Visualizing the beauty in physics and mathematics
Michael Barnsley's fern is a fractal that is named after British Mathematician Michael Barnsley, because he described this fractal in his book “Fractals Everywhere”. It is an example of a so-called “iterated function system” (IFS) fractal, which means that the construction of this fractal makes use of matrices, see also the chaos game fractals.
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. — Wikipedia
From Michael Barnsley's own article we quote:
IFSs provide models for certain plants, leaves, and ferns, by virtue of the self-similarity which often occurs in branching structures in nature. But nature also exhibits randomness and variation from one level to the next; no two ferns are exactly alike, and the branching fronds become leaves at a smaller scale. V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling. These factors allow us to make the hybrid biological models... ...we speculate that when a V -variable geometrical fractal model is found that has a good match to the geometry of a given plant, then there is a specific relationship between these code trees and the information stored in the genes of the plant. — Michael Barnsley, et al. in V -variable fractals and superfractals
As stated above, the Barnsley Fern is created using a matrix transformation:
$f(x, y)=\begin{pmatrix} a & b \ c & d \end{pmatrix}\begin{pmatrix}x \ y\end{pmatrix} + \begin{pmatrix}e \ f\end{pmatrix}$
Four different transformations are used, illustrated in the table below. Variables $a$ to $f$ are the coefficients, and $p$ is the probability factor.
| a | b | c | d | e | f | p | Part generated |
|---|---|---|---|---|---|---|---|
| 0.00 | 0.00 | 0.00 | 0.16 | 0 | 0 | 0.01 | Stem |
| 0.85 | 0.04 | -0.04 | 0.85 | 0 | 1.60 | 0.85 | Small leaflet |
| 0.20 | -0.26 | 0.23 | 0.22 | 0 | 1.60 | 0.07 | Left large leaflet |
| -0.15 | 0.28 | 0.26 | 0.24 | 0 | 0.44 | 0.07 | Right large leaflet |
From this table, we can create the 4 functions for our program:
def transform(self, x, y):
rand = uniform(0, 100)
if rand < 1:
return 0, 0.16 * y
elif 1 <= rand < 86:
return 0.85 * x + 0.04 * y, -0.04 * x + 0.85 * y + 1.6
elif 86 <= rand < 93:
return 0.2 * x - 0.26 * y, 0.23 * x + 0.22 * y + 1.6
else:
return -0.15 * x + 0.28 * y, 0.26 * x + 0.24 * y + 0.44
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