Fractal: Growth and Development
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.
development
The fractal demonstrates Development: this has to do with the increase in complexity of a fractal as it iterates towards fractal equilibrium; it is a qualitative measure of fullness and completeness.
growth
The fractal (also) demonstrates Growth and may be seen as an increase in either the area, number of triangles, or even the perimeter of the snowflake - which is apparently infinite.
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 appear to be limits to growth.
The problem with this approach to 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.
Update May 2017
This is by far my best idea; I have written it up in a working paper at my academia.edu and vixra, and named it: Quantum Mechanics, Information and Knowledge, all Aspects of Fractal Geometry and Revealed in an Understanding of Marginal Economics.
I shall post the abstract, followed by the original post, and then the paper. I hope to collaborate and review my work in time.
Abstract
Fractal geometry is found universally and is said to be one of the best descriptions of our reality – from clouds and trees to market price behaviour. As a fractal structure emerges – repeating a simple rule – it appears to share direct properties familiar to classical economics, including production, consumption, and equilibrium. This paper investigated whether the mathematical principles behind ‘the market’ – known as marginalism – is an aspect or manifestation of a fractal geometry or attractor. Total and marginal areas (assumed to stand for utility) and the cost of production were graphed as the fractal grew and compared to a classical interpretation of diminishing marginal utility theory and the market supply and demand. PED and PES were also calculated and analysed with respect to (iteration) time and decay. It was found that the fractal attractor demonstrates properties and best models classical economic theory. From this, it was deduced that the market is a fractal attractor phenomenon where all properties are inextricably linked. The fractal, at equilibrium, appears to be a convergent – zeta function – series, able to be described by Fourier analysis and involves Pi, i, e, 0, and 1 (of Euler’s identity) in one model. It also demonstrated growth, development, evolution and Say’s Law – production before consumption. Insights from the fractal on knowledge and knowing are also revealed, with implications on the question of what exactly is ‘science’ – and what is ‘art’? A connect between reality and quantum mechanics was identified. It was concluded marginal classical economics is an aspect of fractal geometry.
Keywords
Marginal, Fractal, Elasticity, Utility, Cost, Production, Price, Growth and Development, Say’s Law
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