Cool Stuff: Fractals Alt.Fractals paperback - out January 2011



Alternative Julia-type Sets 

1: Miscellaneous Trigonometric Functions

Having had a look at the fun that can be had with fractals based on z --> z2 + c , and z --> zn + c, we might want to start branching out and trying other types of equation. What happens when we start playing with functions borrowed from trigonometry?

The two obvious functions to investigate are SIN and TAN. When we compare the lengths of a right-angled triangle, the tangent function tells us the ratio between the lengths of the two perpendicular sides (as a function of angle), and the sine and cosine functions tell us the corresponding ratios between one of these two sides and the hypotenuse. 

Although there's not an obvious link between these functions and the usual Julia Set formula, the trig functions are linked by Pythagoras' theorem ("square of the hypotenuse equals the sum of the sqaures of the other two sides"), which again gives us "parameter-squared" properties.

A couple of "trig" fractals

The obvious immediate difference between getting these parameter-squared relationships by writing z^2, or by using a trig function, is that the trig functions cycle and repeat along the x axis, so our resulting fractals will tend to form a strip that repeats forever unless some sort of additional range-limiting factor kicks in.  

With some of these fractals, we find the the intensity of detail is a little overwhelming, since each point accepts contributions from along the entire (infinite) strip. We can winnow down some of this detail by adding a fifth parameter, an "escape function" that tells the calculation to stop when the coordinates jump too far out of our selected range. This fifth parameter crops the image, but also reduces some of the complexity.  

More "trig" fractals

"Cellular" functions

With a bit of trig abuse, we can also generate some interesting "cellular" fractals:

A few "cellular worlds" fractals
... zooming in on image #3 ...
Another "cell" fractal: