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mandelbrot - How to define the Mandelbrot set Math 8 2009...

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Unformatted text preview: How to define the Mandelbrot set Math 8 2009, Chris Lyons The famous Mandelbrot set is a subset of the complex plane: The black points signify the members of the Mandelbrot set. Fix a number c in the complex plane. Here’s how we tell if c belongs to the Mandelbrot set or not: STEP 1: Define z0 = 0. STEP 2: For each n ≥ 0, define 2 zn+1 = zn + c. In other words, we let z1 = c z2 = c2 + c z3 = (c2 + c)2 + c z4 = ((c2 + c)2 + c)2 + c etc... STEP 3: The sequence {zn } is just some sequence of complex numbers, and we care about the size—i.e., the complex absolute value—of the numbers zn as n → ∞. • IF the sequence {zn } is bounded, i.e., if there is some R such that |zn | ≤ R for all n, then we say c belongs to the Mandelbrot set. • IF NOT, then we say that c does not belong to the Mandelbrot set. That’s it! 1 ...
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