Simulating Barnsley’s Fern Fractal

Mathematics can capture nature’s most elegant patterns. In this project, I recreated one of the most striking examples: Barnsley’s Fern, a fractal that perfectly mirrors the geometric beauty found in natural ferns.

Full implementation available here.

What is it?

Named after Dr. Michael Barnsley, who first described it in his book “Fractals Everywhere,” this fern is a perfect example of a self-similar set - a pattern that looks identical at any magnification level. It’s created using an iterated function system (IFS), which builds complex patterns from simple repeated transformations.

The Mathematics Behind the Magic

The fern emerges from four simple coordinate transformations, each chosen with different probabilities:

1. Stem function (1% probability):

   f1: xn+1 = 0, yn+1 = 0.16 yn

2. Smaller leaflet (7% probability):
   f2: xn+1 = -0.15 xn + 0.28 yn, yn+1 = 0.26 xn + 0.24 yn + 0.44

3. Left leaflet (14% probability):

   f3: xn+1 = 0.2 xn - 0.26 yn, yn+1 = 0.23 xn + 0.22 yn + 1.6

4. Main leaflet (85% probability):

   f4: xn+1 = 0.85 xn + 0.04 yn, yn+1 = -0.04 xn + 0.85 yn + 1.6

Implementation in R

Here’s how we bring these mathematical transformations to life:

x <- rep(0, times=max_itr) #make vector of 0s
y <- x #copy the vector

# This creats f(x1,y1) = (0,0)
#run the iteration to generate next points

for (n in 2:(max_itr)) { 

# I have subtracted 1 at every iteration from n so I dont 
# have to add 1 to f(x), basically subtracted it from both 
# sides.

  #generate a random number between 0-100
  rand_num=runif(1, 0, 100) 
  
  if (rand_num < 1) { 
    x[n]<- 0
    y[n]<- 0.16*y[n-1]
  }
  else if (rand_num < 7){
    x[n]<- -0.15*x[n-1]+0.28*y[n-1]
    y[n]<- 0.26*x[n-1]+0.24*y[n-1]+0.44
  }
  else if (rand_num < 14){
    x[n]<- 0.2*x[n-1]-0.26*y[n-1]
    y[n]<- 0.23*x[n-1]+0.22*y[n-1]+1.6
  }
  else {
    x[n]<- 0.85*x[n-1]+0.04*y[n-1]
    y[n]<- -0.04*x[n-1]+0.85*y[n-1]+1.6
  }
  
}

A key detail - points are generated within specific bounds (-2.1820 < x < 2.6558 and 0 ≤ y < 9.9983) to create the proper arch of the fern’s rachis (stalk).

The Beautiful Result The final fern emerges with a simple plot command:

plot(x,y, pch='.', xlab= "", ylab = "", col= color)

Want to see the fern’s creation in action? Check out this interactive visualization. For a deeper dive into the mathematics, visit the Wikipedia article.

NOTE: Found a mistake or have feedback? I’d love to hear from you - please reach out via email!