Barnsley Fern

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Barnsley Fern
Fields: Algebra and Fractals
Image Created By: Michael Barnsley

Barnsley Fern

The Barnsley Fern was created by Michael Barnsley using an iterated function system.

Basic Description

Barnsley's Fern is a iterated function system (IFS) fractal. Being a IFS fractal it exhibits various properties:

  • It is exactly self-similar. If you magnify this fractal, the fractal will look the same because every part of this image looks like the whole.
  • It is infinitely complex. If you continue to magnify this fractal, it will be infinitely complex at an infinite magnification.
  • It is a chaotic IFS fractal. The functions that describe its behavior do not map the points of the fractal in any particular order. The fractal is created using a starting point (x,y) and four systems of equations that are each assigned a probability. These probabilities then determine how often the equations will be used to transform a point.

The construction of Barnsley's Fern is shown above. The fern is animated to approximately the 5,000th iteration, and you can see that the points are plotted chaotically, but slowly form into a visible fern.

Barnsley's Fern animated

A More Mathematical Explanation

Note: understanding of this explanation requires: *Linear Algebra

Making Barnsley's Fern

[[Image:Fern_Process.jpg|thumb|200px|right|Barnsley's Fern at various stag [...]

Making Barnsley's Fern

Barnsley's Fern at various stages of iteration

The process to create Barnsley's Fern is relatively straightforward with some knowledge of matrices. The set of functions that govern Barnsley's Fern include a starting point, four sets of coordinate transformations, and four probabilities - all displayed in the table below. The coordinate transformations can be split into two matrices for transformation and translation. The table also includes an image showing the region of the fern associated with each coordinate transformation. However, there can also be variations to the equations and probabilities used depending on the shape of the fern fractal that is desired.

The procedure to create Barnsley's Fern is as follows:

  • Pick a starting point
  • Choose a coordinate transformation according to probability
  • Multiply the starting point by the matrix transformation
  • Add the result by the matrix translation
  • Repeat or iterate infinitely using the resulting point as the new starting coordinates

Using starting coordinates: \begin{bmatrix} x\\ y \end{bmatrix}

Coordinate Transformation Matrix Transformation Matrix Translation Probability Location of Point
x_{n+1} = 0\,

y_{n+1} = 0.16y_n\,
\begin{bmatrix}0    &0 \\0    &0.16\end{bmatrix} \begin{bmatrix}0\\0\end{bmatrix} 3% FernConstruction4.png
x_{n+1} = 0.85x_n + 0.04y_n\,

y_{n+1} = -0.04x_n + 0.85y_n + 1.6\,
\begin{bmatrix}0.85    &0.04 \\-0.04   &0.85\end{bmatrix} \begin{bmatrix}0\\1.6\end{bmatrix} 73% FernConstruction1.png
x_{n+1} = -0.15x_n + 0.28y_n\,

y_{n+1} = 0.26x_n + 0.24y_n + 1.6\,
\begin{bmatrix}-0.15   &0.28 \\0.26   &0.24\end{bmatrix} \begin{bmatrix}0\\1.6\end{bmatrix} 13% FernConstruction2.png
x_{n+1} = 0.2x_n - 0.26y_n\,

y_{n+1} = 0.23x_n + 0.2y_n + 0.44\,
\begin{bmatrix}0.2    &-0.26 \\0.23   &0.22\end{bmatrix} \begin{bmatrix}0\\0.44\end{bmatrix} 11% FernConstruction3.png

Fractal Dimension

The Fractal Dimension of the Barnsley Fern cannot be calculated by conventional means, and is estimated to be about 1.45.

How the Main Image Relates

The image featured at the top of this page is artistically modified rendering of Barnsley's Fern. Although the equations and probabilities used to make this image are not exactly the same as those given above and used to create typical Barnsley's Ferns, the same concept of an iterated function system was used.

Teaching Materials

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Wikipedia, Iterated Function System Andrew Ho, Spleenwort Fern

Future Directions for this Page

Some sort of interactive/animated feature explaining the matrix transformations and how exactly they affect the points (rotation, contraction).

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