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This page acts as a gateway to the other Mandelbrot pages on this site.
Interactive WebGL Version This version will allow you to zoom in and out and pan around the fractal interactively viewing any of the 6 orientations described on the 4D geometry page. It requires that you have a WebGL enabled browser. It also allows you to generate the equavalent fractals for equations of the form z → zn for 'n' values up to 9. Complexity A discussion of how the relatively simple equation generates these seeminly infinitely complex images. Colouring Algorithms The WebGL and python versions both support 5 different schemes for deciding what colour to shade a pixel. These are described on this page along with a sixth, Python only variant, called "Quadrants". Some images A collection of images generated over the years with various applications. 4-Dimensional Geometry To generate the Mandelrot sets you iterate the equation z -> z2 + c, where 'z' and 'c' are complex numbers. 'z' starts at 0 + 0i and 'c' is allowed to vary over the plane. For Julia sets a fixed 'c' value is chosen and z0 is allowed to vary. All in all there are 4 independent variables. 4-Dimensional Example Images Some examples of 2-dimensional slices through the 4-dimensional object described on the geometry page. Mandelbrots inside the Newton Raphson Fractal The Newton Raphson iteration can be modified using a damping factor I call 'alpha', which is another complex number. If we switch from keeping alpha constant and varying the seed to keeping the seed constant and varying alpha we can produce images that contain mandelbrot like regions. Python Script Describes how to use the python script to produce mandelbrot images. Includes lots of examples. This was written a long time ago (2008) and may not be compatible with the latest versions of Python. I no longer have Python installed to fix this. If you just want to create Mandelbrot images and you have a WebGL compatible system you can use the WebGL version (above) to create images interactively. Some Animations These were created using the Python scripts. A Large Example Image Generated uing the python scripts
Interactive WebGL Version
This version will allow you to zoom in and out and pan around the fractal interactively viewing any of the 6 orientations described on the 4D geometry page. It requires that you have a WebGL enabled browser.
It also allows you to generate the equavalent fractals for equations of the form z → zn for 'n' values up to 9.
Complexity
A discussion of how the relatively simple equation generates these seeminly infinitely complex images.
Colouring Algorithms
The WebGL and python versions both support 5 different schemes for deciding what colour to shade a pixel. These are described on this page along with a sixth, Python only variant, called "Quadrants".
Some images
A collection of images generated over the years with various applications.
4-Dimensional Geometry
To generate the Mandelrot sets you iterate the equation z -> z2 + c, where 'z' and 'c' are complex numbers. 'z' starts at 0 + 0i and 'c' is allowed to vary over the plane. For Julia sets a fixed 'c' value is chosen and z0 is allowed to vary. All in all there are 4 independent variables.
4-Dimensional Example Images
Some examples of 2-dimensional slices through the 4-dimensional object described on the geometry page.
Mandelbrots inside the Newton Raphson Fractal
The Newton Raphson iteration can be modified using a damping factor I call 'alpha', which is another complex number. If we switch from keeping alpha constant and varying the seed to keeping the seed constant and varying alpha we can produce images that contain mandelbrot like regions.
Python Script
Describes how to use the python script to produce mandelbrot images. Includes lots of examples. This was written a long time ago (2008) and may not be compatible with the latest versions of Python. I no longer have Python installed to fix this. If you just want to create Mandelbrot images and you have a WebGL compatible system you can use the WebGL version (above) to create images interactively.
Some Animations
These were created using the Python scripts.
A Large Example Image
Generated uing the python scripts
(c) John Whitehouse 2010 - 2020