## Tau Day MMXVII

Today it is June the 28th which means that it is $\tau$ day!
The irrational and transcendental constant $\tau$ 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 $\tau$ 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 $(0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5)$.

Today’s $\tau$ 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 $\tau$ times.

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8$$q  The formula used for calculating $\tau$ 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 $\tau = 6.283185 \dots$, 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. 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 $\tau$ are The Tau Manifesto and 2 Pi or Not 2 Pi?. I wish everybody a happy $\tau$ day. // C Code // Jonathan Frech, 27th of June 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$$++,,''''......
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..................''''''''^^::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) ## A285494 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 *) Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&] 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]  ## 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 ## T-3PO — Tic-Tac-Toe Played Optimally 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), have this functionality. # Python 2.7.7 Code # Jonathan Frech, 31st of March 2017 # edited 1st of April 2017 ## 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. 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 ## Pi Day MMXVII Every year on March the 14th, for one day the world gets irrationally excited about the famous constant $\pi$. 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A fairly well-known way to approximate pi is to randomly choose points in a square (often thought of as throwing darts at a square piece of cardboard), determine their distance to a circle’s center and do a division, as I did in my π Generator post.

However, $\pi$ does not only appear in the formula for a circle’s area, $A=\pi \cdot r^2$, yet also in the formula for a sphere’s volume, $V=\frac{4}{3} \cdot \pi \cdot r^3$, and for all the infinite hyperspheres above dimension three (view this Wikipedia article for more about volumes of higher-dimensional spheres).

In particular, the formula for the hypervolume of a hypersphere in four dimensions is defined as being $V=\frac{\pi^2}{2} \cdot r^4$. Using this formula, my Python script randomly chooses four-dimensional points (each in the interval $\left[0, 1\right)$), calculates their distance to the point $\left(0.5, 0.5, 0.5, 0.5\right)$ and determines if they are in the hypersphere around that point with radius $0.5$.
By dividing the number of random points which lie in the hypersphere by the number of iterations used ($10^6$ in the example below), the script approximates the hypersphere’s hypervolume. By then rearranging the equation $V=\frac{\pi^2}{2} \cdot r^4$ with $r=0.5$ to $\pi=\sqrt{V\cdot 32}$, the desired constant can be approximated.

\$ python pi.py
3.14196371717

# Python 2.7.7 Code
# Jonathan Frech, 13th of March 2017

## A278328

The On-Line Encyclopedia of Integer Sequences (also known by its acronym, OEIS) is a database hosting hundreds of thousands of — as the name implies — integer sequences. Yet, despite the massive number of entries, I contributed a new integer sequence, A278328.

A278328 describes numbers whose absolute difference to their decimal reverse are square. An example would be $12$ or $21$ (both are the decimal reverse to each other), since $\left|12-21\right|=9$ and $9=3^2$.

Not a whole lot is known about the sequence, partly due to its definition only resulting in the sequence when using the decimal system, though it is known that there are infinitely many numbers with said property. Since there are infinitely many palindromes (numbers whose reverse is the number itself), $\left|n-n\right|=0$ and $0=0^2$.

Due to there — to my knowledge — not being a direct formula for those numbers, I wrote a Python script to generate them. On the sequence’s page, I posted a program which endlessly spews them out, though I later wrote a Python two-liner, which only calculates those members of the sequence in the range from $0$ to $98$ (shown below entered in a Python shell).

>>> import math
>>> filter(lambda n:math.sqrt(abs(n-int(str(n)[::-1])))%1 == 0, range(99))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 22, 23, 26, 32, 33, 34, 37, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 62, 65, 66, 67, 73, 76, 77, 78, 84, 87, 88, 89, 90, 95, 98]

## Maze Solving

Mazes have been a subject of human interest for thousands of years. The Greeks used them to trap a bull-man hybrid, the French built them to show how they could impose order on nature, and even nowadays people enjoy wandering around corn mazes.
The algorithmic art of using computers to solve mazes — and even to find the shortest path through a maze –, however, has only emerged in the last couple of decades.

I was inspired by a recent Computerphile video in which Michael Pound talks about implementing different path finding algorithms for use in maze solving. And as he used Python — one of my favourite languages out there –, I thought I could give it a try and came up with this maze solver.

The mazes given to the solver (through a .png file) have to have a specific form. The maze needs to have a border all around (painted black) with two holes at the top and bottom, marking the maze’s start and exit (all path pixels are white).
Then the solver — using PIL — reads in the maze file, determines start and exit and starts at the maze’s start, labelling each maze path according to its shortest distance to the start. After it has found the exit, it stops looking at the maze and traces its origins back from the exit, marking the path it goes along as the maze’s optimal solution (highlighted in red).
The different hues of blue indicate the tile’s distance to the start, the white tiles are tiles the solver did not even look at.
The different shadings also reveal information about the maze. Mazes with only one solution tend to have sharp changes as there are parts of the maze separated by only one wall, yet separated by a huge walk distance through the maze. The one maze with multiple solutions (see upper right image below) — in contrast — has only gradual changes in hue.

To solve a 4 megapixel maze, the solver takes around 3 seconds, for a 16 megapixel maze around 14 seconds and for a 225 megapixel maze around 7 minutes and 22 seconds.
Performance was measured on an Intel Core i7 (4.00 GHz).

All mazes shown were downloaded from Michael Pound’s mazesolving GitHub repository, which were mostly generated using Daedalus.

The solver’s source code is listed below, though you can also download the .py file.

# Python 2.7.7 Code
# Jonathan Frech, 25th of Feburary 2017
#          edited 26th of February 2017
#          edited 27th of February 2017
#          edited 22nd of March    2017
#          edited 29th of March    2017

## 4096

4096 is a Java-based clone of the well-known web and mobile game 2048, which itself clones 1024 and is similiar to THREES. The naming trend is quite obvious, though note that $2^{12}$ is a power of two where the exponent is divisible by three, futher connecting to the aforementioned game.

In the game, you are faced with a 4×4 matrix, containing powers of two. By swiping in the four cardinal directions (e.g. pressing the arrow keys), you shove all the non-empty cells to that side. When two equal powers of two collide, they fuse together, adding. Once you shoved, an empty tile pseudo-randomly transforms to either a two-tile (90%) or a four-tile (10%).
Your objective at first is to reach the tile 4096, though the real goal is to achieve the highest score. Your score is the sum of all the collisions you managed to cause.

To play 4096, you can either download the .jar file or review and compile the game for yourself, using the source code listed below.

#### Controls

• Up, down, left or right arrow key shoves the tiles
• Escape restarts the game upon a loss
• F11 toggles fullscreen

// Java 1.8 Code
// Jonathan Frech,  5th of December 2016
//          edited  6th of December 2016
//          edited  7th of December 2016
//          edited  8th of December 2016
//          edited  9th of December 2016
//          edited 19th of February 2017
//          edited 24th of February 2017
//          edited 28th of February 2017
//          * gave the 4096 tile a color
//          edited 22nd of April    2017
//          * fixed window positioning by changing
//            frame.setLocationRelativeTo(null); to
//            frame.setLocationByPlatform(true);`