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	D2/4 - D/2 + c = 0
	R2.1/2a	D/4 - 1 - c = 0
	R2F(1/2B1)Sa, R2.1/3a and R2.2/3a	c3 + 2 c2 - (D/8-1) c + (D/8-1)2 = 0
	Period-4 mu-atoms	c6 + 3 c5 + (D/16+3) c4 + (D/16+3) c3 - (D/16+2) (D/16-1) c2 - (D/16-1)3 = 0
	Period-5 mu-atoms	c15 + 8 c14 + 28 c13 + (mu + 60) c12 + (7 mu + 94) c11 + (3 mu2 + 20 mu + 116) c10 + (11 mu2 + 33 mu + 114) c9 + (6 mu2 + 40 mu + 94) c8 + (2 mu3 - 20 mu2 + 37 mu + 69) c7 + (3 mu - 11) (3 mu2 - 3 mu - 4) c6 + (mu - 1) (3 mu3 + 20 mu2 - 33 mu - 26) c5 + (3 mu2 + 27 mu + 14) (mu - 1)2 c4 - (6 mu + 5) (mu - 1)3 c3 + (mu + 2) (mu - 1)4 c2 - (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) 1987-2012.     Mu-ency index

---

Robert Munafo's home pages on HostMDS   © 1996-2012 Robert P. Munafo.   about   contact   Google+   mrob27   @mrob_27

This work is licensed under a Creative Commons Attribution 2.5 License. Details here s.13