Thursday, January 27, 2011

Marginal Analysis of the Fractal

Marginalism and Marginal Analysis : derived from the fractal.
Here I preform basic marginal analysis on the Koch Snowflake

Intro
For some time now, I have pondered on how the Koch Snowflake fractal (seen below) demonstrates, uncannily, key features in economic theory.
Is it that the economy - as we experience it - is a fractal? or, is the fractal an economy?
I am now confident - after doing the following basic math's - that fractal geometry is the foundation to the social science, Economics. 



Koch Snowflake animation:


Towards Equilibrium
From the outset, the concept of Marginalism and fractals have something in common, they are both about the next -the next unit, or the next iteration.

Once a rule, in this case new triangles,  is set in action and allowed to progress (iterate) step by step, it will go on to eventually form a snowflake.  After iteration 4, the shape will set and no longer change - at least from the viewers' static perspective.

Koch Snowflake development

If one were to continue the iterations beyond the point of equilibrium (4.) the computer generating the pattern would soon crash:  the effort (cost) would multiply while the extra benefit to the viewer - in terms of the shape – would fall close to, but never, zero.
Take the time to look at this webpage http://mathforum.org/mathimages/index.php/Koch_Curve where you can experience the fill working of the Koch Curve first hand.


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:

Taking s 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

To show that the fractal Koch Snow Flake demonstrates marginal utility theory, I substituted the Economists' utility (satisfaction or benefit gained from another iteration) with the area of the curve: reasoning that after more iterations, one gets more total satisfaction or in this case Total Area (TA). One gets (pleasingly) closer to the true shape of the snowflake.
I have used no units to measure Area, as the area in the diagram corresponds to a curve I used as a reference, http://math.rice.edu/~lanius/frac/koch3.html on the internet.

Results Table





This result table will be central to my analysis of the fractal.

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