Binary Decomposition

Robert P. Munafo, 2023 Jun 10.

A Representation Function which represents the Mandelbrot Set in a way that lends itself well to casual study of External Angles.

The Level Sets and Field Lines are superimposed, creating a sort of grid, and the "squares" of the grid are filled with N-digit binary numbers giving the first N binary digits of the external angles of field lines passing through the square.

(Since these squares generally get smaller as N increases it is more common to just display the Nth binary digit within each square, using two colors, a brightness variation or some other simple visual distinction.)

Each level set (dwell band) is divided into 2ⁿ squares. It is easy to "read" the external arguments of points in the boundary of the Mandelbrot Set using a binary decomposition. This can be used as the basis of a practical computational technique for tracing external rays (see that article for details).

The simplest way to create a binary decomposition plot is to use the ordinary escape-iterations algorithm with a very large escape radius, note whether the escaped value of Z (the N^th iteration) has a positive imaginary component. Use this to choose a lighter or darker color; the result shows the N^th bits of the binary decomposition within the N^th dwell band.

The following images show R2 (the entire Mandelbrot set) showing dwell information, showing binary decomposition, and showing both:

dwell only

binary decomposition

both binary decomposition and dwell

(The image on the right also shows the filaments using a distance estimator algorithm.)

revisions: 19980504 oldest on record; 20110228 add illustrations; 20230610 add another sentence linking to the external rays article

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 11.

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