Thursday, April 12, 2012

Fractal Koch Snowflake Spiral

The Koch Snowflake Spiral


On route to understanding if  fractals have a 'wavy' like nature, I finally put pencil to paper and drew what I've been thinking.
I have been pondering on what effect a 'mutation' or change to one triangle - a dot at the apex of the triangle as shown below - would have or show on the iterating Koch Snowflake. I envisaged that it would spiral to infinity as shown below. By tracing a smooth curve through the (red) 'dots' series of iterated mutant triangles, I would develop - what looks like - a kind of logarithmic spiral, or a wave or pulse if this spiral curve was viewed in the front elevation view as opposed to the plan elevation (above).




Method
As a circles radius (what the compass is set to) scribes 6 times around the circles circumference, the three corners of the 1st equilateral triangle (iteration 1) can be produced, and the apex of the 2nd triangle (iteration 2) can also be marked. With the compass still at this 1st setting, a segment of the Koch Spiral - from apex 1 to apex 2 (iteration 1 to iteration 2) - is created. The apexes of every triangle in the iteration series are the tangent points of the spiral.

 Find the centre the 2nd triangle, and repeat the process. The process can be repeated - both zooming in or zooming out - to infinity.




Monday, April 2, 2012

Fractal: Multiplier


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

The (Keynesian) Multiplier shows how an initial injection of expenditure into an economic system goes on to create more income. This is because added expenditure – in itself – 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 use of the fractal:
In the diagram below: income is represented by the area of each and every triangle, and the rounds of spending 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 area of the triangle – after each iteration  changes the total area of the snowflake no more. In the diagram above the total area reaches 1.6, at around the 12th iteration. 
The Keynesian Multiplier equals: change in income divided by change in injection.
The Fractal Multiplier thus equals: change in total area divided by change in initial area.

The (Koch Snowflake) fractal multiplier is equal to 1.6  (1.6 divided 1).


Notes:
  • From this it can be inferred or deduced – due to its principle similarity – that the multiplier effect is a universal fractal phenomena, 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 of my mathematician colleague’s, 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 below photo) careful 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 of the 1st half.