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Barnsley fern
The Barnsley Fern is a fractal named after the British mathematician Michael Barnsley who first described it in his book Fractals Everywhere.^{[1]} He made it to resemble the Black Spleenwort, Asplenium adiantumnigrum.
History
The fern is one of the basic examples of selfsimilar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's book about fractals is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology, called Fractal Geometry. After publishing the book, a second course was developed, called Fractal Measure Theory.^{[1]} Barnsley's work has been a source of inspiration to graphic artists attempting to imitate nature with mathematical models.
The fern code developed by Barnsley is an example of an iterated function system (IFS) to create a fractal. He has used fractals to model a diverse range of phenomena in science and technology, but most specifically plant structures.
"IFSs provide models for certain plants, leaves, and ferns, by virtue of the selfsimilarity 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. Vvariable 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...—Michael Barnsley et al.^{[2]}...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.
Construction
Barnsley's fern uses four affine transformations. The formula for one transformation is the following:
 <math>
f(x,y) = \begin{bmatrix}a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} e \\ f \end{bmatrix} </math> Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table.^{[1]} In the table, the columns "a" through "f" are the coefficients of the equation, and "p" represents the probability factor.
w  a  b  c  d  e  f  p  Portion generated 

ƒ_{1}  0  0  0  0.16  0  0  0.01  Stem 
ƒ_{2}  0.85  0.04  −0.04  0.85  0  1.60  0.85  Successively smaller leaflets 
ƒ_{3}  0.20  −0.26  0.23  0.22  0  1.60  0.07  Largest lefthand leaflet 
ƒ_{4}  −0.15  0.28  0.26  0.24  0  0.44  0.07  Largest righthand leaflet 
These correspond to the following transformations:
 <math>
f(x,y) = \begin{bmatrix} \ 0.00 & \ 0.00 \ \\ 0.00 & \ 0.16 \end{bmatrix} \begin{bmatrix} \ x \\ y \end{bmatrix} </math>
 <math>
f(x,y) = \begin{bmatrix} \ 0.85 & \ 0.04 \ \\ 0.04 & \ 0.85 \end{bmatrix} \begin{bmatrix} \ x \\ y \end{bmatrix} + \begin{bmatrix} \ 0.00 \\ 1.60 \end{bmatrix} </math>
 <math>
f(x,y) = \begin{bmatrix} \ 0.20 & \ 0.26 \ \\ 0.23 & \ 0.22 \end{bmatrix} \begin{bmatrix} \ x \\ y \end{bmatrix} + \begin{bmatrix} \ 0.00 \\ 1.60 \end{bmatrix} </math>
 <math>
f(x,y) = \begin{bmatrix} \ 0.15 & \ 0.28 \ \\ 0.26 & \ 0.24 \end{bmatrix} \begin{bmatrix} \ x \\ y \end{bmatrix} + \begin{bmatrix} \ 0.00 \\ 0.44 \end{bmatrix} </math>
Computer generation
Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory. Many different computer models of Barnsley's fern are popular with contemporary mathematicians. As long as the math is programmed correctly using Barnsley's matrix of constants, the same fern shape will be produced.
The first point drawn is at the origin (x_{0} = 0, y_{0} = 0) and then the new points are iteratively computed by randomly applying one of the following four coordinate transformations:^{[3]}^{[4]}
ƒ_{1}
 x_{n + 1} = 0
 y_{n + 1} = 0.16 y_{n}.
This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem. This part of the figure is the first to be completed in during the course of iterations.
ƒ_{2}
 x_{n + 1} = 0.85 x_{n} + 0.04 y_{n}
 y_{n + 1} = −0.04 x_{n} + 0.85 y_{n} + 1.6.
This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represented by the red triangle to a point inside the opposite, smaller leaflet represented by the blue triangle in the figure.
ƒ_{3}
 x_{n + 1} = 0.2 x_{n} − 0.26 y_{n}
 y_{n + 1} = 0.23 x_{n} + 0.22 y_{n} + 1.6.
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (it flips it).
ƒ_{4}
 x_{n + 1} = −0.15 x_{n} + 0.28 y_{n}
 y_{n + 1} = 0.26 x_{n} + 0.24 y_{n} + 0.44.
This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (without flipping it).
The first coordinate transformation draws the stem. The second generates successive copies of the stem and bottom fronds to make the complete fern. The third draws the bottom frond on the left. The fourth draws the bottom frond on the right. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note that the complete fern is within the range −2.1820 < x < 2.6558 and 0 ≤ y < 9.9983.
Mutant varieties
By playing with the coefficients, it is possible to create mutant fern varieties. In his paper on Vvariable fractals, Barnsley calls this trait a superfractal.^{[2]}
One experimenter has come up with a table of coefficients to produce another remarkably naturally looking fern however, resembling the Cyclosorus or Thelypteridaceae fern. These are:^{[5]}^{[6]}
w  a  b  c  d  e  f  p 

ƒ_{1}  0  0  0  0.25  0  −0.4  0.02 
ƒ_{2}  0.95  0.005  −0.005  0.93  −0.002  0.5  0.84 
ƒ_{3}  0.035  −0.2  0.16  0.04  −0.09  0.02  0.07 
ƒ_{4}  −0.04  0.2  0.16  0.04  0.083  0.12  0.07 
References
 ^ ^{a} ^{b} ^{c} Fractals Everywhere, Boston, MA: Academic Press, 1993, ISBN 0120790629
 ^ ^{a} ^{b} Michael Barnsley, et al.,"Vvariable fractals and superfractals" PDF (2.22 MB)
 ^ Barnsley, Michael (2000). Fractals everywhere. Morgan Kaufmann. p. 86. ISBN 0120790696. Retrieved 20100107.
 ^ Weisstein, Eric. "Barnsley's Fern". Retrieved 20100107.
 ^ Other fern varieties with supplied coefficients, retrieved 201017
 ^ A Barnsley fern generator
