Friday, October 30, 2015

Improved Fractal Lorenz Curve



Four years ago I conducted a 'loose' experiment on a Christmas tree to test the weight distribution of branches for Lorenz curve wealth distribution similarities in an economy. This week I finished the Figure 3 chart (below) after modelling the Koch Snowflake fractal for (Lorenz) area distribution. Took me hours. 
I found the area distribution not only follows what we observe with wealth distribution, but expands as the fractal grows (or develops) with time (Table 1). I also found different area groups (triangle sizes) grow with time (from an arbitrary size), and accelerate apart from each other with time. This is a geometry and can be observed in any fractal structure. I have updated my post at academia.edu
Abstract:

Global income has increased exponentially over the last two hundred years; while, and at the same time respective Gini coefficients have also increased: this investigation tested whether this pattern is a property of the mathematical geometry termed a fractal attractor. The Koch Snowflake fractal was selected and inverted to best model economic production and growth: all triangle area sizes in the fractal grew with iteration-time from an arbitrary size – growing the total set. Area of triangle the ‘bits’ represented wealth. Kinematic analysis – velocity and acceleration – was undertaken, and it was noted growing triangles propagate in a sinusoidal spiral. Using Lorenz curve and Gini methods, bit size distribution – for each iteration-time – was graphed. The curves produced matched the regular Lorenz curve shape and expanded out to the right with fractal growth – increasing the corresponding Gini coefficients: contradicting Kuznets cycles. The ‘gap’ between iteration triangle sizes (wealth) was found to accelerate apart, just as it is conjectured to do so in reality. It was concluded the wealth (and income) Lorenz distribution – along with acceleration properties – is an aspect of the fractal. Form and change of the Lorenz curve are inextricably linked to the growth and development of a fractal attractor; and from this – given real economic data – it can be deduced an economy – whether cultural or not – behaves as a fractal and can be explained as a fractal. Questions of the discrete and wave properties and the accelerated expansion – similar to that of trees and the conjectured growth of universe at large – of the fractal growth, were discussed.













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