Fractal: Multiplier

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 Development and growth of the fractal demonstrate the (money and Keynesian) Multiplier.

The (Keynesian) Multiplier shows how an initial injection of expenditure into an economic system creates more income. This is because added expenditure sets off additional rounds of expenditure with each and every hand or round this income passes.
This principle of 'multiplying' the initial injection can be demonstrated by using the fractal.
In the diagram below, income is represented by each triangle's area, and the spending rounds by every iteration of the rule.  
 The initial injection (iteration 1) is represented by the first triangle, which has an arbitrary area of 1. The next round adds 3 extra triangles, increasing the total area of the snowflake. This (principle) process continues until the changes in the area of the triangle—after each iteration—no longer change the total area of the snowflake. In the diagram above, the total area reaches 1.6 at around the 12th iteration. 
The Keynesian Multiplier equals the change in income divided by the change in injection.
The Fractal Multiplier thus equals the change in the total area divided by the change in the initial area.

The (Koch Snowflake) fractal multiplier equals 1.6  (1.6 divided by 1).


Notes:
  • From this it can be inferred or deduced – due to its principle similarity – that the multiplier effect is a universal fractal phenomenon, and that the Keynesian observation stands as further evidence that fractals are our reality – that reality is best understood by understanding fractal geometry.
  • It is of interest to me (the author) and my mathematician colleagues that the Koch snowflake fractal multiplier is 1.6. This 1.6 is very close to (but not the same as!) the Fibonacci or Golden ratio of 1.618.

A classroom demonstration:
(In class) The multiplier can be demonstrated by having the students pass on (for example) 50% of a piece of sheet paper (see the photo below) carefully to save the other 50%. The 1st and largest half represents the initial injection, and the 11th and smallest half (the last the paper can be divided). The iterations multiply a factor of 2 times the 1st half.



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