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).

f_c(z)=z^2+cHowever, 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.

f_c(z)=z^3+cThe 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.

f_c(z)=z^4+cLuckily 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).

f_c(z)=z^5+cUsing 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.

f_c(z)=z^6+cThe 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!)

f_c(z)=z^10+cTo 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.

Multibrot animation (probably loading...)


// 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

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JClock VIII

Interpreting the hour hand on a clock as a two-dimensional object on a plane, the hand’s tip can be seen as a complex number.
This clock converts the hour hand’s position into a complex number, sets the number’s length to the current minutes and displays it in the form a + b \cdot i.
The angle \phi is determined by the hours passed (\frac{2 \cdot \pi \cdot \text{hour}}{12} = \frac{\pi \cdot \text{hour}}{6}) but has to be slightly modified because a complex number starts at the horizontal axis and turns anti-clockwise whilst an hour hand starts at the vertical axis and turns — as the name implies — clockwise.
Thus \phi = (2 \cdot \pi - \frac{\pi \cdot \text{hour}}{6}) + \frac{\pi}{2} = (\frac{15 - \text{hour}}{6}) \cdot \pi.
The complex number’s length is simply determined by the minutes passed. Because the length must not be equal to 0, I simply add 1. |z| = k = \text{minute} + 1.
Lastly, to convert a complex number in the form k \cdot e^{\phi \cdot i} into the form a + b \cdot i, I use the formula k \cdot (\cos{\phi} + \sin{\phi} \cdot i) = a + b \cdot i.

17:04


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 29th of July, 2016

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Mandelbrot Set

Working with complex numbers and iterating the function f_c(z) = z^2 + c, I created this program which lets you explore the depth of this self-repeating fractal. The different colors are caused by different color schemes.

Controls

  • Left mouse click and mouse movements control the zoom frame
  • Middle mouse click zooms in
  • Space takes a screenshot
  • F1 moves a zoom back
  • F2 zooms to default zoom

Mandelbrot set from afar [-6.75, -4.5] to [6.75, 4.5] An eye in the Mandelbrot set [-0.17466172461436666, -1.0720840874732396] to [-0.1746617232046618, -1.0720840865334365] A gap in the Mandelbrot set [-0.16101577503429365, -1.0369693644261544] to [-0.16019958847736634, -1.0364252400548697]Another gap in the Mandelbrot set [0.25, -0.04999999999999982] to [0.40000000000000036, 0.04999999999999982]A spiral in the Mandelbrot set [0.18276898469650216, -0.5821251321730682] to [0.18311316336591232, -0.5818956797267948]


# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech  4th of December, 2015
#         edited  5th of December, 2015
#         edited 11th of December, 2015

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