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@@ -77,7 +80,7 @@ <h1>Lindenmayer Grew a Forest</h1>
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You start with an axiom — a seed string like <code>F</code> — and a table of rewrite rules. At each step, every symbol in the string expands according to its rule. Apply four iterations of <code>F → FF+[+F-F-F]-[-F+F+F]</code> and the string grows to roughly 14,000 characters. Feed that string to a turtle that treats <code>F</code> as "draw forward," <code>+</code>/<code>-</code> as turns, and <code>[</code>/<code>]</code> as push/pop stack, and you get a branching tree indistinguishable from hand-drawn botanical illustration.
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You start with an axiom, a seed string like <code>F</code>, and a table of rewrite rules. At each step, every symbol in the string expands according to its rule. Apply four iterations of <code>F → FF+[+F-F-F]-[-F+F+F]</code> and the string grows to roughly 14,000 characters. Feed that string to a turtle that treats <code>F</code> as "draw forward," <code>+</code>/<code>-</code> as turns, and <code>[</code>/<code>]</code> as push/pop stack, and you get a branching tree indistinguishable from hand-drawn botanical illustration.
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@@ -111,7 +114,7 @@ <h1>Lindenmayer Grew a Forest</h1>
@@ -100,7 +103,7 @@ <h1>Space by Proximity Only</h1>
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The buffer renders at 1/4 resolution (<code>SCALE = 4</code>) and gets scaled back up with <code>imageSmoothingEnabled = false</code> to preserve the pixelated cell edges. For 30 seeds at a 400×225 buffer, this is around 2.7 million distance comparisons per frame — fast enough at 60 fps, clearly not the approach you'd take with 500 seeds. Fortune's algorithm would finish the diagram in a single sweep regardless of seed count. The brute force version only survives because small seed counts and the quarter-resolution buffer keep the constant small.
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The buffer renders at 1/4 resolution (<code>SCALE = 4</code>) and gets scaled back up with <code>imageSmoothingEnabled = false</code> to preserve the pixelated cell edges. For 30 seeds at a 400×225 buffer, this is around 2.7 million distance comparisons per frame, fast enough at 60 fps, clearly not the approach you'd take with 500 seeds. Fortune's algorithm would finish the diagram in a single sweep regardless of seed count. The brute force version only survives because small seed counts and the quarter-resolution buffer keep the constant small.
@@ -78,11 +81,11 @@ <h1>Cayley Never Saw This</h1>
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In 1879, Arthur Cayley wrote a short paper asking a simple question: given Newton's method applied to a complex polynomial, which starting points converge to which root? For a quadratic, it turns out to be clean. The complex plane splits in two along a straight line, and each half flows to one root without ambiguity. Cayley extended the analysis to cubics — a polynomial with three roots — and found the problem "seems to present considerable difficulty." He published the quadratic case and left the rest open. He had stumbled into a fractal boundary and had no tools to describe it.
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In 1879, Arthur Cayley wrote a short paper asking a simple question: given Newton's method applied to a complex polynomial, which starting points converge to which root? For a quadratic, it turns out to be clean. The complex plane splits in two along a straight line, and each half flows to one root without ambiguity. Cayley extended the analysis to cubics, a polynomial with three roots, and found the problem "seems to present considerable difficulty." He published the quadratic case and left the rest open. He had stumbled into a fractal boundary and had no tools to describe it.
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The difficulty Cayley noticed: a third basin appears near the boundary between any two basins of attraction. And near that boundary, the first two reappear. The structure repeats at every scale. In the 1970s and 1980s, as raster displays arrived and Benoit Mandelbrot was cataloguing fractal geometry, John Hubbard worked out the topology of Newton basins and showed that the boundary between any two basins is simultaneously the boundary of every other basin. Every boundary point is a limit point of all regions at once. The fractal is not just complicated — it is, in a precise sense, irreducible.
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The difficulty Cayley noticed: a third basin appears near the boundary between any two basins of attraction. And near that boundary, the first two reappear. The structure repeats at every scale. In the 1970s and 1980s, as raster displays arrived and Benoit Mandelbrot was cataloguing fractal geometry, John Hubbard worked out the topology of Newton basins and showed that the boundary between any two basins is simultaneously the boundary of every other basin. Every boundary point is a limit point of all regions at once. The fractal is not just complicated, it is, in a precise sense, irreducible.
@@ -183,7 +186,7 @@ <h2>What the gallery is building</h2>
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<h3>Slime Mold</p>
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<pclass="post-artifact-desc">4,000 agents navigating by chemical trail sampling, self-organizing into vascular networks — Physarum polycephalum in software.</p>
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<pclass="post-artifact-desc">4,000 agents navigating by chemical trail sampling, self-organizing into vascular networks. Physarum polycephalum in software.</p>
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