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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

The unique point within any mu-atom which has the property of belonging to its own limit cycle. This point is called the superstable point.

This use of the word 'nucleus' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature.

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.

R2a.n (nucleus of R2a) is the origin, 0 + 0i. R2.1/2a.n (the nucleus of R2.1/2a) is -1 + 0i.

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)

An exact solution to the Cubic equation gives the exact albeit cumbersome formula shown in the exact coordinates article.

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

I also sometimes use "nucleus" as a colloquial term for the feature with 4-fold rotational symmetry at the center of an embedded Julia set or paramecium. See paramecia.

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) 1987-2024.

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