## 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.

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.

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
• Fractal: extra info about current fractal rendering
• Color Scheme: change color scheme and maximum iteration depth
• HD: controls for high definition rendering

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.

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.

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.

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

## Mandelbrot Set ASCII Viewer

The Mandelbrot Set is the set of all complex points which, when one iteratively and infinitely applies the function $f_c(z)=z^2+c$, converge to a value. This simple rule results in stunning complexity and beauty.
Many Mandelbrot Set animations use regularly colored pixels to represent the number of iterations needed at the fractal’s edges to escape converging. Yet this mathematical object can also be represented as ASCII characters — similar to what I did in my Curses Cam post. The characters are chosen according to their opaqueness. A full stop (‘.’) looks lighter than a dollar sign (‘$’), so they represent a smaller or larger number of iterations needed. The order of characters used is taken from this post by Paul Borke. As there are only 70 characters used, each frame is being rendered twice to determine the minimum number of iterations needed by every point in that frame. Thereby the full visual character range is used. The characters shown below represent a Mandelbrot Set still. To see the zoom in action, either run the program (listed below) or take a look at this Mandelbrot Set ASCII journey.  ..................''''''''"">>II''''...... ..................''''''''^^,,ii::^^''''...... ..................''''''''^^::ww$$++,,''''...... ................''''''''^^^^""::$$::""^^''''...... ..............''''''""{{;;XXuuUU,,,,""''...... ............''''^^,,rr<<$$--........ ........''^^""LL$$$$__""''...... ..''''''^^!!"""",,""""::__$$ll""''........ ''''^^::__IIYYii::llpp^^''........ ''"";;[[$$++__$$^^''''...... ^^^^,,;;>>ww''''...... "",,,,II$$nn$$$$""''''...... "",,,,II$$nn""''''...... ^^^^,,;;>>ww''''...... ''"";;[[$$++__$$^^''''...... ''''^^::__IIYYii::llpp^^''........ ..''''''^^!!"""",,""""::__$$ll""''........ ........''^^""LL$$$$__""''...... ............''''^^,,rr$$$$<<$$--........ ..............''''''""{{;;XXuuUU,,,,""''...... ................''''''''^^^^""::$$::""^^''''...... ..................''''''''^^::ww$$++,,''''...... ..................''''''''^^,,ii::^^''''...... The fractal viewer is written in Python 2.7 and works by determining the terminal’s size and then printing a string of according size. This creates the illusion of a moving image, as the terminal will hopefully always perfectly scroll so that only one frame is visible at a time. In the code’s first non-comment line one can change the complex point at the image’s center, (really, its conjugate, which is partially irrelevant as the set is symmetric along the real axis) the initial zoom value (complex distance above the image’s center), the zoom factor (the factor by which the zoom value gets multiplied after a frame), the total number of frames (-1 means there is no upper limit), the delay between frames (in seconds, can be floating-point) and the color characters used. The program’s source code may not be particularly easy to read, yet it does its job and only requires seven non-comment lines! The code is shown below, though the .py file can also be downloaded. To achieve the JavaScript animation linked to above, I wrote a simple Python converter which takes in the fractal renderer’s output and it spits out an HTML page. This converter’s code is not listed, though the .py file can be downloaded. Instructions on how to use the converter can be seen in its source code. # Python 2.7 Code; Jonathan Frech, 15th and 16th of June 2017 P,Z,F,N,D,K=-.707+.353j,3,.9,-1,.1," .'^\",:;Il!i><~+_-?][}{1)(|\\/tfjrxnuvczXYUJCLQ0OZmwqpdbkhao*#MW&8%B@$"
def C(c):
global m;z,i=0j,-1
while abs(z)<=2 and i<len(K)-1+M:z,i=z*z+c,i+1
m=min(m,i);return K[i-M]*2
while n<N or N==-1:h=Z*2.;w=h*W/H;R=lambda:"\n\n"*(n!=0)+"\n".join("".join(C(P-complex(w/2-w*x/W,h/2-h*y/H))for x in range(W))for y in range(H));M,m=0,len(K);R();M=max(M,m);S.write(R());S.flush();Z,n=Z*F,n+1;time.sleep(D)

