Although small and tucked away in the corner, Pattern is the single
most important control. It selects the basic way that Polynomial and
Parameter are used to form the fractal equation. There are 4 patterns
to choose from, but to explain all the details of how they work is
a little too much for the scope of this help page. For now just
the class names that commonly appear in fractal literature, Mandelbrot,
Julia and Newton are used to describe them here.
There are lots of good sources of information about fractals on the web.
Try these links for an introduction to fractal geometry and plotting
and some great samples images:
Each of the patterns has so many possible variations, nearly infinite
for all practical purposes, that it's difficult to say that any
particular image is normal or even typical for any one of them.
The PhotoSwizzle Fractal Gallery has several
samples of each and that's as good a place as any to start. The following
table lists which is which in the gallery.
Pattern Characteristics

Pattern 

1 Mandelbrot variation 1 
2 Mandelbrot variation 2 
3 Julia 
4 Newton 
Samples 
1,
2,
3,
4,
5

6,
7,
8

9,
10

11,
12,
13


Primary Equation 
Polynomial 
Yes 
No 
Yes 
Yes 
Parameter 
No 
No 
Yes 
No 

Initial Value 
Polynomial 
No 
No 
No 
No 
Parameter 
No 
Yes 
Yes 
No 

Hidden Function 
Polynomial 
No 
Yes 
No 
No 
Parameter 
Yes 
No 
No 
No 
One more thing you probably noticed already is that the table above has
more things in it besides the sample numbers. Here's what all of that is
about. Each pattern has a primary equation and an initial value. The
Polynomial and the
Parameter
are used in various combinations in the patterns for either the primary
equation, the intial value or both. The table lists which patterns use
which of these features.
Also for some patterns (currently 1 and 2) there are "hidden" functions
that add more variety to the way the image can be created.
A hidden function is a combination of the trigonometric functions sin
and cos that makes the initial values periodic over a range
significantly less than the image dimensions. This changes the rendered
image from a single large pattern to multiple repetitions of a smaller
pattern.
The hidden function may be controlled by either the Parameter or
the Polynomial. When it's controlled by the Parameter, the Parameter
X value operates on the horizontal dimension and the Parameter Y value
operates on the vertical dimension. When it's controlled by the
Polynomial, the x^{2} term operates on the horizontal
dimension and the x^{1} term operates on the vertical dimension.
The Polynomial C term is also active in a different way and does some
interesting things for Pattern 2. There are 10 different selections for
the hidden function, 1 through 10. Values 0 does nothing. Larger numbers
will also do something but the hidden function just repeats after 10,
i.e. 11 is the same as 0, 12 the same as 1, etc.
Sounds confusing? It can be at first. Just look at the Hidden
Function part of the table. Wherever it says Yes there
is a hidden function available. The feature that controls it is the
one named at the far left of the corresponding table row.
Once you start to experiment with them you can see how they work.
For example, Pulsar in the
Fractal Gallery was created with Pattern 2 using the hidden functions.
In the hidden function section of the table there is a Yes for
Pattern 2 in the Polynomial row, so the Polynomial controls the hidden
functions.
