Barnsley Fern
From Math Images
Barnsley Fern 

Barnsley Fern
 The Barnsley Fern was created by Michael Barnsley using an iterated function system.
Contents 
Basic Description
Barnsley's Fern is a iterated function system (IFS) fractal. Being a IFS fractal it exhibits various properties: It is exactly selfsimilar. 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.
A More Mathematical Explanation
 Note: understanding of this explanation requires: *Linear Algebra
Making Barnsley's Fern
[[Image:Fern_Process.jpgthumb200pxrightBarnsley's Fern at various stag [...]Making Barnsley's Fern
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:
Coordinate Transformation  Matrix Transformation  Matrix Translation  Probability  Location of Point 


3%  

73%  

13%  

11% 
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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References
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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