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Mandelbrot Set II

2017-01-14, post № 157

Java, mathematics, programming, #compiled, #filled Julia set, #generalized filled Julia set, #generalized Mandelbrot set, #Julia, #speed increase, #viewer

Over a year ago, I published my first Mandelbrot Set viewer, a Python program using Pygame. Since then, I have published a rather short program highlighting errors that can occur when calculating the set (Mandelbrot Set Miscalculations).
Since my first viewer was written in Python, which is an interpreted programming language, and I wanted to make my viewer faster, I decided to write one in Java. I was hoping for a speed increase since Java is compiled and thus should run at higher speeds. I was not disappointed. The new Java-based viewer runs noticeably faster and additionally I added a lot of new features, all listed below.

Controls

  • Left-clicking and dragging draws a zoom frame, single left-clicking removes the frame,
  • Right-clicking (and optionally dragging) moves the zoom frame,
  • ‘Space’ zooms into the zoom frame,
  • ‘F1’ moves one step back the zoom history,
  • ‘F2’ shows the path a complex number at the cursor’s position follows when the function is iteratively applied,
  • ‘F3’ shows the \mathbb{R} and i\,\mathbb{R} axis,
  • ‘F4’ displays the current cursor’s corresponding complex number,
  • ‘F5’ toggles between showing and hiding the menu (text in the left upper corner describing the viewer’s functions and current states),
  • ‘F6’ increments the exponent (going from f_c(z)=z^2+c to f_c(z)=z^5+c in whole-number steps),
  • ‘F7’ toggles between the Mandelbrot set and the filled Julia set,
  • ‘F8’ toggles between previewing a small filled Julia set at the cursor’s position based upon the cursor’s complex number,
  • ‘F9’ completely resets the zoom and zoom history,
  • ‘F11’ (or ‘F’) toggles between fullscreen and windowed mode,
  • ‘F12’ quits the application,
  • ‘L’ increases the color depth (starting at 𝟤𝟧𝟨 and increasing in steps of 𝟤𝟧𝟨),
  • ‘Q’ saves the current image to disk.

To use this application, you can either trust me and download the .jar-file or view the source code listed below, verify it and compile the program yourself.
The program will start in fullscreen mode, to change to windowed mode, just press ‘F11’ (as listed above).

mandelbrot-set-ii-0_mandelbrot_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png
mandelbrot-set-ii-1_julia_-0.5629629629629629+0.47777777777777786i_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png
mandelbrot-set-ii-2_mandelbrot_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png
mandelbrot-set-ii-3_mandelbrot_tl0.732913241572962-0.151610056431476i_br0.7334855905959307-0.15128791342783118i.png
mandelbrot-set-ii-4_julia_0.7332134267115711+0.1514433175990339i_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png
Source code: Main.java
Extra assets: mandelbrot-set-ii-5_mandelbrot_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png, mandelbrot-set-ii_extra_00.png, mandelbrot-set-ii_extra_01.png, mandelbrot-set-ii_extra_02.png, mandelbrot-set-ii_extra_03.png, mandelbrot-set-ii_extra_04.png, mandelbrot-set-ii_extra_05.png, mandelbrot-set-ii_extra_05_a-half.png, mandelbrot-set-ii_extra_06.png, mandelbrot-set-ii_extra_06_a-half.png, mandelbrot-set-ii_extra_07.png, mandelbrot-set-ii_extra_07_a-half.png, mandelbrot-set-ii_extra_08.png, mandelbrot-set-ii_extra_09.png, mandelbrot-set-ii_extra_10.png, mandelbrot-set-ii_extra_11_Fractal-at-25fe8ea8.png, mandelbrot-set-ii_extra_12_mandelbrot_tl-1.8-1.2i_br1.8+1.2i.png, mandelbrot-set-ii_extra_13_mandelbrot_tl-1.8-1.2i_br_1.8+1.2i.png, mandelbrot-set-ii_extra_14_julia_c-0.4800000000000002+0.6366666666666666i_tl-1.8-1.2i_br1.8+1.2i.png, mandelbrot-set-ii_extra_15_mandelbrot_tl0.05333333333333323-0.6333333333333333i_br0.3599999999999999-0.4622222222222222i.png, mandelbrot-set-ii_extra_21_julia_c-0.5266666666666668+0.4033333333333333i_tl-1.8-1.2i_br1.8+1.2i.png, mandelbrot-set-ii_extra_22_julia_c-0.5466666666666666+0.2844444444444445i_tl-2.1333333333333333-1.2i_br2.1333333333333333+1.2i.png, mandelbrot-set-ii_extra_23_julia_c-0.5466666666666666+0.2844444444444445i_tl0.8800000000000003-0.046666666666666634i_br1.1911111111111112+0.1266666666666667i.png, mandelbrot-set-ii_extra_24_mandelbrot_tl-0.2174531103962005-0.7228248243121531i_br-0.21745053052484012-0.7228231210961094i.png, mandelbrot-set-ii_extra_25_mandelbrot_tl-0.21745176527993842-0.7228241924663124i_br-0.21745173948122481-0.7228241754341519i.png, mandelbrot-set-ii_extra_26_mandelbrot_tl-3.5555555555555554-2.0i_br3.5555555555555554+2.0i.png, mandelbrot-set-ii_extra_27.png, mandelbrot-set-ii_extra_28.png

New Year

2016-12-31, post № 156

art, haiku, poetry, #2017, #Silvester

Aiming at the sky.
Stunningly empty vastness …
Sound of explosion.

Christmas

2016-12-24, post № 155

art, haiku, poetry, programming, Python, #ascii, #ascii art, #star tree, #tree, #xmas

Christmas tree gets chopped,
Excitement fills the people.
Forming winter mood.

         *         
        ***        
       *****       
      *******      
     *********     
    ***********    
   *************   
  ***************  
 ***************** 
       *****       

Advent IV

2016-12-18, post № 154

art, haiku, poetry, #nature, #sleep

The lakes are frozen.
All world seems barren and still.
Nature is asleep.

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
Jonathan Frech's blog; built 2022/01/16 06:13:11 CET