Island Mu-Molecule

Robert P. Munafo, 2012 Aug 18.

Definition :
Any mu-molecule that is not the continent.

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

Island Mu-Molecules are usually the third thing one notices (after the continent itself and the filaments; see exploring). It is an unexpected surprise: "Hey! there are little Mandelbrot Sets here!" and the dedicated explorer will then find that the islands are all miniatures of the entire Mandelbrot Set down to the last filament. See the largest islands entry for a list.

It is also noticed fairly early on that the islands are everywhere, and most of them are very small and difficult to find. As Milnor stated:

Every neighborhood of a point on the boundary of the Mandelbrot set contains infinitely many embedded copies of the Mandelbrot Set.

In other words, every tiniest bit of filament has infinitely many islands embedded in it, although most are very tiny.

The islands are at the center of most of the interesting filament features, including the filament bifurcation that makes the external angles visible, and embedded Julia sets.

Colloquial names for island mu-molecules: Bug, Island, Mandelbrotie, Minibrot.

The largest islands are R2F(1/2B1)S and R2F(1/3B1)S; see the largest islands article for more.

Images and Links

R2F(1/2B1)S, the largest island

A cluster of islands in the R2.2/5 radical

R2F(1/15B1)S, located in the cusp R2.C(0), a highly distorted island