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Orange

2016-12-17, post № 153

art, haiku, poetry, #color, #falling, #falling up, #fruit, #gif, #screw gravity

Orange on the floor,
Grows a tall tree beside it.
Falls up to its home.

orange.gif

Advent III

2016-12-11, post № 152

art, haiku, poetry, #Christmas, #fir, #tree

As a fir cone born.
As a goal to reach the sky.
As destiny death.

Advent II

2016-12-04, post № 151

art, haiku, poetry, #ice

Soft ice from above
Falls upon the barren land.
Coats the world in white.

Mandelbrot set miscalculations

2016-12-03, post № 150

Java, mathematics, programming, Python, #assignment, #complex numbers

While developing a Java program to create an image of the Mandelbrot set, I stumbled upon a small error which completely changes the set’s look. To fix this bug, you need to swap two lines of code.

The bug arises when trying to convert convenient Python features to Java.
To iteratively apply the function z\mapsto z^2+c, you update your complex number 𝑧 a certain amount of times. When you are not using a complex number class, but instead you use two floating point numbers (in Java doubles to gain precision) a and b to define the real and imaginary part (z=\texttt{a}+\texttt{b}\cdot i), logically both numbers need to be updated.
In Python you may write the following, when 𝑐 is defined as being a complex number with parts c and d (c=\texttt{c}+\texttt{d}).

a, b = a**2 - b**2 + c, 2 * a * b + d

Which seems to be very similar to those two lines.

a = a**2 - b**2 + c
b = 2 * a * b + d

But notice that in the first code snippet you define a tuple consisting of the real and imaginary part and then assign it to the variables. The first snippet really looks like this.

t = (a**2 - b**2 + c, 2 * a * b + d)
a, b = t

Using this assignment of variables, which corresponds to z\mapsto z^2+c, you get an image of the correct Mandelbrot set.

mandelbrot-set-miscalculations_mandelbrot-set-correct.png

In contrary, the second code snippet assigns the old a its new value, then uses this new a to define the value of new b, thus does not calculate z\mapsto z^2+c, which is equivalent to z\mapsto (\texttt{a}^2-\texttt{b}^2+\texttt{c})+(2\cdot\texttt{a}\cdot\texttt{b}+\texttt{d})\cdot i, but rather z\mapsto (\texttt{a}^2-\texttt{b}^2+\texttt{c})+(2\cdot\texttt{a}^2\cdot\texttt{b}-2\cdot\texttt{b}^3+2\cdot\texttt{b}\cdot\texttt{c}+\texttt{d})\cdot i.

In Java it would look like this.

a = a*a - b*b + c;
b = 2 * a * b + d;

Which results in this rather unusual depiction of the famous fractal.

mandelbrot-set-miscalculations_mandelbrot-set-miscalculation.png

You can easily avoid this bug when using two sets of variables to define old 𝑧 and new 𝑧, as shown in the following.

_a = a*a - b*b + c;
_b = 2 * a * b + d;
a = _a;
b = _b;

Or you can define variables \texttt{asqr}=\texttt{a}^2 and \texttt{bsqr}=\texttt{b}^2 and swap the assignment. Using variables for the squares of the parts of 𝑧 also helps to improve performance [1].

b = 2 * a * b + d;
a = asqr - bsqr + c;
asqr = a*a;
bsqr = b*b;
Source code: mandel.java

MMXVI

2016-12-01, post № 149

mathematics, #all days of December

The idea is to only use the year’s digits — preferably in order — and mathematical symbols (+,-,\cdot,\sqrt{},\lfloor\rfloor,\lceil\rceil,\dots) to create an equation that evaluates to a specific day of the month.
The 0th of December, 2016 would, for example, be 2\cdot 0\cdot 1\cdot 6, 2^0-1^6 or \lfloor\frac{2}{0+16}\rfloor.

Advent I

2016-11-27, post № 148

art, haiku, poetry, #winter

First candle is lit,
Advent season is now here.
Charming wintertime.

Praiku

2016-11-19, post № 147

art, haiku, mathematics, poetry, programming, Python, #prime

While you have no primes,
While you would like to know them.
If, if you could print…

brainfuck

2016-11-05, post № 146

brainfuck, programming, Python, #esoteric, #interpreter, #minimalism

Usually, programming languages are designed to be efficient, understandable and usable. Their concepts are well thought-out, giving the programmer powerful tools to create their application.
Esoteric programming languages have a slightly different aim. Instead of focusing on the product, they focus on the programmer’s journey and try new approaches to building a program.
One well-known esoteric programming language is Urban Müller’s brainfuck. It is a Turing-complete programming language — meaning that it could with infinite memory do what a Turing machine can do —, which practices extreme minimalism. The language knows of only eight characters (<>+-[].,).
The language’s usage is very similar to a Turing machine. Starting the program, the pointer is at cell zero and can be moved left (<) or right (>) by one cell.
The cells’ values are all starting at 𝟢, but can be either increased (+) or decreased (-) by one. Because the cells can only store one unsigned byte, adding one to 𝟤𝟧𝟧 yields in 𝟢 and subtracting one from 𝟢 yields in 𝟤𝟧𝟧.
Also, a loop is possible by using square brackets. An open square bracket ([) starts a loop and a closed square bracket (]) ends a loop. When the end of a loop is reached, the interpreter will jump back to its start if and only if the currently selected cell’s value is 𝟢. [1]
The only way to communicate with the user is to print the currently selected cell’s ASCII character to the screen (.) and get a user input which will be stored in the currently selected cell as its ASCII value (,).

Because of its minimalistic design, writing an interpreter is not particularly hard. Listed below is the Python code of my interpreter, which I used to execute my own brainfuck “Hello World.” program (𝟣𝟣𝟪 characters).
Online interpreters include for example bf.doleczek.pl and sange.fi.

Useful Wikipedia articles include Esoteric Programming Language, Brainfuck and Turing Machine.

$ python brainfuck.py
Hello World.
Source code: brainfuck.py
Jonathan Frech's blog; built 2021/10/02 17:36:09 CEST