Last week I did attend to a “Flower Festival”, in the city of Joinville. I had the opportunity to admire several of the most beautiful flowers, orchids, and decoration plants. After few hours of contemplation, I realized I never had thought on flowers to be fractals.

Fractals attract lots of interest, especially because of their structures and combinations made by simple equations. But what exactly makes something a fractal?

A fractal is defined as an object that displays self-similarity on all scales. In fact, it doesn’t need to have exactly the same structure at all scales, but the pattern must be visible or recognizable any how.

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Mandelbrot

The structure or the equation that defines a fractal is most of the time very simple. For instance, the formula for the famous Mandelbrot is $z_{n+1}=z_n^2+c$.

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How can we start?

We start by plugging in a constant value for c, z can start at zero. After one iteration, the equation gives us a new value for z; then we plug this back into the equation at old z and iterate it again, it can proceed infinitely.

As a very simple example, let’s start this with c = 1.

Graphing these results against n would create an upward parabolic curve because the numbers increase exponentially (to infinity). But if we start with c = -1 for instance, z will behave completely different. That is, it will oscillate between two fixed points as:

And this movement back and forth will continue forever as we can imagine. I figured out, that the Mandelbrot set is made up of all the values for z that stay finite, thus a solution such as the first example for c = 1 is not valid and will be thrown out because z in those cases goes to infinity and Mandelbrot lives in a finite world. The following is an example of such set.

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The script for this set:

library(caTools) 
# caTools is handy because it provides write.gif function

cols <- colorRampPalette(c("#00007F", "brown", "blue", "#007FFF", "green", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000", "magenta"))
m <- 1200 # define size
C <- complex( real=rep(seq(-1.8,0.6, length.out=m), each=m ),
              imag=rep(seq(-1.2,1.2, length.out=m), m ) )
C <- matrix(C,m,m) # reshape as square matrix
Z <- 0 # initialize Z to zero
X <- array(0, c(m,m,20)) # initialize output 3D array
for (k in 1:20) { # loop with 20 iterations
    Z <- Z^2+C # The equation
    X[,,k] <- exp(-abs(Z)) # capture results
}
write.gif(X, "Mandelbrot.gif", col=cols, delay=1000)

#reproducible