Fractals Help - Pattern


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
variation 1
variation 2
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 x2 term operates on the horizontal dimension and the x1 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.


Copyright © 1994-2013 Tom Carlino