Island

Robert P. Munafo, 2023 Jun 16.

Colloquial abbreviation of island mu-molecule, or of island mu-unit.

In Mu-Ency, island is more often used to refer to the island's mu-unit, because that is the more useful distinction when looking at the mathematical properties. However, when Benoit Mandelbrot used the term in his book The Fractal Geometry of Nature, he used it to refer to a mu-molecule.

Ubiquity

Given any point c on the boundary of the Mandelbrot set, find an external ray that leads to the point (there may be more than one; choose any one). Take the external angle of that ray, expressed as a binary fraction, and truncate it after an arbitrary number of digits d. Then form another binary fraction by repeating those d digits over and over again, indefinitely. The result is a fraction with denominator 2^*d-1. The external ray for that fraction lands at the cardioid cusp of an island of period d.

This process can be performed for any arbitrarily large (but finite) number of digits d, yielding a ray that leads to an island of arbitrarily large (but finite) period.

In practice, it seems that one can choose any arbitrarily small neighborhood of c and select a suitably large number of digits d such that the external ray leads to an island within that neighborhood. However, proving this for every boundary point c is equivalent to the MLC conjecture. If MLC turns out to be true then we could say that any neighborhood of a point on the boundary of the Mandelbrot set contains an infinite number of embedded copies of the Mandelbrot set. Regarding the more limited set of points c for which this is known, see the boundary article, in the section "Density of Islands".

See also the largest islands and distortion articles.

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

revisions: 20031223 oldest on record; 20101021 add images and links; 20230616 ubiquity of small islands near any point

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