Lorenz Curve of the Koch Snowflake Fractal

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Update 2015, I have published/posted.
I have found that Lorenz distribution is a fractal phenomenon. The Lorenz Curve and Gini coefficients increase in the fractal models as the fractal grows and develops. The distribution between groups accelerates with growth and development. Lorenz distribution is universal: income and wealth inequality is a scale-invariant aspect of a universal phenomenon.
 https://www.academia.edu Demonstrating_Lorenz_Wealth_Distribution_and_Increasing_Gini_Coefficient_with_the_Iterating_Koch_Snowflake_Fractal_Attractor

Also see: Improved Fractal Lorenz Curve





 Wealth Distribution: a (universal) fractal phenomena 
I was teaching income distribution recently, and I thought the (Koch Curve) fractal may demonstrate the Lorenz Curve.
After doing this entry I analysed a (Xmas) tree for Lorenz distribution. Click to see.

I could not make any obvious connection between the Lorenz Curve's income distribution and the Koch snowflake development: maybe because income is a flow concept and the Koch snowflake doesn't seem to demonstrate this.  So I thought: what distribution does it demonstrate? I turned my attention to the concept of wealth distribution. I assumed that Wealth corresponds better with the Area  - reasoning that both are a 'stock' concept.
I did the calculations (taken from the area calculation table in my first blog 1.1, seen below): per cent of the number of triangles and per cent of the area for each triangle, and then corrected the results so that they resembled the original Lorenz Curve.
Koch Snowflake (fractal) development 


Results
Analysis of the Koch Snowflake

Lorenz Curve calculation

Diagram:

Lorenz Curve for the Koch Snowflake
Thinking that this Lorenz Curve was exaggerated to the right - with a very large Gini coefficient  - I searched for a reference to agree with this extreme position - in 'nature' or the real world. In one of my economics texts, I found the following wealth Lorenz Curve reference for the USA economy.



Discussions and Conclusions
The wealth and income  Lorenz Curves are fractal phenomena and are best demonstrated using the (Koch Snowflake) fractal.
Lorenz distribution is a universal phenomenon inherent in all fractal systems and at all scales. Income and wealth distribution are just two manifestations of this structure, and thus, they are not directly determined by economic output or growth—they are 'natural'.  
I look forward to making real Lorenz curves from real fractals: trees, rivers, and the like.  

 2011-01-15
BBC article on Wikipedia:
Listening to the podcast below, I heard a reference to the concept of article posting distribution —most of the articles are added by a very few.

http://www.bbc.co.uk/worldservice/documentaries/2011/01/110111_wikipedia_at_10.shtml


Lorenz Curve from Book: Economics, Micheal Parkin, Addison-Wesley 1990
2012-12-10
http://www.whypoverty.net/en/theme/inequality/
It interested me when I saw this image on facebook: why poverty now :


My mind immediately saw the Fibonacci resemblance.








Comments

  1. SageApril 16, 2018 at 3:38 AM

    It is not natural.

    Humans are individuals, not branches on a tree caste structure.

    ReplyDelete

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