The set obtained by the Quadratic Recurrence

(1) |

J. Hubbard and A. Douady proved that the Mandelbrot set is Connected. Shishikura (1994) proved that the boundary of the Mandelbrot set is a Fractal with Hausdorff Dimension 2. However, it is not yet known if the Mandelbrot set is pathwise-connected. If it is pathwise-connected, then Hubbard and Douady's proof implies that the Mandelbrot set is the image of a Circle and can be constructed from a Disk by collapsing certain arcs in the interior (Douady 1986).

The Area of the set is known to lie between 1.5031 and 1.5702; it is estimated as 1.50659....

Decomposing the Complex coordinate and gives

(2) | |||

(3) |

In practice, the limit is approximated by

(4) |

(5) | |||

(6) | |||

(7) | |||

(8) |

When written in Cartesian Coordinates, the first three of these are

(9) | |||

(10) | |||

(11) |

which are a Circle, an Oval, and a Pear Curve. In fact, the second Lemniscate can be written in terms of a new coordinate system with as

(12) |

The kidney bean-shaped portion of the Mandelbrot set is bordered by a Cardioid with equations

(13) |

(14) |

Generalizations of the Mandelbrot set can be constructed by replacing with or , where is a Positive Integer and denotes the Complex Conjugate of . The following figures show the Fractals obtained for , 3, and 4 (Dickau). The plots on the right have replaced with and are sometimes called ``Mandelbar Sets.''

**References**

Alfeld, P. ``The Mandelbrot Set.'' http://www.math.utah.edu/~alfeld/math/mandelbrot/mandelbrot1.html.

Branner, B. ``The Mandelbrot Set.'' In *Chaos and Fractals: The Mathematics Behind the Computer Graphics,
Proc. Sympos. Appl. Math., Vol. 39* (Ed. R. L. Devaney and L. Keen). Providence, RI: Amer. Math. Soc., 75-105, 1989.

Dickau, R. M. ``Mandelbrot (and Similar) Sets.'' http://forum.swarthmore.edu/advanced/robertd/mandelbrot.html.

Douady, A. ``Julia Sets and the Mandelbrot Set.'' In *The Beauty of Fractals: Images of Complex Dynamical Systems*
(Ed. H.-O. Peitgen and D. H. Richter). Berlin: Springer-Verlag, p. 161, 1986.

Eppstein, D. ``Area of the Mandelbrot Set.'' http://www.ics.uci.edu/~eppstein/junkyard/mand-area.html.

Fisher, Y. and Hill, J. ``Bounding the Area of the Mandelbrot Set.'' Submitted.

Hill, J. R. ``Fractals and the Grand Internet Parallel Processing Project.'' Ch. 15 in *Fractal Horizons:
The Future Use of Fractals.* New York: St. Martin's Press, pp. 299-323, 1996.

Lauwerier, H. *Fractals: Endlessly Repeated Geometric Figures.* Princeton, NJ: Princeton University Press,
pp. 148-151 and 179-180, 1991.

Munafo, R. ``Mu-Ency--The Encyclopedia of the Mandelbrot Set.'' http://home.earthlink.net/~mrob/muency.html.

Peitgen, H.-O. and Saupe, D. (Eds.). *The Science of Fractal Images.* New York: Springer-Verlag, pp. 178-179, 1988.

Shishikura, M. ``The Boundary of the Mandelbrot Set has Hausdorff Dimension Two.'' *Astérisque*, No. 222, **7**, 389-405, 1994.

© 1996-9

1999-05-26