Switching and Switch Formula Classes

Switching

Those who are already familiar with the mathematics and mechanics of switching from Mandelbrots to Julias may like to skip to the section on Switch Formula Classes.

If you're not into the mathematics, then the best way to explain switching is to simply say that most formulas can be viewed in one of two ways - as a Mandelbrot or as a Julia. (Note that the original Mandelbrot iteration replaces z by z² + c; here we're using "Mandelbrot" and "Julia" in a more general sense as modes of iteration.)

What's interesting and artistically useful about this? For every location on a Mandelbrot there is a corresponding Julia, and the geometry of the Julia corresponding to a given point on a Mandelbrot matches the local geometry around the point on the Mandelbrot. Hence if you find an area of a Mandelbrot that's visually interesting, then you can get more of the same by switching to a Julia from that area of the Mandelbrot.

Another way to put this is that a Mandelbrot is like a "map" of the Julia for the same formula.

The mechanics of switching in Ultra Fractal are simple: just press F7 or click on the "Switch" icon in the top toolbar and then move your mouse pointer over a switchable Mandelbrot. Previews of the corresponding Julias will appear; to actually switch to one of the Julias click the left mouse button and a new fractal window will appear.

Let's backtrack a bit to review what UF formulas do and how do they do it. Formula iterations depend on two complex numbers, z and c. UF does the following for each pixel in the image.

•  Initialize: Set the initial values of z and c.

•  Iterate: Each iteration generates a new value for z, generating a sequence z[0], z[1], z[2], ..., with each new z[n] depending on the previous value z[n-1] and on c. (It's also possible to change c, but this is not usually done.) Iterations continue until one of three things happen:

1. the maximum iteration limit is reached; or

2. if Periodicity Checking (in the Formula tab) is On, periodicity is established; or

3. z is such that the Bailout criterion is reached

The mathematical difference between Mandelbrots and Julias is straightforward. For a Mandelbrot of the form z[n] = f(z[n-1],c), where f(z,c) is some function such as z² + c, the initial value of z is the same fixed value for every pixel (often it's zero), and the initial value of c is the pixel value. The corresponding Julia uses the same f(z,c) but different initial values; the initial value of z is the pixel value and the initial value of c is the same for all pixels. The initial c value for a Julia is often called the seed.

In UF, when you use F7 or the Switch icon to switch from a Mandelbrot to a Julia, the place where you click in the Mandelbrot image is used for the Julia seed, the initial c. Similarly, when you switch from a Julia to a Mandelbrot, the place where you click is used for the initial value of z.

Switch Formula Classes

One of the limitations of UF5's new class formulas was that they did not allow the use of Ultrafractal's built-in switching facility. To get around this limitation, Dave Makin wrote mmf5.ufm:Generic Switch Formula, which makes it possible for formula authors to write class formulas that allow switching.

To use Switch Formula objects simply choose mmf5.ufm:Generic Switch Formula as the main fractal formula (instead of using Standard.ufm:Generic Formula) and then you can plug in any "Switch" class formula in the usual way. At the moment there are Switch formulas in mmf.ulb and jlb.ulb.

The following upr is a simple example. You'll see the usual Mandelbrot image, but it's generated in two ways. the bottom layer is Mandelbrot done the standard way. The upper layer uses Generic Switch Formula  and Switch Transform Formula from mmf.ulb to get the same result. (There's no reason why you would want to do that, except as a starting point.) Try switching using each of the layers and see that you get the same results (if the Julia seed in the Standard method matches the Julia Constant in the New method, of course).

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Formula Tutorial 1

Let's look more closely at the Formula tab on the New layer, at The Formula Method. The Fractal Type is tx(z)+c. This means that the iteration transforms z and adds c to get the next z. (There are other possibilities in the drop-down list. Try them.) The "Initial Zold" isn't of interest now. Below that is The Formula Transform, a plug-in; the current plug-in is Power. Power transforms z in two steps; first apply the function from the drop-down list to z, then raise the result to the power shown.

Since the function is the identity, ident, applying the function just gives z. Since the power is 2, the transform result is z². Since the Fractal Type is tx(z)+c, the iteration formula is z²+c, the standard Mandelbrot formula.

Try changing the function and the power to see what happens. Try switching to Julia mode.

Here's an example where one layer is Mandelbrot mode and one is Julia mode. Note that the Mandelbrot layer has four-fold symmetry and the Julia has three-fold symmetry. In the Mandelbrot layer, try switching to Julia mode and see if you can duplicate the Julia layer. In the Julia layer, try switching to Mandelbrot mode; notice that the Mandelbrot Start value is no longer zero, and that the symmetry is lost for most start values.

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Formula Tutorial 2

Other Switch formulas can be plugged into the slot in Generic Switch Formula. Another from Dave Makin is Switch Formula Transformations, in mmf.ulb. (Don't confuse Switch Formula Transformations [SFT] with Switch Transform Formula [STF].)

SFT's iteration formula is in three parts, each a plug-in. SFT transforms z first by a transform, the Pre-Transform; then by a Switch
formula; and then by another transform, the Post-Transform. Either (or both) of Pre-Transform and Post-Transform can be Null Transforms, and do nothing.

The next upr is a Mandelbrot mode fractal with

• Pre-Transform: Null

• Formula: Switch Standard from mmf.ulb. This transform does z²+c.

• Post-Transform: Newton from jlb.ulb. The transform does one iteration of the Newton formula.

Try zooming into this Mandelbrot mode fractal. You'll find various modification of mini-Mandelbrots throughout. Try changing the power in the Switch Standard formula and the parameters in the Newton transform.

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Formula Tutorial 3

The above fractal is from a composite formula. In UF4, a formula writer would have had to write this. In UF5, a user of UF who is not a formula writer can generate such a formula. Knowing UF programming, or even knowing the mathematics of fractals, is not necessary. (Admittedly, knowing UF programming and mathematics does help.)

 From the previous upr, you can switch and find Julia fractals. Here's one. From here, you can use Explore on the Julia Constant (the seed) or change any of the formula parameters.

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Formula Tutorial 4