Robert P. Munafo, 2023 Aug 5.
The term "nucleus" has an official definition and a colloquial definition specific to Mu-ency.
Nucleus of a Mu-Atom
If you set the polynomial formula for a lemniscate ZN equal to zero and solve for C (to get the roots of the polynomial), the roots are the nuclei of the mu-atoms of period N, plus any mu-atoms of periods that divide evenly into N. This procedure has been used numerically by Jay Hill to find all mu-atoms for periods up to about 16.
The R2 System suffix .n designates the nucleus of a given mu-atom.
Finding Nucleus Coordinates Exactly, and Numerically
The 3rd lemniscate is a 4th-order polynomial, the product of c and a 3rd-order polynomial :
Z3 = C4 + 2 C3 + C2 + C
= C (C3 + 2 C2 + C + 1)
Beyond period 3, numerical techniques can be used, as demonstrated here using maxima:(%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) n(p) := allroots(L(p)); (%o2) n(p) := allroots(L(p)) (%i3) n(3); (%o3) [c = 0.0, c = 0.7448617666197442 %i - 0.1225611668766536, c = (- 0.7448617666197442 %i) - 0.1225611668766536, c = - 1.754877666246693] (%i4) n(4); (%o4) [c = 0.0, c = 0.5300606175785253 %i + 0.2822713907669139, c = 0.2822713907669139 - 0.5300606175785253 %i, c = - 0.9999999999999986, c = 1.032247108922832 %i - 0.1565201668337551, c = (- 1.032247108922832 %i) - 0.1565201668337551, c = - 1.310702641336835, c = - 1.940799806529483]
Nucleus of a Paramecium
revisions: 20030922 oldest on record; 20120316 describe R2 suffix .n; 20120421 add headings; 20230805 numerical solutions
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Aug 05. s.27