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The Mandelbrot set M is defined by a family of complex quadratic polynomials given by:

Pc: z -> z^2 + c,

where c is a complex parameter.

A mathematician's depiction of the Mandelbrot set M, a point c is coloured black if it belongs to the set, and white if not.

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by colouring all the points c which belong to M black, and all other points white. The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials Pc(z), that is, it is the subset of the complex plane consisting of those parameters c for which the Julia set of Pc is connected.

The more colourful pictures usually seen are generated by colouring points not in the set according to how quickly or slowly the sequence diverges to infinity



In the Mandelbrot Blue Series 1 I Have Tried To Show The Self-Similarity a Fractal Displays Under Magnification:

Blue Mandelbrot Series 1 Part 1 : [link]
Blue Mandelbrot Series 1 Part 2 : [link]
Blue Mandelbrot Series 1 Part 3 : [link]
Blue Mandelbrot Series 1 Part 4 : [link]

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