Exact Coordinates
Robert P. Munafo, 2023 Aug 5.
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).
Exact mu-Atom Nucleus Locations
The symbolic maths program Maxima can find the exact locations of the nuclei of R2F(1/2B1)S and R2.1/3a by setting the 3rd Lemniscate equal to zero and solving for c:
(%i1) L(n) := if n=0 then 0 else L(n-1)^2+c; 2 (%o1) L(n) := if n = 0 then 0 else L (n - 1) + c (%i2) solve(L(3)=0, c); sqrt(3) %i - 1 ---------- + --- 2 2 sqrt(23) 25 1/3 - 1 sqrt(3) %i 2 (%o2) [c = -------------------- + (-------- - --) (--- - ----------) - -, sqrt(23) 25 1/3 3/2 54 2 2 3 9 (-------- - --) 2 3 3/2 54 2 3 - 1 sqrt(3) %i --- - ---------- sqrt(23) 25 1/3 sqrt(3) %i - 1 2 2 2 c = (-------- - --) (---------- + ---) + -------------------- - -, 3/2 54 2 2 sqrt(23) 25 1/3 3 2 3 9 (-------- - --) 3/2 54 2 3 sqrt(23) 25 1/3 1 2 c = (-------- - --) + -------------------- - -, c = 0] 3/2 54 sqrt(23) 25 1/3 3 2 3 9 (-------- - --) 3/2 54 2 3 (%i3) rectform(float(%o2)); (%o3) [c = (- 0.7448617666197486 %i) - 0.1225611668766516, c = 0.7448617666197486 %i - 0.1225611668766516, c = - 1.754877666246696, c = 0.0]The final command shows that the answers are equivalent (except for roundoff error) to the numerical results in the nucleus article.
See Also
See also rational coordinates.
revisions: 20030922 oldest on record; 20230805 exact coordinates
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Aug 05.
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