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Barnsley fern

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Barnsley fern

The Barnsley fern FFF is the only non-empty compact subset of ℝ2superscriptℝ2\mathbb{R}^{2} satisfying the relation

F=⋃i=14Ti⁢(F)Fsuperscriptsubscripti14subscriptTiFF=\bigcup_{{i=1}}^{4}T_{i}(F)

where Ti:ℝ2→ℝ2normal-:subscriptTinormal-→superscriptℝ2superscriptℝ2T_{i}\colon\mathbb{R}^{2}\to\mathbb{R}^{2} are the following linear mappings:

T1⁢(x,y)subscriptT1xy\displaystyle T_{1}(x,y) =(0.85⁢x+0.04⁢y,-0.04⁢x+0.85⁢y+1.6),absent0.85x0.04y0.04x0.85y1.6\displaystyle=(0.85x+0.04y,-0.04x+0.85y+1.6),
T2⁢(x,y)subscriptT2xy\displaystyle T_{2}(x,y) =(0.2⁢x-0.26⁢y,0.23⁢x+0.22⁢y+1.6),absent0.2x0.26y0.23x0.22y1.6\displaystyle=(0.2x-0.26y,0.23x+0.22y+1.6),
T3⁢(x,y)subscriptT3xy\displaystyle T_{3}(x,y) =(-0.15⁢x+0.28⁢y,0.26⁢x+0.24⁢y+0.44),absent0.15x0.28y0.26x0.24y0.44\displaystyle=(-0.15x+0.28y,0.26x+0.24y+0.44),
T4⁢(x,y)subscriptT4xy\displaystyle T_{4}(x,y) =(0,0.16⁢y).absent00.16y\displaystyle=(0,0.16y).
Major Section: 
Reference
Type of Math Object: 
Example
Parent: 
self-similar fractals

Mathematics Subject Classification

28A80 Fractals

Comments

Submitted by rm50 on Thu, 07/20/2006 - 22:02

Interesting post. It could be made more interesting by providing some background: for example: Are there similar results for other coefficient values? How would one go about proving this result? Who discovered this and in what context?

Submitted by ratboy on Fri, 07/21/2006 - 03:30

I think this field has its roots in the 19th century. Of course, Mandelbrot first popularized fractals. Michael Barnsley is the man behind fractals generated by iterated function systems. He has a textbook _Fractals Everywhere_. He used the technique to implement an image compression scheme in which only the coefficients of the transformations are stored. This is treated in his book _Fractal Image Compression_. The hard part is finding the coefficients that will yield a given image.

Submitted by silverfish on Fri, 07/21/2006 - 09:08

Fascinating. I have a different question: are things like the Mandlebrot set "self-similar fractals" with this definition? If so, can the contractions be written down?

Submitted by paolini on Tue, 07/25/2006 - 05:51

No, Mandelbrot is not this kind of fractal!

Submitted by paolini on Tue, 07/25/2006 - 05:53

I think some questions are answered in the parent object.

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Info

Owner: paolini
Added: 2006-07-20 - 09:26
Author(s): paolini

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(v7) by paolini 2013-03-22
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