internet

The entertainment, media industries or anything subject to file sharing or digital copying in any form, have slowly been re-positioned from being private or club/congestion goods, to their now extreme opposite, public goods – as shown below in figure. 

Figure 1. Public Goods including effects of the Internet.

1.3      The free rider (free copy) problem

The proof of this 'public good' claim is the presence of the free rider. Quite literally – when asked to pay by a market provider – consumers simply find a substitute or find a free copy elsewhere (on the internet). We quite literally get a free ride – or in this case, a free copy. And for music and films, copies are made 'free' by file sharing and the like.

1.4      Lack of Rivalry

Unlike the use of other traditional Private or Common goods – for example print newspapers – there is no rivalry: there is little bothered by the 'stealing' or copying of the good, and they do not show real concern – or fight – for the media – as they would for the exploitation of Common's goods, e.g. the of ocean's or atmosphere. 

2        Discussions

The problem is public goods tend to fail as a business operation, and are almost always supported by Government. Traditional examples are lighthouses, and streetlights. So this leaves a great paradox: we have the Internet, and do well for it, but on the other hand maybe our own, slow undoing. Democracy demands or depends on a 'free from Government' media; and firms in the arts and entertainment industry demand revenue if they are to survive.
It is as if the Internet has formed some kind of an economic black hole. Is there any escaping? Our own privacy on the Internet may well fall under this public goods paradox also: including – in the future – ones genome.


1.         Gillespie A. Advanced Economics Through Diagrams. Oxford University Press; 2001.