2017-06-28, post № 173

C, mathematics, programming, #constant

Today it is June the 28th which means that it is 𝜏 day!
The irrational and transcendental constant 𝜏 is what defines \pi=\frac{\tau}{2}, which obviously makes it an important constant. To celebrate this day, I created a C program which calculates 𝜏 by randomly creating 𝟫-dimensional points inside the 𝟫-dimensional hypercube and testing if they are inside the 𝟫-dimensional hypersphere with its center located at (0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5) [1].

Today’s 𝜏 time is 3:18:53 as \tau=6.2831853\dots. As one does not know if the time is specified as ante or post meridiem, there are actually two perfectly acceptable 𝜏 times.

 )$$$m             $$$$$$$$                    
+$$c              q$$$$$$$>                    
$$                $$$$$$$$                     
           $$$$$$$$;             $$~           
           $$$$$$$$             $$$            
          '$$$$$$$$C          "$$$             
           $$$$$$$$$w       '$$$$              

The formula used for calculating 𝜏 is derived from a 𝟫-dimensional hypersphere’s hypervolume formula V=\frac{32\cdot\pi^4}{945}\cdot R^9 (see this Wikipedia article).

    V = \frac{2^5 \cdot \tau^4 \cdot R^9}{945 \cdot 2^4} = \frac{2 \cdot R^9}{945} \cdot \tau^4\\
    \tau^4 = \frac{V \cdot 945}{2 \cdot R^9}; R = 0.5\\
    \tau^4 = \frac{V \cdot 945 \cdot 2^9}{2}\\
    \tau = \sqrt[4]{V \cdot 241920}

The constant gets calculated to \tau^*=6.293700. The real value is approximately 𝜏 = 𝟨.𝟤𝟪𝟥𝟣𝟪𝟧…, which makes the percent error

\left|\frac{\tau^*-\tau}{\tau}\right|=\left|\frac{6.293700-6.283185}{6.283185}\right|=0.001673514 = 0.167\%.

Thereby, this C program’s approximation is not too far off. [2] The source code is listed below and can also be downloaded here. Instructions on how to compile it using GCC can be seen below or in the source code.

gcc tau.c -o tau -lm; ./tau
tau = 6.293700

Resources worth checking out regarding 𝜏 are The Tau Manifesto and 2 Pi or Not 2 Pi? I wish everybody a happy 𝜏 day.

Source code: tau-day-mmxvii.c

Mandelbrot Set ASCII Viewer

2017-06-17, post № 172

ascii, HTML, JavaScript, mathematics, programming, Python, #animation, #fractal

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 Bourke.
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.


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.

mandelbrot-set-ascii-viewer.py; Python2, 593 bytes, 8 lines
# 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)


2017-06-03, post № 171

mathematics, OEIS, programming, Python, Wolfram Language, #decimal, #decimal sum, #digit sum, #distinct prime, #int, #integer, #number, #number theory, #prime, #prime factors, #primes

The On-Line Encyclopedia of Integer Sequences gets regularly updated with new integer sequences. One of the recent updates was contributed by me, A285494.

A285494 is the list of all numbers 𝑘 so that its digit sum equals its number of distinct prime factors.
A number’s digit sum is the sum of all of its decimal digits. The number 𝟨𝟤𝟪𝟥𝟣𝟪𝟧𝟥, for example, has a digit sum of 𝟨 + 𝟤 + 𝟪 + 𝟥 + 𝟣 + 𝟪 + 𝟧 + 𝟥 = 𝟥𝟨.
A number’s number of distinct prime factors is the number of different prime numbers that multiply together to result in the original number. As an example, 62831853=3^2\cdot 7\cdot 127\cdot 7853, so it has five prime factors of which four are distinct.
Thereby one can conclude that 𝟨𝟤𝟪𝟥𝟣𝟪𝟧𝟥 is not an entry in this sequence, as 36\neq 4.

The sequence is certainly infinite, as the number k=2\cdot 10^n with n\in \mathbb{N}^* has a digit sum of 2+(0\cdot n)=2 and — because k=2^{n+1}\cdot 5^n — exactly two distinct prime factors.

In the encyclopedia entry, I provided a Mathematica one-liner to compute the first few entries of this sequence. Since then, I have also written a Python two-liner to achieve the same goal.

(* Mathematica *)
Out = {20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010}
# Python 2.7
>>> def p(n):exec"i,f=2,set()\nwhile n>1:\n\tif n%i==0:f.add(i);n/=i;i=1\n\ti+=1";return len(f)
>>> print filter(lambda n:p(n)==sum(map(int,str(n))),range(2,10001))
[20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010]


2017-05-20, post № 170

games, Java, programming, #2016, #classic, #clone, #mine, #Minesweeper, #pixel, #sweeping, #Windows

Adding to my collection of clones of popular, well-known games, I created back in November of 2016 a Java-implementation of the all-time Windows classic game, Minesweeper.

