General information

Overview

Fraxion is a multi-threaded fractal exploration application, mainly providing Mandelbrot/Julia-type, Newton/Raphson-type, Magnet-type, and Markus/Lyapunov-type fractals. It started as a way for plotting the classic Mandelbrot and interactively viewing the various dual Julia sets, by iterating the following equations:

To construct the Mandelbrot set as a main fractal, we iterate over all pixels in the shown region, each time starting with z0 equal to zero and c equal to the relative location of the screen pixel in the complex plane. So during iterations, z is variable and c is taking as a constant.

Similarly, the Julia set as a dual fractal is constructed by fixing c to a constant value and iterating over all pixels in the shown region, each time starting with z0 equal to the relative location of the screen pixel in the complex plane. Note that we typically speak of the filled-in Julia set, as the Julia set itself is the boundary of the latter. The complement of a Julia set is called a Fatou set. The Mandelbrot set is located in the c complex parameter plane, the Julia set in the z complex dynamic plane. The fundamental dichotomy states that Julia sets are either connected, or composed of infinitely many separate points (i.e., a Cantor set). More so, if a point lies inside the Mandelbrot set, its corresponding Julia set is connected (becaues of a bound orbit). If conversely a point lies outside the Mandelbrot set, then its corresponding Julia set dissolves into fractal dust (because of a diverging orbit).

Every point is thus classified as being either inside or outside the Mandelbrot and Julia sets, depending on whether or not its orbit never escapes (converging), implying that it is attracted to a single fixed point or enters a periodic cycle, or in constract escapes to infinity (diverging). This is shown in the following examples, whereby we each time show the forward orbit in the complex plane for the Mandelbrot set and an analysis panel containing the time series of the log-log modulus (upper part) and the angle (lower part).

Single fixed point attractor
Periodic cycle (polygon-shaped)
Periodic cycle (star-shaped)
Escaping to infinity

Behind the scenes, the Fraxion application divides the screen into equally sized blocks (the amount can be controlled via the Set number of blocks to use option in the Multithreading menu). Each block is then randomly selected, so as to spread the load between rapidly diverging and slowly converging regions. Fraxion then spawns multiple threads that are executed on the available number of CPU cores in the system, including hyperthreads (the amount can be controlled via the Set number of CPU cores to use option in the Multithreading menu). In case of multiple available CPU cores, Fraxion leaves 1 unused by default in order to leave to some processing power for other applications. Fraxion calculates and stores up to 15 parameters for each point in the complex plane, explaining its strenuous computation times. All results are stored in an iteration result buffer. Subsequently, the application uses the selected colouring algorithms to create an image buffer, which is then displayed on the screen. If the colouring is changed later on, then the application only needs to recreate the image buffer instead of recalculating the entire result buffer.

Fraxion currently supports converging algorithms (based on escape times) for Mandelbrot/Julia-type fractals, diverging algorithms (based on root detection) for Newton/Raphson-type fractals, both converging/diverging algorithms for Magnet-type fractals, and explicit bifurcation calculations for Markus/Lyapunov-type fractals.

Contact

The Fraxion (version 1.3.5) application was designed for the Java 2 Platform Standard Edition 1.8.0 or higher. When using this application, please use the following reference:

Fraxion (2003-2016)
Sven Maerivoet
URL: http://fraxion.maerivoet.org/
E-mail: sven.maerivoet@gmail.com
Copyright 2003-2016 Sven Maerivoet

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

References

Future work

The original Mandelbrot

Images reproduced after Heinz-Otto Peitgen and Peter H. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, 1986.

First Mandelbrot island molecule (1 March 1980)
Successive zooms in the Mandelbrot set (April 1980)

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