Mu-atom
Robert P. Munafo, 2023 Feb 18.
Definition : math. Any one of those areas, resembling either a circle or a cardioid, which is the largest connected superset of a subset of M, and possessing the property that all internal points have the same period.
This term was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature. On page 183, he defines a mu-atom as one of the "maximal connected sets", no two of which overlap, except perhaps in a single point (a bond). The mu-atoms make up Mu-molecules. All mu-atoms are shaped either like a disk or like a cardioid. The former are descendants; the latter are seeds.
For examples of mu-atoms, see the entry R2 or the figure in the child article.
A mu-atom is also called an attractive component, e.g. by Douady in The Beauty of Fractals page 165.
Some colloquial names for mu-atoms are ball, bud, bulb, decoration, lake and lakelet.
The R2 System suffix a designates the mu-atom that is the owner of a given mu-unit.
To the eye, it appears that all mu-atoms are either perfect circles or perfect cardioids. However, it now known that none except R2a and R2.1/2a are perfect (see Fractal Horizons).
The boundary of a mu-atom can be located, and numerically traced, using the Newton-Raphson method. This was used by Jay Hill in his estimation of the area of the Mandelbrot set
If the location and size of a mu-atom are known well enough to determine a rectangle that is about the same size as the mu-atom and that contains its nucleus, the Jordan curve method can be used to determine the mu-atom's period.
Acknowledgements
non-circular property: Ken Shirriff
revisions: 20000204 oldest on record; 20111228 add references to boundary tracing and area estimate; 20120316 describe R2 suffix a
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 Apr 18.
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