Tag Archives: Julia

Flowers for Julia

Drawings, Fractals, , , , ,

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.

Playing With Julia (Set)

Drawings, Fractals, Image Processing, , , , ,

Viento, me pongo en movimiento y hago crecer las olas del mar que tienes dentro (Tercer Movimiento: Lo de Dentro, Extremoduro)

I really enjoy drawing complex numbers: it is a huge source of entertainment for me. In this experiment I play with the Julia Set, another beautiful fractal like this one. This is what I have done:

  • Choosing the function f(z)=exp(z3)-0.621
  • Generating a grid of complex numbers with both real and imaginary parts in [-2, 2]
  • Iterating f(z) over the grid a number of times so zn+1 = f(zn)
  • Drawing the resulting grid as I did here
  • Gathering all plots into a GIF with ImageMagick as I did in my previous post: each frame corresponds to a different number of iterations

This is the result:

julia

I love how easy is doing difficult things in R. You can play with the code changing f(z) as well as color palettes. Be ready to get surprised:

library(ggplot2)
library(dplyr)
library(RColorBrewer)
setwd("YOUR WORKING DIRECTORY HERE")
dir.create("output")
setwd("output")
f = function(z,c) exp(z^3)+c
# Grid of complex
z0 <- outer(seq(-2, 2, length.out = 1200),1i*seq(-2, 2, length.out = 1200),'+') %>% c()
opt <-  theme(legend.position="none",
              panel.background = element_rect(fill="white"),
              plot.margin=grid::unit(c(1,1,0,0), "mm"),
              panel.grid=element_blank(),
              axis.ticks=element_blank(),
              axis.title=element_blank(),
              axis.text=element_blank())
for (i in 1:35)
{
  z=z0
  # i iterations of f(z)
  for (k in 1:i) z <- f(z, c=-0.621) df=data.frame(x=Re(z0), y=Im(z0), z=as.vector(exp(-Mod(z)))) %>% na.omit() 
  p=ggplot(df, aes(x=x, y=y, color=z)) + 
    geom_tile() + 
    scale_x_continuous(expand=c(0,0))+
    scale_y_continuous(expand=c(0,0))+
    scale_colour_gradientn(colours=brewer.pal(8, "Paired")) + opt
  ggsave(plot=p, file=paste0("plot", stringr::str_pad(i, 4, pad = "0"),".png"), width = 1.2, height = 1.2)
}
# Place the exact path where ImageMagick is installed
system('"C:\\Program Files\\ImageMagick-6.9.3-Q16\\convert.exe" -delay 20 -loop 0 *.png julia.gif')
# cleaning up
file.remove(list.files(pattern=".png"))