Demonstrating Lorenz Curve Distribution and Increasing Gini Coefficient with the Iterating (Koch Snowflake) Fractal Attractor.

Global income has increased exponentially over the last two hundred years, while the 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: at each iteration, the fractal's triangle areas grew from an arbitrary size, increasing the total set. The area of the triangle, the ‘bits’, represented wealth. Kinematic analysis of velocity and acceleration was conducted, and the growing triangles propagated in a sinusoidal spiral. Using Lorenz curve and Gini methods, bit-size distribution was graphed for each iteration time. The curves matched the regular Lorenz curve shape and expanded to the right with fractal growth, increasing the corresponding Gini coefficients and contradicting Kuznets cycles. The ‘gap’ between iteration triangle sizes (wealth) widened, or accelerated apart, as conjectured in reality. It was concluded that the wealth (and income) Lorenz distribution and its acceleration properties are aspects of the fractal. The form and change of the Lorenz curve are inextricably linked to the growth and development of a fractal attractor; therefore, given real economic data, one can deduce that an economy, whether cultural or not, behaves as a fractal and can be explained as such. The discrete and wave properties of fractal growth, and its accelerated expansion, similar to that of trees and the conjectured growth of the universe at large, were discussed.