First, what is the boundary
of a set S in the plane?. |

The boundary of the Mandelbrot set looks very crinkly. |

To get an idea of just how complicated the boundary is, we introduce
two special types of points in the Mandelbrot set. |

Centers of Mandelbrot set components are c values with
z = 0 belonging to a cycle. |

Misiurewicz points are c values with
z = 0 iterating in a finite number of steps to a cycle not containing z = 0. |

At the Mandelbrot set boundary, the Julia sets and the Mandelbrot set appear
to look more and more alike. |

The boundary of the Mandelbrot set is so fuzzy it is
2-dimensional. |