Fractals related to the Mandelbrot set#
See also: http://www.juliasets.dk/Pictures_of_Julia_and_Mandelbrot_sets.htm
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
See also: http://www.dhushara.com/DarkHeart/quasif/quasi.htm (includes Kleinian fractals)
See also: https://en.wikipedia.org/wiki/Mandelbrot_set
Difference Method: https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/demm
MandelbrotSetExterior: https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/MandelbrotSetExterior#Real_Escape_Time
Fractals/fractalzoomer: https://en.wikibooks.org/wiki/Fractals/fractalzoomer
Mandelbrot set, example 1, classical set: \(z_{n+1} = z_n^2 + c\)#
Classical formula: \(z_{n+1} = z_n^2 + c\).
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D01a_Mandelbrot1.py, and the XML code in the file D01a_Mandelbrot1.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D01b_Mandelbrot1.py, and the XML code in the file D01b_Mandelbrot1.2D.xml.
Mandelbrot set, example 2: higher powers#
Recursion formula: \(\displaystyle z_{n+1} = z_n^5 + c\).
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D02a_Mandelbrot2.py, and the XML code in the file D02a_Mandelbrot2.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D02b_Mandelbrot2.py, and the XML code in the file D02b_Mandelbrot2.2D.xml.
See also: https://paulbourke.net/fractals/mandelpower/
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
Mandelbrot set, example 3: \(z_{n+1} = z_n^5 + \frac{c}{z_n}\)#
Recursion formula: \(\displaystyle z_{n+1} = z_n^5 + \frac{c}{z_n}\).
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D03a_Mandelbrot3.py, and the XML code in the file D03a_Mandelbrot3.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D03b_Mandelbrot3.py, and the XML code in the file D03b_Mandelbrot3.2D.xml.
Mandelbrot set, example 4: \(z_{n+1} = z_n^5 + \frac{c}{z_n^5}\)#
Recursion formula: \(\displaystyle z_{n+1} = z_n^5 + \frac{c}{z_n^5}\).
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D04a_Mandelbrot4.py, and the XML code in the file D04a_Mandelbrot4.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D04b_Mandelbrot4.py, and the XML code in the file D04b_Mandelbrot4.2D.xml.
Mandelbrot set, example 5, “Inverse Mandelbrot”#
Recursion formula: \(\displaystyle z_{n+1} = z_n^2 + \frac{1}{c}\).
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D05a_Mandelbrot5.py, and the XML code in the file D05a_Mandelbrot5.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D05b_Mandelbrot5.py, and the XML code in the file D05b_Mandelbrot5.2D.xml.
Mandelbrot set, example 6, “Marek Dragon Fractal”:#
Compare this with “Phoenix Julia fractals”
Recursion formula: \(\displaystyle z_{n+1} = \exp(2 \pi i r) z_n + z_n^2\) where r is an irrational number, in this example it is \(1 - (\sqrt{2} - \sqrt{3} + \sqrt{5} )/2\).
See also: https://paulbourke.net/fractals/marek/
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D06a_Mandelbrot6.py, and the XML code in the file D06a_Mandelbrot6.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D06b_Mandelbrot6.py, and the XML code in the file D06b_Mandelbrot6.2D.xml.
Mandelbrot set, example 7, “Burning Ship Fractal”#
Recursion formula: \(\displaystyle z_{n+1} = \left(|\Re(z_n)| + i|\Im(z_n)| \right)^2 + c\).
See also: http://www.3d-meier.de/, Mandelbrot und Julia Mengen
See also: https://paulbourke.net/fractals/burnship/
The Python code for the example below, without using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D07a_Mandelbrot7.py, and the XML code in the file D07a_Mandelbrot7.2D.xml.
The Python code for the example below, using numba, can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D07b_Mandelbrot7.py, and the XML code in the file D07b_Mandelbrot7.2D.xml.
