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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