Fractals: The Math Behind Nature's Patterns

Through researching Thomasina's method of iterating mathematical functions and graphing the result, I have discovered that she was correct in assuming that there are ways of mathematically determining even the most complex patterns in nature. The method that she used is much easier today with new computers which can iterate functions in mere seconds, while it took here much longer doing it by hand in the margins of her math book. When a function is iterated over and over again (that is, each output is put back into the function as a new input) an IFS (Iterative Function Series) fractal is created, which is a self similar, never ending image. Thomasina remarked to Septimus that the equations he had her graph were too industrial, too plain, that for her theory of a formula for nature to be true, there must be something more complicated, and fractals are just that. Patterns in nature that seem too complex to have a mathematical explanation can be explained through fractals, rivers and fjords, canyons, leaves, lightning, and many more seemingly random patterns in nature all fall under the category of fractals.

Here are just a couple:

This website shows many natural fractals with their explanations:
17 Captivating Fractals Found in Nature
Here is a website that explains what a fractal is better than I can:
What are Fractals?