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[parent] Barnsley fern (Example)

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

$\displaystyle F = \bigcup_{i=1}^4 T_i(F) $
where $ T_i\colon \mathbb{R}^2\to \mathbb{R}^2$ are the following linear mappings:
$\displaystyle T_1(x,y)$ $\displaystyle =(0.85 x + 0.04 y, -0.04 x + 0.85 y+1.6),$    
$\displaystyle T_2(x,y)$ $\displaystyle =(0.2 x - 0.26 y, 0.23 x + 0.22 y + 1.6),$    
$\displaystyle T_3(x,y)$ $\displaystyle =(-0.15 x +0.28 y, 0.26 x+0.24 y +0.44),$    
$\displaystyle T_4(x,y)$ $\displaystyle =(0,0.16 y).$    

\includegraphics{felcex}



"Barnsley fern" is owned by paolini.
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Cross-references: linear mappings, relation, compact subset
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This is version 4 of Barnsley fern, born on 2006-07-20, modified 2006-09-09.
Object id is 8154, canonical name is BarnsleyFern.
Accessed 1128 times total.

Classification:
AMS MSC: 28A80 (Measure and integration :: Classical measure theory :: Fractals)
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What's with the fern? by rm50 on 2006-07-20 18:02:22
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?
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