Animating the Quantum Drunkard’s Walk

A recent PPCG challenge titled The Quantum Drunkard’s Walk was about a tiny drunken person for which quantum mechanics apply and who — being drunk — will randomly walk in one of the four cardinal directions each step they take.
As they experience the strange world of quanta, they can simultaneously take every path at once and their paths influence each other. The challenge then asked to output all possible positions the quantum drunkard could occupy, all paths they could take in ASCII representation.

The question also states this problem’s equivalence to a cellular automaton, when one removes the story boilerplate.
Cells in this cellular automaton are square and can occupy one of three states: empty, awake or sleeping. Each iteration, all cells change according to three rules.

  • An empty cell wakes up iff there is exactly one awake cell amongst its cardinal neighbours, else it stays empty.
  • An awake cell goes to sleep.
  • A sleeping cell continues to sleep.

Being code golf, the aim was to come up with a minimally sized source code; my attempt required 214 bytes and prints a nested array containing one-length strings (characters), as this output method is cheaper than concatenating all characters to one string.

python quantum.py -rmi 200
python quantum.py -rmi 200

However, one user took the challenge idea a bit further and created an animated gif showing the walk’s, or cellular automaton’s, progression over time with an increasing number of iterations. My Python program shown in this post does exactly that, generating an animated gif showing the automaton’s progression. I even implemented rainbow support, possibly improving the automaton’s visual appearance.
Python source code can be downloaded and seen below.

I use the Python Imaging Library to produce all frames and use a shell call to let ImageMagick convert all frames to an animated gif. Animation parameters are taken via shell arguments, a full list of features follows (also available via the -h flag).

  • --iterations N Number of iterations (initial frame not counted)
  • --colorempty C Empty cell color (#rrggbb)
  • --colorawake C Awake cell color (#rrggbb)
  • --colorsleeping C Sleeping cell color (#rrggbb)
  • --rainbow Use rainbow colors (overrides color settings)
  • --scale N Cell square pixel size
  • --convert Execute ImageMagick’s convert command
  • --delay N Delay between frames in output image.
  • --loop N Gif loops (0 means for indefinite looping)
  • --keepfiles Do not delete files when converting
python quantum.py -s 25 -md 50
python quantum.py -s 25 -md 50

# Python 2.7 code; Jonathan Frech; 1st of December 2017

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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$$++,,````''''......                      
................''''''''``^^^^""::$$$$$$::""^^``''''......                    
..............''''''````""{{;;XX$$$$$$$$uuUU,,,,""''......                    
............''''``````^^,,rr$$$$$$$$$$$$$$$$<<$$--``........                  
........''``````````^^""LL$$$$$$$$$$$$$$$$$$$$__""``''......                  
..''''''^^!!"""",,""""::__$$$$$$$$$$$$$$$$$$$$$$ll""''........                
''''````^^::__IIYYii::ll$$$$$$$$$$$$$$$$$$$$$$$$pp^^''........                
''``````"";;[[$$$$$$++__$$$$$$$$$$$$$$$$$$$$$$$$$$^^''''......                
``^^^^,,;;>>$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ww``''''......                
"",,,,II$$nn$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$""``''''......                
"",,,,II$$nn$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$""``''''......                
``^^^^,,;;>>$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ww``''''......                
''``````"";;[[$$$$$$++__$$$$$$$$$$$$$$$$$$$$$$$$$$^^''''......                
''''````^^::__IIYYii::ll$$$$$$$$$$$$$$$$$$$$$$$$pp^^''........                
..''''''^^!!"""",,""""::__$$$$$$$$$$$$$$$$$$$$$$ll""''........                
........''``````````^^""LL$$$$$$$$$$$$$$$$$$$$__""``''......                  
............''''``````^^,,rr$$$$$$$$$$$$$$$$<<$$--``........                  
..............''''''````""{{;;XX$$$$$$$$uuUU,,,,""''......                    
................''''''''``^^^^""::$$$$$$::""^^``''''......                    
  ..................''''''''``^^::ww$$++,,````''''......                      
    ..................''''''''``^^,,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@$"
import os,time,sys;H,W,S,n=map(int,os.popen("stty size").read().split())+[sys.stdout,0];W/=2
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).

f_c(z)=z^2+cHowever, 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.

f_c(z)=z^3+cThe 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.

f_c(z)=z^4+cLuckily 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).

f_c(z)=z^5+cUsing 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.

f_c(z)=z^6+cThe 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!)

f_c(z)=z^10+cTo 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.

Multibrot animation (probably loading...)


// 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

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