| Mandelbrot set definition: simple iteration involving only multiplication and addition can generate the most complicated object ever seen. | |
| Small copies of the Mandelbrot set: the Mandelbrot set is surrounded by (infinitely many) small copies of the Mandelbrot set, each of which is surrounded by infinitely many still smaller copies of the Mandelbrot set, and so on, forever. | |
| are peninsulas, not islands: all these copies are connected together, despite the appearance that they are islands. | |
| The Mandelbrot set boundary: is so remarkably wiggly that it fills up a 2-dimensional space. | |
| The main unsolved problem: take a point on the boundary of the Mandelbrot set. Does a small enough circle around that point cut off pieces of the Mandelbrot set that cannot be joined back to the point without leaving the circle? | |
| Some recent results: the Mandelbrot set has inspired a great collection of subtle mathematical work. | |
Much more information about the Mandelbrot set can be found at http://classes.yale.edu/fractals/MandelSet/welcome.html.