# Nucleus

Robert P. Munafo, 2012 Apr 21.

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 Z_{N} 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.

### 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

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023. Mu-ency index

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Apr 18. s.27