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MAKE A MEME View Large Image $Peano-curve animated (140x120).gif below 2008-03-23 self is for the lower left triangle <math>C \ 0 \scriptstyle \frac 1 2 \mapsto \ x y \;/\; x\geqslant 0 \; y\geqslant 0 \; x+y \leqslant 1 \ </math> For a point <math>p x y </math>in the ...$

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Keywords: Peano-curve animated (140x120).gif below 2008-03-23 self is for the lower left triangle <math>C \ 0 \scriptstyle \frac 1 2 \mapsto \ x y \;/\; x\geqslant 0 \; y\geqslant 0 \; x+y \leqslant 1 \ </math> For a point <math>p x y </math>in the image of <math>C</math> i e in the lower left triangle we construct its preimage <math>\lambda</math> using the notation <math>\begin align r \bot r_x r_y \bot r_y -r_x \end align </math> Starting with <math>\begin align p x y \\ r_0 \scriptstyle \frac 1 2 \scriptstyle \frac 1 2 \\ q_0 0 0 \end align </math> we set <math>\begin align \alpha_n p-q_n \cdot r_n \bot\\ \beta_n \begin cases -1 \text if \alpha_n < 0\\ 1 \text if \alpha_n \geqslant 0 \end cases \\ q_ n+1 q_n+r_n\\ r_ n+1 \frac 1 2 \begin pmatrix 1 -\beta_n \\ \beta_n 1 \end pmatrix r_n\\ \lambda \frac 1 2 \sum_ n 0 \infty \frac 1+\beta_n -1 n 2 \cdot \frac 1 2 n \end align </math> We then get <math>C \lambda p</math> Given <math>\lambda_0</math> in 0 ½ all pixels with <math>\lambda p < \lambda_0</math> are colored in scarlet and all pixels with <math>\lambda p > \lambda_0</math> are colored in white Animations of fractals Peano curve