Mandelbrot Set III

I wrote my first ever Mandelbrot Set renderer back in 2015 and used Python to slowly create fractal images. Over a year later, I revisited the project with a Java version which — due to its code being actually compiled — ran much faster, yet had the same clunky interface; a rectangle the user had to draw and a key they had to press to the view change to the selected region.
In this post, over half a year later, I present my newest Mandelbrot Set fractal renderer (download the .jar), written in Java, which both runs fast and allows a much more intuitive and immersive walk through the complex plane by utilizing mouse dragging and scrolling.
The still time demanding task of rendering fractals — even in compiled languages — is split up into a low quality preview rendering, a normal quality display rendering and a high quality 4K (UHD-1 at 3840×2160 pixels to keep a 16:9 image ratio) rendering, all running in seperate threads.

Rainbow spiral
Rainbow spiral

The color schemes where also updated, apart from the usual black-and-white look there are multiple rainbow color schemes which rely on the HSB color space, zebra color schemes which use the iterations taken modulo some constant to define the color and a prime color scheme which tests if the number of iterations taken is prime.

Zebra spiral
Zebra spiral

Apart from the mouse and keyboard control, there is also a menu bar (implemented using Java’s JMenuBar) which allows for more conventional user input through a proper GUI.

Controls

  • Left mouse dragging: pan view
  • Left mouse double click: set cursor’s complex number to image center
  • Mouse scrolling: zoom view
  • Mouse scrolling +CTLR: pan view
  • ‘p’: render high definition fractal
  • ‘r’: reset view to default
  • ‘w’, ‘s’: zoom frame
  • Arrow keys: pan view
  • Arrow keys +CTRL: zoom view
  • Menu bar
    • Fractal: extra info about current fractal rendering
    • Color Scheme: change color scheme and maximum iteration depth
    • HD: controls for high definition rendering
    • Extra: help and about
Blue spiral
Blue spiral

A bit more on how the three threads are implemented.
Whenever the user changes the current view, the main program thread renders a low quality preview and immediately draws it to the screen. In the background, the normal quality thread (its pixel dimensions match the frame’s pixel dimensions) is told to start working. Once this medium quality rendering is finished, it is preferred to the low quality rendering and gets drawn on the screen.
If the user likes a particular frame, they can initiate a high quality rendering (4K UHD-1, 3840×2160 pixels) either by pressing ‘q’ or selecting HD->Render current frame. This high quality rendering obviously takes some time and a lot of processing power, so this thread is throttled by default to allow the user to further explore the fractal. Throttling can be disabled through the menu option HD->Fast rendering. There is also the option to tell the program to exit upon having finished the last queued high definition rendering (HD->Quit when done).
The high definition renderings are saved as .png files and named with their four defining constants. Zim and Zre define the image’s complex center, Zom defines the complex length above the image’s center. Clr defines the number of maximum iterations.

Another blue spiral
Another blue spiral

Just to illustrate how resource intensive fractal rendering really is.
A 4K fractal at 3840×2160 pixels with a iteration depth of 256 would in the worst case scenario (no complex numbers actually escape) require 3840 \cdot 2160 \cdot 256 \cdot 4 = 8493465600 double multiplications. If you had a super-optimized CPU which could do one double multiplication a clock tick (which current CPUs definitely cannot) and ran at 4.00 GHz, it would still take that massively overpowered machine \frac{8493465600}{4 \cdot 10^9} = 2.123 seconds. Larger images and higher maximum iterations would only increase the generated overhead.
The program’s source code is listed below and can also be downloaded (.java), though the compiled .jar can also be downloaded.

Green self-similarity
Green self-similarity

Unrelated to algorithmically generating fractal renderings, I recently found a weed which seemed to be related to the Mandelbrot Set and makes nature’s intertwined relationship with fractals blatently obvious. I call it the Mandel Weed.

Mandel Weed
Mandel Weed
// Java Code; Jonathan Frech; 22nd, 23rd, 24th, 25th, 26th, 27th of July 2017

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Bifurcation Diagram

Generating the famous fractal, which can be used to model populations with various cycles, generate pseudo-random numbers and determine one of nature’s fundamental constants, the Feigenbaum constant \delta.
The fractal nature comes from iteratively applying a simple function, f(x) = \lambda \cdot x \cdot (1-x) with 0 \leq \lambda \leq 4, and looking at its poles.
The resulting image looks mundane at first, when looking at 0 \leq \lambda \leq 3, though the last quarter section is where the interesting things are happening (hence the image below only shows the diagram for 2 \leq \lambda \leq 4).
From \lambda = 3 on, the diagram bifurcates, always doubling its number of poles, until it enters the beautiful realm of chaos and fractals.

Bifurcation Diagram lambda in range [2; 4]
For more on bifurcation, fractals and \delta, I refer to this Wikipedia entry and WolframMathworld.


# Python 2.7.7 Code
# Jonathan Frech, 24th of March 2017

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Mandelbrot Set II

Over a year ago, I published my first Mandelbrot Set viewer, a Python program using pygame. Since then, I have published a rather short program highlighting errors that can occur when calculating the set (Mandelbrot Set Miscalculations).
Since my first viewer was in Python, which is an interpreted programming language, and I wanted to make my viewer faster, I decided to write one in Java. I was hoping for a speed increase since Java is compiled and thus should run at higher speeds. I was not disappointed. The new Java-based viewer runs noticeably faster and additionally I added a lot of new features, all listed below.

Controls

  • Left-clicking and dragging draws a zoom frame, single left-clicking removes the frame
  • Right clicking (and optionally dragging) moves the zoom frame
  • Space zooms into the zoom frame
  • F1 moves one step back the zoom history
  • F2 shows the path a complex number at the cursor’s position follows when the function is iteratively applied
  • F3 shows the \mathbb{R} and \mathbb{R}i axis
  • F4 displays the current cursor’s corresponding complex number
  • F5 toggles between showing and hiding the menu (text in the left upper corner describing the viewer’s functions and current states)
  • F6 increments the exponent (going from f_c(z)=z^2+c to f_c(z)=z^5+c in whole-number steps)
  • F7 toggles between the Mandelbrot set and the filled Julia set
  • F8 toggles between previewing a small filled Julia set at the cursor’s position based upon the cursor’s complex number
  • F9 completely resets the zoom and zoom history
  • F11 (or F) toggles between fullscreen and windowed mode
  • F12 quits the application
  • L increases the color depth (starting at 256 and increasing in steps of 256)
  • Q saves the current image to disk

To use this application, you can either trust me and download the .jar-file or view the source code listed below, verify it and compile the program yourself.
The program will start in fullscreen mode, to change to windowed mode, just press F11 (as listed above).

The standard Mandelbrot setA Filled Julia setA Mandelbrot set using the fourth powerA deeper zoom into a fourth power Mandelbrot setA filled Julia set of the fourth power


// Java Code
// Jonathan Frech 14th of September, 2016
//         edited 15th of September, 2016
//         edited 16th of September, 2016
//         edited 17th of September, 2016
//         edited 18th of September, 2016
//         edited 19th of September, 2016
//         edited 23rd of September, 2016
//         edited 24th of September, 2016
//         edited 26th of September, 2016
//         edited 27th of September, 2016
//         edited 28th of September, 2016
//         edited 29th of September, 2016
//         edited 30th of September, 2016
//         edited  1st of October  , 2016
//         edited  2nd of October  , 2016
//         edited  3rd of October  , 2016
//         edited  4th of October  , 2016
//         edited 21st of November , 2016
//         edited 23rd of November , 2016
//         edited 14th of December , 2016
//         edited 13th of January  , 2017

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