Click on the image to see its parameters.

The Mandelbrots are of the standard variety, iterating the function z = z^{n}
+ c, where z is initialized to a constant value (typically, 0) and c takes on the
value of the pixel. Each Mandelbrot can have its own power, n.

Likewise, the Julias are of the standard variety. They iterate the function z =
z^{n} + c, where z is initialized to the pixel value and c takes on a specific
parameter value. Each Julia can have its own power, n, and parameter, c.

To see why there are four Newton functions, consider how the Newton function is
typically used. A standard Newton fractal could be created by using Newton's method
to find the fourth roots of 1, that is, find z such that z^{4} = 1. This is
a Julia form of Newton fractal; the initial value of z is taken from the pixel and
the value c is a constant for all pixels (1, in this example). In this formula, this
function is denoted "Newton J." There is also a Mandelbrot form of the Newton fractal,
denoted here as "Newton M." With a Newton M, the initial value of z is an input parameter
and c is taken from the pixel value. See the below figure for examples of each. The
Alternating Functions fractal formula allows for two different Newton M functions and
two different Newton J functions. Click on the images to see their parameters.

z ^{4} = pixelinitial z = 1 |
z ^{4} = 1initial z = pixel |

There are also four Popcorn functions, two each of two different types. The basic Popcorn algorithm splits z into its real and imaginary parts, x and y, respectively. Then, each component is iterated independently and the new components are combined into the new z:

- x
_{old}= the real part of z_{old}and

y_{old}= the imaginary part of z_{old}. - x
_{new}= x_{old}- step × f_{outer}(y_{old}+ f_{inner}(frequency × y_{old})) and

y_{new}= y_{old}- step × f_{outer}(x_{old}+ f_{inner}(frequency × x_{old})). - z
_{new}= x_{new}+ (0, 1) × y_{new}.

Up to six of the 12 functions can be used in the alternating pool. You choose the number of functions in the pool and then define the functions. For example, you could have three functions, #1 being a Mandelbrot, #2 being a Newton M, and #3 being a Popcorn R. Then, you can specify how the functions are alternated. The "ramp" oscillator just repeatedly runs up through the numbers: 1, 2, 3, 1, 2, 3, 1, 2, 3, etc. The "sine" and "cosine" methods tend to put more emphasis are some of the functions. With three functions, the sine method would use them in this order for the first 12 iterations: 2, 3, 3, 2, 1, 1, 1, 3, 3, 3, 1, and 1. The cosine method yields: 3, 3, 1, 1, 1, 3, 3, 3, 2, 1, 1, and 2. The last two methods, "random msb" and "random lsb" are two different ways to randomly choose functions, based on Ultra Fractal's random() function.

Once the function is chosen and iterated, you can determine what value is sent to the coloring formula. The idea for this was taken from Newton fractals. With the Newton formula, z usually settles down to one of a few values. However, the difference between the old and new values of z can be interesting. So, this formula allows you to send the new z (iterate), the difference between the old and new values, their ratio, or some other combination.

General Parameters

- initial z type: Choose "manual" to enter the real and imaginary components of the initial z value, or "pixel" to have the pixel be the intial z value.
- bailout: For the Mandelbrot, Julia, and Popcorn functions, bailout is based on
the value of z
_{new}. For Newton functions, bailout is based on the reciprocal of the difference between z_{old}and z_{new}. (This allows the Newton iteration to converge while still using a large bailout value.) - z type: The number sent to the coloring formula. Choose from "iterate"
(z
_{new}), the "difference" between z_{old}and z_{new}, the "ratio" of z_{new}to z_{old}, and a "weird" combination of the two z values.

- # of functions: How many functions are in the alternating pool, from 1 to 6. If you choose 1, then the same function is used for every iteration.
- function 1: The type of formula for function 1. Choose from:
- (a blank choice--this is explained below)
- Mandelbrot 1 or Mandelbrot 2,
- Julia 1 or Julia 2,
- Newton M 1 or Newton M 2,
- Newton J 1 or Newton J 2,
- Popcorn C 1 or Popcorn C 2, or
- Popcorn R 1 or Popcorn R 2.

- specific parameters for each function: These include the powers for Mandelbrot, Julia, and Newton types, specific c parameters for the Julia and Newton J types, and the step, frequency, outer function and inner function for the Popcorn types. This also includes weights for the Mandelbrot and Julia types and nova factors (weights) for the Newton functions.

- oscillator type: The choices are: ramp, sine, cosine, random msb, and random lsb.
- initial seed: The initial seed for the random number generator, for either random mode.

- If you want to play with just one type of function, set the number of functions to 1.
- Mandelbrot/Julia, Newton, and Popcorn formulas have very different dynamics. Thus, combining them in this way will probably lead to very unpredictable results. It may be easier to figure out what is going on if you start with just a few functions of the same basic type.
- Because you can combine Mandelbrot and Julia types into the same image, there's no ability to "switch" as with a standard Mandelbrot to a standard Julia.
- With the "function 1," "function 2," etc., parameters, the first choice in the pull-down menu is blank. Selecting this will remove their specific parameters from the window. For example, let's say that you were using three functions with Newton J 1 as the third. If you then reduced to two functions, the Newton J 1 parameters will still be visible. To prevent them from showing up, select the blank choice for function 3 and then change the number of functions to 2.
- Use the weights to reduce the effect of an added function (e.g., the second with 2 alternating functions) to understand how combining it will change the fractal. To do this, set the weight of the added function to a very small number (like 0.001). Setting it to 0 will turn it off. For the Mandelbrot and Julia functions, these are the weights (e.g., "Mandelbrot 1 weight"). For the Newton functions, the weights are the nova factors (e.g., "Newton M1 nova factor"). With the Popcorn functions, the step size works as a weight.