# Exact Coordinates

Robert P. Munafo, 2003 Sep 22.

Several features of the Mandelbrot Set can be calculated exactly.

Using the Brown method, the boundary of many small period cardioids and circular mu-atoms can be found. For example, the following formulas express the relation between a point c on the boundary of a mu-atom and a point D on the unit disk:

R2a | D^{2}/4 - D/2 + c = 0
| |

R2.1/2a | D/4 - 1 - c = 0 | |

R2F(1/2B1)Sa, R2.1/3a and R2.2/3a |
c^{3} + 2 c^{2} - (D/8-1) c + (D/8-1)^{2} = 0
| |

Period-4 mu-atoms |
c^{6} + 3 c^{5} + (D/16+3) c^{4} + (D/16+3) c^{3}
- (D/16+2) (D/16-1) c^{2} - (D/16-1)^{3} = 0
| |

Period-5 mu-atoms |
c^{15} + 8 c^{14} + 28 c^{13} + (mu + 60) c^{12} + (7 mu + 94) c^{11}
+ (3 mu^{2} + 20 mu + 116) c^{10} + (11 mu^{2} + 33 mu + 114) c^{9}
+ (6 mu^{2} + 40 mu + 94) c^{8}
+ (2 mu^{3} - 20 mu^{2} + 37 mu + 69) c^{7}
+ (3 mu - 11) (3 mu^{2} - 3 mu - 4) c^{6}
+ (mu - 1) (3 mu^{3} + 20 mu^{2} - 33 mu - 26) c^{5}
+ (3 mu^{2} + 27 mu + 14) (mu - 1)^{2} c^{4}
- (6 mu + 5) (mu - 1)^{3} c^{3} + (mu + 2) (mu - 1)^{4} c^{2}
- (mu - 1)^{5} c + (mu - 1)^{6} = 0
where mu = D/32 |

The formulas for period-1 through period-3 mu-atoms can be solved explicitly; the others are evaluated numerically using Newton's method or a similar technique (see derivative).

See also rational coordinates.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022. Mu-ency index

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2022 Mar 28. s.27