Minesweeper was pre-installed on every installation of Windows up to and including Windows 7 and has been ported to a variety of different systems. Because of this, nearly everyone has at least once in their life played Minesweeper or at least heard of it.
In Minesweeper you are presented with a square grid of covered tiles containing either numbers or mines. Your task is it to uncover all tiles which are not mines in the least amount of time. When you uncover a mine, it explodes and the game is lost. To aid in figuring out which tiles are mines and which are not, every tile that is not a mine tells you how many mines are in the neighbouring eight tiles. Tiles which have no neighbouring mines are drawn gray and uncover neighbouring non-mine tiles once uncovered.
More on Minesweeper can be found in this Wikipedia article — I am linking to the German version, as the current English version has major flaws and lacks crucial information. If you are so inclined, feel free to fix the English Minesweeper Wikipedia article.

In my clone, there are three pre-defined difficulty levels, directly ported from the original Minesweeper game, and an option to freely adjust the board’s width and height as well as the number of bombs which will be placed. Gameplay is nearly identical to the original, as my clone also uses a square grid and the tile’s numbers correspond to the number of bombs in the eight tiles surrounding that tile.
The game has a purposefully chosen pixel-look using a self-made font to go along with the pixel-style.


  • Arrow keys and enter to navigate the main menu,
  • Arrow keys or mouse movement to select tiles,
  • ‘Space’, enter or left-click to expose a tile,
  • ‘f’ or right-click to flag a tile,
  • ‘r’ to restart game when game is either won or lost,
  • ‘Escape’ to return to the main menu when game is either won or lost,
  • ‘F11’ toggles fullscreen.

To play the game, you can either download the .jar file or compile the source code for yourself. The source code is listed below and can be downloaded as a .java file.

Source code: Main.java

Pinhole Photographs MMXVII

2017-05-06, post № 169

art, #light, #nature, #photography, #picture, #tulip, #World-Wide Pinhole Day, #WWPD

Purple Tulip
Crimson Tulip
Light Interference

Multibrot Set

2017-04-22, post № 168

Java, mathematics, programming, #animated gif, #animation, #Cartesian, #complex, #complex arithmetic, #fractal, #generalization, #gif, #Mandelbrot set, #multi-threading, #polar, #reals, #threading

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} [1]. 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 [2] to use more memory).
If you are a sole Windows user, I recommend installing the Windows 10 Bash Shell.

Source code: multibrot.java


2017-04-16, post № 167

art, ascii, haiku, poetry, #ascii egg, #celebration, #easter egg, #egg

Winds swirl through the air,
Water sloshes at the shore;
A peaceful island.


T-3PO — Tic-Tac-Toe Played Optimally

2017-04-08, post № 166

games, HTML, JavaScript, mathematics, programming, Python, #computer player, #game ai, #perfect, #perfect play, #three in a row, #three to win, #three

Tic-Tac-Toe, noughts and crosses, Xs and Os, three in a row or whatever you want to call it may be the simplest perfect information game that is enjoyable by humans. Two players set their pieces (X or O) on an 𝟥 ⨉ 𝟥 grid, alternating their turns. The first player to get three of their pieces in a line, wins. If no player succeeds to get a line, the game ends in a draw.

Tic-Tac-Toe’s simplicity may become clear, if you consider that skilled players — people who have played a few rounds — can reliably achieve a draw, thereby playing perfectly. Two perfect players playing Tic-Tac-Toe will — whoever starts — always tie, so one may call the game virtually pointless, due to there practically never being a winner.
Because of its simple rules and short maximal number of turns (nine) it is also a game that can be solved by a computer using brute-force and trees.

The first Tic-Tac-Toe-playing program I wrote is a Python shell script. It lets you, the human player, make the first move and then calculates the best possible move for itself, leading to it never loosing. On its way it has a little chat whilst pretending to think about its next move. The Python source code can be seen below or downloaded here.

The second Tic-Tac-Toe-playing program I wrote uses the exact same method of optimizing its play, though it lets you decide who should begin and is entirely written in JavaScript. You can play against it by following this link.

Both programs look at the entire space of possible games based on the current board’s status, assumes you want to win and randomly picks between the moves that either lead to a win for the computer or to a draw. I did not include random mistakes to give the human player any chance of winning against the computer. Other Tic-Tac-Toe-playing computers, such as Google’s (just google the game [1]), have this functionality.

Source code: t-3po.py
Jonathan Frech's blog; built 2022/01/16 06:13:11 CET