Second-Order Embedded Julia Set

Robert P. Munafo, 2002 Apr 18.

Truly the most stunning features of the Mandelbrot Set, when viewed with the aid of the Distance-Estimator function, are the higher-order Embedded Julia Sets.

Here is an example (click on it to enlarge):

A second-order embedded Julia set

Just as Embedded Julia Sets are found by exploring very close to an island Mu-molecule, the second-order Embedded Julia Sets are found by exploring very close to an island — namely, an island within an embedded Julia set.

The order indicates how many islands you have gone close to while zooming in. An island counts only if it is not part of the mu-unit of a larger island which has already been counted. Every new mu-unit you get close to introduces a new "distortion field" (more accurately, tuning) that creates Julia Sets, and the place you explore in relation to this mu-unit determines the shape of the Julia set for that order. Each time you go to a higher order, another Julia Set shape is added to the nested symmetric filaments around the island at the center.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index

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