R2a  

Robert P. Munafo, 2012 Apr 16.



R2a is the largest mu-atom in the Mandelbrot Set, and it is the seed of the continent. It is the large cardioid; see the R2 entry for a picture identifying its location and links to other attached features.

Colloquial names for R2a include body, main cardioid, fixed-point cardioid.

Formulas

The boundary of R2a can be precisely described by a parametrized relation. It is the set of all points P given by the expression

P = ((A + i)/2)2 + 1/4

where A is a point on the unit circle, i.e. |A| = 1. From this formula it is also possible to derive the location of all secondary continental mu-atoms; see the secondary continental mu-atom article for details.

Unique Properties

All of the points in the interior of R2a have a period of 1. Equivalently, the iterates of any point in R2a approach a limit cycle that contains just one point. (This is the reason for the name fixed-point cardioid).

The Julia sets of points in the interior of R2a have an interior that is connected. No other Julia Sets have this property.


revisions: 20020508 oldest on record; 20120416 elaborate on fixed point property




From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.     Mu-ency index
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