### Fractals - Marginal Analysis of the Koch Snowflake

**Marginalism and Marginal Analysis are derived from the fractal (?)**

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 have some collaboration 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, and 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 connection 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

**Original Post**

**Intro**

For some time now, I have pondered the fractal, particularly the development of the Koch Snowflake fractal (seen below). I have noticed how it uncannily demonstrates many of the key features of economic theory.

Is the economy - as we experience it - a fractal? Is it that the fractal can be understood as an economy?

I am now confident, after doing the following analysis of the (Koch Snowflake) fractal, that fractal geometry is the foundation of neoclassical economics and our reality.

The following is the first in a series of insights I have found within the fractal:

__The marginal utility (or benefit) of additional triangle area - after each iteration__To show that the fractal demonstrates marginal utility (or benefit) I substituted area for utility, and then analysed the change in

**area**over the development of the Koch Snowflake fractal - from a triangle (iteration 1 below), to the complete shape of the snowflake at iteration 4 (below). The shape is usually formed at and around 7 plus or minus 2 iterations: it is at iteration 4 (below) because of its size and the relative thickness of the line.

**Towards (Fractal) Equilibrium**

Koch Snowflake development: iteration 1 to 3 |

**fractal equilibrium**: no more iterations will change the shape after this iteration - assuming a static viewer's perspective, ie no zooming.

Koch Snowflake development animation: iteration 1 to 6 |

If one were to continue the iterations beyond the point of fractal equilibrium (iteration 4 in the figure above and iteration 6 in the animation above), the computer generating the pattern would soon crash: the effort (cost) would multiply. At the same time, the extra benefit to the viewer - in terms of the shape – would fall close to, but never equal to, zero.

Take the time to look at this webpage http://mathforum.org/mathimages/index.php/Koch_Curve where you can experience the full working of the Koch Curve first-hand.

*Marginal Analysis: Method.*I analysed the growth of the Koch Snowflake using marginal analysis (a method found in any basic economic textbook), which looks at (in this case) the extra 'area' added - iteration by iteration.

Working from internet sources, I constructed a table showing the area progression of the curve - iteration after iteration:

*Takings as the side length, the original triangle area is. The side length of each successive small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is proportional to the square of its side length, the area of each triangle added in the nth step is 1/9th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are added as in the previous iteration; because the first iteration adds 3 triangles, the nth iteration will add triangles. Combining these two formulae gives the iteration formula:*

Area (A) of each triangle (as in column 2) = (s^2 √(3))/4

Results

*Results Table*

__The Fractal Total and Marginal Area Curves__Fig 1. shows each iteration's total area (TA) and marginal area (MA).

Notice, from the start, that Fig.1 resembles any marginal utility diagram found in any elementary economic textbook.

Fig 1. is derived from the fractal - not from any (fictitious) made-up numbers, as in textbooks. All fractals - whether computer-generated or not - will chart as Fig 1.

**Diminishing Marginal Utility.**

fig 4-1 |

The MA curve is a 'Power law' function that demonstrates the Pareto distribution—

**the 80/20 rule.**

As the total area of the Koch Snowflake increases, the marginal or extra area (also) diminishes after each iteration.

The fractal is the perfect demonstration of diminishing marginal utility; it could be interpreted as a perfect demonstration of the (universal) fractal - that both are the same thing?

Here are some internet references on marginal utility:

http://www.investopedia.com/university/economics/economics5.asp

http://www.google.se/imgres?

**A window into reality:**

My fractal-marginal analysis diagram is a snap-shot, taken somewhere (anywhere) in the set: this insight, in turn, has opened a window of enormous insight into reality and has led me to the world of quantum mechanics. The fractal (above) shows no scale - the numbers assigned are purely arbitrary and serve merely to show the relationship between the total area, 160, and the first triangle area, 100. In fact, triangles range in size from small, (but never reaching 0), to infinitely large.

From this early insight (in this blog) I have gone on to unlock the quantum features of the fractal and have recently suggested a direct link between the demand curve and the de Broglie wave function.

**negative marginal utility misattributed**

Next: Marginal cost, Fractal production.

I'm doing the International Baccalaureate course for my senior years, and I came across this for my Maths Exploration. This is actually very interesting, thanks for the inspiration!

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