Binary Decomposition  

Robert P. Munafo, 2011 Feb 28.

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.

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 Nth iteration) has a positive imaginary component. Use this to choose a lighter or darker color; the result shows the Nth bits of the binary decomposition within the Nth dwell band.

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

dwell only
dwell only
binary decomposition
binary decomposition
both binary decomposition and dwell
both binary decomposition and dwell

(All three images also show the filaments using a distance estimator algorithm.)

revisions: 19980504 oldest on record; 20110228 add illustrations

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.     Mu-ency index

Robert Munafo's home pages on HostMDS   © 1996-2020 Robert P. Munafo.
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