Flowers for Julia

November 19, 2018

(This article was first published on R – Fronkonstin, and kindly contributed to R-bloggers)

No hables de futuro, es una ilusión cuando el Rock & Roll conquistó mi corazón (El Rompeolas, Loquillo y los Trogloditas)

In this post I create flowers inspired in the Julia Sets, a family of fractal sets obtained from complex numbers, after being iterated by a holomorphic function. Despite of the ugly previous definition, the mechanism to create them is quite simple:

  • Take a grid of complex numbers between -2 and 2 (both, real and imaginary parts).
  • Take a function of the form  f(z)=z^{n}+c setting parameters n and c.
  • Iterate the function over the complex numbers several times. In other words: apply the function on each complex. Apply it again on the output and repeat this process a number of times.
  • Calculate the modulus of the resulting number.
  • Represent the initial complex number in a scatter plot where x-axis correspond to the real part and y-axis to the imaginary one. Color the point depending on the modulus of the resulting number after applying the function f(z) iteratively.

This image corresponds to a grid of 9 million points and 7 iterations of the function f(z)=z^{5}+0.364716021116823:

To color the points, I pick a random palette from the top list of COLOURLovers site using the colourlovers package. Since each flower involves a huge amount of calculations, I use Reduce to make this process efficiently. More examples:

There are two little Julias in the world whom I would like to dedicate this post. I wish them all the best of the world and I am sure they will discover the beauty of mathematics. These flowers are yours.

The code is available here.

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