# 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 2^{n} 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

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Jun 11. s.27