Wednesday, April 6, 2016

Fun with Fractals


Fun with Fractals

By Hope Roberts
George Mason University
Math 493
Mathematics through 3D Printing

What are Iterated Functions & Fractals?

Iterated functions are a way of constructing Fractals.
An iterated function  takes a function  X → X and composes another function of itself a repeated amount of times. (Wikipedia).
Repeating the same function over and over is known as iteration.
After so many iterations the structure is seen to shrink. This is known as limiting behavior. The set will converge to a single fixed point, which is known as the attractive fixed point.

Example of a self-similar repeating pattern.

Fractals are patterns that repeat infinitely. The patterns repeated are complex and self-similar. The repeated patterns are created by recursion, so repeating a pattern in a similar way.



Brief History on Iterated Functions

Iterated functions are credited to John E Hutchinson discovered in 1981.
Michael Barnsley’s is credited with making them popular in a book he wrote titled Fractals.


Making of my Fractal

Using OpenSCAD with code given by Dr. Sander I changed the design of the Fractal by adjusting the set of  rules of the fractal, specifically changing r, s and theta in the code. Changing the r,s and theta of the code by any amount yielded a different shape.
I obtained these values from a yale.edu (reference at bottom)
The type picked from this website was (o).


Original structure before the code was adjusted.

I then adjusted the length and width in the code so that the platform and bases were supporting the structure in such a way that the printing would have support in all areas of the structure. Finally I added one additional layer of height to the structure from 5 levels to 6 levels also increasing the height of each layer, which increased the thickness of each layer.


My Fractal after adjustments to r,s, theta, length, width and height.


In MakerBot I adjusted the size to a four-hour approximate time for printing, making sure the structure was flat on the surface with supports and rafts.

Equations & Code Changes:

Below is the basic equation for an iterated function.

if for all components of  f, it is in X

Below are my values for r, s and theta for my Fractal:

scale1x = -.5; scale2x = .5; scale3x = .5;  scale1y = .5; scale2y = .5; scale3y = .5;  theta1  = 90;  theta2 =  -90; theta3 =  0;

Below are my values for the changes to length, width and height in my code.

len = 45; // length of the first segment wid = 45; // width of the first segment height  = 3.0; // height of the first cube.

Finished Product:





The finished product was successful and there was relatively no clean up of the structure involved.


Sources:







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