This article assumes you've read the Julia
set article under real and complex fractals. It includes the
algorithm for Julia sets using transcendental functions which is also
repeated here This article fills in some details about Julia sets with
functions other than z^2 as their basic function. The fractal on the
left, above uses the complex cosine function as its basic function.
The algorithm for this type of fractal is given below. The parameter
for the cosine fractal shown above is **0.38**+**1.8***i*.
It's view window is the 6x6 square with (0,0) at its center. The
complex exponential also works in the algorithm below with the escape
test |z|<12.

## Julia set membership algorithm for cos(z). |

Choose the parameter mu=a+b |

The polynomial fractal above has
mu=**0.635**+**0.8418***i* and is based on the function
f(z)=z^4+mu. The algorithm for such fractals is given below. Notice
that the escape condition |z|<2 is the same as for the quadratic Julia
sets. If you use a more complex polynomial then the escape condition
becomes |z-w|<2 where w is some constant you need to discover
experimentally. For f(z)=z(1-z) for example it's about
w=**0.5**+**0***i*.

## Polynomial Julia set membership algorithm. |

Choose the parameter mu=a+b |

One thing you will notice about the Julia sets based on f(x)z^n+mu is that they have a overall n-fold symmetry. Examine the two fractals below which are based on f(z)=z^3 and f(z)=z^5. The three-fold and five-fold symmetries are easy to see. The article on how Mandelbrot sets index Julia sets is a good one to read for planning these sort of Julia sets.