Complex Numbers

Mandelbrot Set Intuition

Fractal from complex iteration

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  • about 5 min
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|z|=2z0z1z2z3z4z5z6z7z8z9z10z11z12z13z14z15z16z17z18z19z20c
Re(c) = -0.70
Im(c) = 0.27
zn+1=zn2+c,c=−0.7+0.27iz_{n+1} = z_n^2 + c,\quad c = -0.7+0.27i
∣zn∣>2⇒ extescapes|z_n| > 2 \Rightarrow \ ext{escapes}

Stays bounded (50 iters)

The Mandelbrot set = all c for which the sequence zₙ₊₁=zₙ²+c stays bounded. Each pixel in the Mandelbrot image is one complex c.