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.
However, one can extend the fractal’s definition beyond only having the exponent 𝟤 in the function to be with . The third post I mentioned actually has some generalization as it allows for , 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 for each 𝑛. The polynomial , for example, would be expanded to . 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 and polar form . Converting from Cartesian to polar form is simply done by finding the number’s vector’s magnitude and its angle to the 𝑥-axis . (The function is used in favor of 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
With the exponentiated in polar form, it can be converted back in Cartesian form with
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.