Mandelbrot Set drawn by a Turing Machine

Abstract

the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial z_(n+1)=z_n^2+c remains bounded.

That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. 

Screenshots

The Turing Machine runs around the Mandelbrot Set

The Mandelbrot Set completed
a click into the Mandelbrot Set draws the Julia Set

a click into the Mandelbrot Set draws the Julia Set

Running the Java Application

git clone https://github.com/phasenraum2010/mandelbrot.git
cd mandelbrot
mvn clean install exec:java

or Download the jar and double click on it to start the Application:
http://www.thomas-woehlke.de/a/mandelbrot/mandelbrot-1.1-SNAPSHOT.jar

Running the JavaScript Application

goto: http://woehlke.org/html5-lab/mandelbrot/

The Turing Machine to Compute the Mandelbrot Set

The Complex Number Plane is divided into Cells for the two dimensional Tape of the Turing-Machine.

Starting with Complex Number of Cell right from the Mandelbrot Set the Turing Machine goes one Step to the left and computes, wether the Complex Number of the Cell is inside the Set. If not, it continues to go left.

The Turing-Machine then goes around the Set by computing wether the Complex Number of the Cell is inside the Set. If not it turns left, else it turns right. After that it goes one step forward.

After the Turing Machine finished to run around the Set, the Application paints the Inside of the Set Black und computes the Colors of the Outside.

Julia Set

The Mandelbrot Set is drawn into the Complex Number Plane and so clicking into it, it delivers the Complex Number to calculate a Julia Set for it.

The most beautiful Julia Sets are drawn by clicking into the Border of the Mandelbrot Set.

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