The Bodaciously Excellent Blog of Doctorwhen

Mandelbrot & Julia Sets

The Mandelbrot Set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map

{\displaystyle z_{n+1}=z_{n}^{2}+c}

remains bounded. That is, a complex number c is part of the Mandelbrot Set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.

Images of the Mandelbrot Set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot Set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization and mathematical beauty.

Each point in the Mandelbrot Set defines a corresponding Julia Set.

The images in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.