Creating Fractal Images w/ Ultra Fractal

Hi!

To create a fractal image one could understand the relevant math (i.e., Fractal Geometry) and the functionality of the software that generates the image. The relevant math is beyond the grasp of most artists and the functionality of complex software such as Ultra Fractal is, at least initially, also beyond the grasp of most artists.

As an alternative, one could ignore the math and identify the bare bones elements of the software that can be tweaked to create a fractal image that can be exported.

In Ultra Fractal (v. 6.02), rightly or wrongly, I have tentatively identified the following elements: (a) the Fractal Formula, which generate a set of numbers, (b) the Inside/Outside Coloring Algorithms, which assign a value to the numbers, which I believe is called the Index Value, and (c) the Gradient, which assign a color to the values.

Therefore, the elements that can be tweaked are (a1) selecting the Fractal Formula and (a2) changing the parameters of the formula, (b1) selecting the Inside/Outside Coloring Algorithms and (b2) changing the parameters of the algorithms, and (c) changing the parameters of the Gradient.

Two other elements, Transformations and Plugins, are still not clear. Are Transformations like “independent” add-ons that change the image? If so, here’s how I view Ultra Fractal:

INPUTS:
Fractal Formula and parameters
Inside Coloring Algorithm and parameters
Outside Coloring Algorithm and parameters
Gradient parameters
Transformations and parameters

OUTPUT:
Fractal image

About Plugins, can someone explain, in layman language, how Plugins fit in my simplistic view of Ultra Fractal?

Thank you!

Dr. T

One small augmentation to your explanation: the Fractal Formula generates a sequence of numbers for each pixel in the image, called the "orbit". It also determines whether the pixel is inside or outside so it can use the correct Coloring Algorithm.

Transformations transform the pixel to a new value before iterating the Fractal Formula (which is why the Mapping tab is before the Formula tab). This makes global changes to the resulting fractal shape. For example, the Lake transform doesn't change points in the top half of the image, but changes points in the bottom half to their corresponding points in the top half modulated by a wave function to give the appearance of the top half of the fractal being reflected in a lake. To learn more about them, go to Help->Tutorials and expand "Learning about transformations".

Think of plugins as a special kind of parameter for formulas, coloring algorithms, and transformations. Instead of just a value, they implement code, so can be very powerful. Plugins often have their own parameters, which can even be plugins themselves! For more information and some examples, go to the Ultra Fractal Help, click on the Contents tab, and expand "Plug-ins".

Dear rsidwell,

Thank you for your reply.

(1) Please clarify something. As I understand it, a fractal formula generates a set of numbers, which based on the formula’s definition, may be said to be inside or outside. And, I thought there was a one-to-one relationship between the set of numbers and the image’s pixels. That is, that one number is associated with one pixel.

You said that “the Fractal Formula generates a sequence of numbers for each pixel in the image.” That is, that there is a many-to-one relationship between the set of numbers and the image’s pixels. In other words, that more than one number is associated with one pixel. Did I misunderstand you?

You also said that the image (i.e., the collection of the pixels) is called the orbit. Is that right?

(2) About Plugins, as I understood you, Plugins may be viewed as sub-elements that modify formulas, algorithms, and transformations and that, themselves, may have their own parameters. Got it!

I know that transformations are added in the Mapping tab, formulas are selected in the Formula tab, algorithms are selected in the Inside/Outside tabs, and the Gradient window is displayed from the Toolbar.

However, it is not clear how Plugins are implemented.

Dr. T

Dear rsidwell,

After doing some readings about plugins, I now understand that UF plugins are not sub-elements but rather higher level mini-programs that add functionality to UF.

Please read my post titled "Creating Fractal Images w/ Ultra Fractal - Part 2."

Thank you!

Dr. T

Let me be a little more precise. An image is a collection of pixels. To determine the color of a pixel, Ultra Fractal picks a point (x,y) for it, and treats that point as a complex number x+yi. That number is first passed through the transformations (if any) to get the starting number for the formula; it represents a point, so let's call it p0. The formula then maps the starting number to a new number, p1. That number is then passed back to the formula to get p2, which is in turn passed to the formula to get p3, and so on. Each time the formula is executed is called an iteration. The sequence of numbers/points p1, p2, p3, ... is called the orbit of p0.

The process continues until one of two things happens:

1. The bailout condition is reached. This is defined by the formula, but a typical condition is when the magnitude of the number (the distance of the point from the origin) exceeds a Bailout parameter specified in the formula. When this happens, we say the orbit "escapes", and the point is considered Outside. (This is the origin of the term "escape-time" fractal; the escape-time is the number of iterations completed before escaping).

2. The maximum number of iterations specified in the Maximum Iterations field have been done without the orbit escaping. In this case, the point is considered Inside.

I've been trying to keep the explanation simple, but be aware of one complexity: When anti-aliasing is used, Ultra Fractal will actually pick multiple points for each pixel and iterate the formula for each one. The color for each point is also determined, and the results for all of the points are averaged to determine the final color for the pixel. This generally improves the quality of the result, but obviously will take a lot more time, so there is a trade-off.