Barnsley's Fern

Barnsley's fern is an example of an Iterated function system, generated by repeatedly choosing one of four affine transformations and plotting the resulting point. A simple local rule, applied thousands of times, produces a fern.

Animation of Barnsley's fern emerging point by point
50,000 points, drawn one transformation at a time.

Algorithm

Pseudocode

(x, y) ← (0, 0)

repeat N times:
    r ← random number in [0, 1)
    # apply (affine transform with probability, r)
    if r < 0.01:
        (x', y') ← F₁(x, y)
    else if r < 0.86:
        (x', y') ← F₂(x, y)
    else if r < 0.93:
        (x', y') ← F₃(x, y)
    else:
        (x', y') ← F₄(x, y)

    plot(x, y)
    
    x = x' 
    y = y'

The four rules

Affine transformations

Each rule maps the current point (x, y) to a new point (x′, y′).

F₁ · Stem

1%
x' = 0 y' = 0.16y

F₂ · Successive leaflets

85%
x' = 0.85x + 0.04y y' = -0.04x + 0.85y + 1.6

F₃ · Left leaflet

7%
x' = 0.20x - 0.26y y' = 0.23x + 0.22y + 1.6

F₄ · Right leaflet

7%
x' = -0.15x + 0.28y y' = 0.26x + 0.24y + 0.44