## Multibrot Set

The Mandelbrot Set is typically defined as the set of all numbers $c \in \mathbb{C}$ for which — with $z_0 = 0$, $z_{n+1} = f_c(z_n)$ and $f_c(z) = z^2 + c$ — the limit $\lim\limits_{n \to \infty} z_n$ converges. Visualizations of this standard Mandelbrot Set can be seen in three of my posts (Mandelbrot Set, Mandelbrot Set Miscalculations and Mandelbrot Set II).

However, one can extend the fractal’s definition beyond only having the exponent $2$ in the function to be $f_c(z)=z^\text{exp}+c$ with $\text{exp} \in \mathbb{R}$. The third post I mentioned actually has some generalization as it allows for $\text{exp} \in \{2,3,4,5\}$, although the approach used cannot be extended to real or even rational numbers.

The method I used in the aforementioned post consists of manually expanding $(a+b\cdot i)^n$ for each $n$. The polynomial $(a+b\cdot i)^3$, for example, would be expanded to $(a^3 - 3 \cdot a \cdot b^2) + (3 \cdot a^2 \cdot b - b^3) \cdot i$.
This method is not only tedious, error-prone and has to be done for every exponent (of which there are many), it also only works for whole-number exponents. To visualize real Multibrots, I had to come up with an algorithm for complex number exponentiation.

Luckily enough, there are two main ways to represent a complex number, Cartesian form $z = a+b\cdot i$ and polar form $z = k\cdot e^{\alpha\cdot i}$. Converting from Cartesian to polar form is simply done by finding the number’s vector’s magnitude $k = \sqrt{a^2+b^2}$ and its angle to the x-axis $\alpha = \mbox{atan2}(\frac{a}{b})$. (The function $\mbox{atan2}$ is used in favor of $\arctan$ to avoid having to divide by zero. View this Wikipedia article for more on the function and its definition.)
Once having converted the number to polar form, exponentiation becomes easy as $z^\text{exp} = (k \cdot e^{\alpha\cdot i})^\text{exp} = k^\text{exp} \cdot e^{\alpha \cdot \text{exp} \cdot i}$. With the exponentiated $z^\text{exp}$ in polar form, it can be converted back in Cartesian form with $z^\text{exp} = k^\text{exp} \cdot (\cos{(\alpha \cdot \text{exp})} + \sin{(\alpha \cdot \text{exp})} \cdot i \big)$.

Using this method, converting the complex number to perform exponentiation, I wrote a Java program which visualizes the Multibrot for a given range of exponents and a number of frames.
Additionally, I added a new strategy for coloring the Multibrot Set, which consists of choosing a few anchor colors and then linearly interpolating the red, green and blue values. The resulting images have a reproducible (in contrast to randomly choosing colors) and more interesting (in contrast to only varying brightness) look.

The family of Multibrot Sets can also be visualized as an animation, showing the fractal with an increasing exponent. The animated gif shown below was created using ImageMagick’s convert -delay <ms> *.png multibrot.gif command to stitch together the various .png files the Java application creates. To speed up the rendering, a separate thread is created for each frame, often resulting in 100% CPU-usage. (Be aware of this should you render your own Multibrot Sets!)

To use the program on your own, either copy the source code listed below or download the .java file. The sections to change parameters or the color palette are clearly highlighted using block comments (simply search for ‘/*’).
To compile and execute the Java application, run (on Linux or MacOS) the command javac multibrot.java; java -Xmx4096m multibrot in the source code’s directory (-Xmx4096m tag optional, though for many frames at high quality it may be necessary as it allows Java to use more memory).
If you are a sole Windows user, I recommend installing the Windows 10 Bash Shell.

// Java 1.8 Code
// Jonathan Frech, 11th of September 2016
//          edited 17th of April     2017
//          edited 18th of April     2017
//          edited 20th of April     2017
//          edited 21st of April     2017
//          edited 22nd of April     2017`