Fractals/The Newton-Raphson fractal

   "The "flowers" are beautiful to behold but totally abhorrent from the numerical point of view." Ramillies[1]

The newton fractal is calculated by iterating the Newton-Raphson root finding method,[2] and coloring the sample point in various ways based on various properties of how the point is attracted to the various 'roots' or 'solutions' to the method. For example how fast or at what angle the orbit is attracted.

Newton Family of fractals edit

An example showing the PhoenixDoubleNova Fractal, in a complex composite by Dan Wills

There has been much development using this method to develop fractal formulae in the UltraFractal formula database, and in other fractal-rendering software. One notable family of newton style fractals is the Nova fractal family, with formulae that combine aspects of other well known fractals such as Mandelbrot, Phoenix and Halley's fractal - which is another fractal that is similar to the newton in that it implements a root finding method known as Halley's method.

An entire family of fractal formulae have been derived this way and given names that reflect their parent formulae, for example:

  • Nova Julia
  • Nova Mandelbrot
  • Phoenix Nova
  • Halley Nova
  • Phoenix Double Nova
  • Phoenix Halley Nova

As the development of these fractal formula authors has been quite prolific, there are a great many variations and combinations.

Links edit

  1. math.stackexchange question : why-does-the-newton-raphson-method-not-converge-for-some-functions
  2. Newton’s Method as a Dynamical System by Johannes Ruckert