The Collatz Fractal

It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing (Sofia Kovalevskaya)

How beautiful is this fractal! In previous posts I colored plots using module of complex numbers generated after some iterations. In this occasion I have used the escape-time algorithm, a very well known coloring algorithm which is very easy to implement in R.
Collatz07
Those who want to know more about this fractal can go here. For coloring, I chose a simple scale from red to yellow resulting a fractal interpretation of my country’s flag. You can choose another scale or use a RColorBrewer palette as I did in this previous post. Choosing another x or y ranges you can zoom particular areas of the fractal.

Try yourself and send me your pictures!

library(ggplot2)
xrange <- seq(-8, 8, by = 0.01)
yrange <- seq(-3, 3, by = 0.01)
f  <- function (z) {1/4*(2+7*z-(2+5*z)*cos(pi*z))}
z <- outer(xrange, 1i*yrange,'+')
t <- mat.or.vec(nrow(z), ncol(z))
for (k in 1:10)
{
  z <- f(z)
  t <- t + (is.finite(z)+0)
}
## Supressing texts, titles, ticks, background and legend.
opt <- theme(legend.position="none",
             panel.background = element_blank(),
             axis.ticks=element_blank(),
             axis.title=element_blank(),
             axis.text =element_blank())
z <- data.frame(expand.grid(x=xrange, y=yrange), z=as.vector(t))
ggplot(z, aes(x=x, y=y, color=z)) + geom_tile() + scale_colour_gradient(low="red", high="yellow") + opt
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4 thoughts on “The Collatz Fractal

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