Alternative
Juliatype Sets
1: Miscellaneous Trigonometric Functions
Having had a look at the fun that can be had with
fractals based on z > z^{2 }+ c
, and z
> z^{n} + 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 rightangled 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 "parametersquared"
properties.
A couple of "trig" fractals 

The obvious immediate difference between getting
these parametersquared 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
rangelimiting 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.
"Cellular" functions
With a bit of trig abuse, we can also generate
some
interesting "cellular" fractals:
A few "cellular worlds" fractals 

Copyright ©
Eric Baird 2009
