FormulaX Plug-Ins

I’ll start with this quotation from the UF help file:

Since the first release of Ultra Fractal in 1998, the formulas written for it have become larger and more complex. Even though many formulas are quite similar, they could not share any common code, but had to be written as stand-alone formulas. For example, if you wanted to add just one new feature to an existing formula written by someone else, you had to create a new formula, duplicate all the code from the original formula, and only then add some new things.
To improve this situation, Ultra Fractal 5 introduced the ability for formulas to re-use common code via plug-ins. You can view a plug-in as a “black box” that adds functionality to a formula. Via plug-in parameters, a formula enables you to select the actual plug-in that performs an action for the formula.
The formula declares what kind of functionality it needs and you can select any compatible plug-in to actually implement it. This makes it possible for formula authors to write a new plug-in to implement just the new feature that they want to add to an existing formula. The formula doesn't have to be changed at all. You just select the new plug-in when you use the formula.
For everyday users, plug-ins make it possible to combine existing elements in new ways without needing to do any programming. For formula authors, plug-ins make it easier to write and manage complex formulas.

More than 15 years later, I have seen few users and formula authors take advantage of plug-ins. For example, on the excellent site https://www.deviantart.com/ultra-fractal-redux, many fractals are available along with their parameters but few of them use plug-ins.

Perhaps the reason for not using plug-ins is that formula writers have not written their formulas with sufficient flexibility. I have been working on flexible plug-in formulas since the early days of UF5, with many tries that I’ve since rejected (note the large number of “NotRecommended” lines in jlb.ulb). This tutorial covers the latest (though perhaps not the last) version.

This is a long tutorial and you don’t have to do it all at once. The images are meant only as illustrations, not for any artistic merit. Artistry is up to the user.

While writing about my programs, I realized that I needed to make many minor improvements. Therefore you’ll need to download the latest versions.

This tutorial only discusses plug-ins for the Formula tab. A later tutorial will discuss plug-ins for the Coloring/Outside tab.

Start with this upr:

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The Formula tab uses “Plug-In Formula” from Standard.ufm, “Switch Formula” from jlb.ulb, and “Standard_Mandelbrot” from Standard.ulb. This shows the usual Mandelbrot Set, the red part (red is the “Solid Color” set on the Inside tab). Everything outside the red part diverges.

For the rest of this tutorial I won’t change the gradient or the coloring and will work with only one layer. Unless I say otherwise, I’ve used the default parameters

Switch Formula (SF) allows two types of formula plug-ins. If “Does Formula do its own bailout test?” is “Yes”, SF looks for formulas of the Formula class (defined in common.ulb), such as those in Standard.ulb, mt.ulb, and om.ulb. These use the bailout test defined in the Formula class and set their own “c” values. If the answer is “No”, SF looks for formulas of the FormulaX class, currently only found in jlb.ulb. SF handles bailout as well as setting the “c” value.
Experiment with Yes formulas and then set the answer to No. Make sure that the “Divergent bailout method” is set to “B00 |z|”, as this corresponds to the method used in Standard_Mandelbrot. The difference in the coloring is because Standard_Mandelbrot starts with z of 0 and SF starts with z equal to the pixel. (In SF, this can be changed; see later in this tutorial.)

The next line is “Bailout type”. SF allows this to be Divergent, Convergent, or Either. With Divergent, we see the usual Mandelbrot set. Set the Bailout type to Convergent.

The yellow area that’s inside the Mandelbrot set converges. (Yellow is the color at position zero of the gradient.) The outside seems to converge, but before we saw that it diverged. What happened? The outside pixels, given enough iterations, go to INF, a value too large for the computer to handle, so Ultra Fractal (UF) bails out. Change the “Periodicity Checking” to Normal. Note that UF’s periodicity checking can’t deal properly with much of the inside area.

Change the “Periodicity Checking” to Off and set the “Bailout type” to Divergent.

The next line down is “Divergent bailout method”. Click on that line’s Browser button and a browser pops up with thumbnails showing the effect of each of the 10 possible bailout methods. For this fractal they all look alike, but that’s not true for all fractals. Note: If you click on View, Preferences, Browser, you can change the size of the thumbnails that are shown. Mine are set to 200 by 150, larger than the default value.

The next line down is “Bailout value”. Here it’s set to 100. With z = x+iy, or (x,y), “B00 |z|” bails out when (x^2+y^2) is larger than 100. Similarly, “B04 |Real| and |Imag|” bails out when both abs(x) and abs(y) are larger than 10. Most users will just look at the thumbnails to decide which method to use.

Note that all FormulaX plug-ins can do both M-type, with c the pixel value, and “J-type”, with c the seed value. There’s no need to have two different plug-ins. Uncheck M-type to see a Julia fractal with the default seed of (1, -0.5).

Unchecking “Basic Mandelbrot” gives you the option to change the power from 2 to any complex number. That’s within the Mandelbrot plug-in.

Unchecking the “Basic version of Switch Formula” will unlock the advanced features of SF. For example, it’s possible to tweak z before the first iteration, and it’s possible to tweak z after each iteration. In each case this is done with the Tweak plug-in. (Tweak is probably an inaccurate name, as the plug-in can make a small change to z or completely change z.) Replacing the initial value of z with (0,0) will give the same coloring as with Standard_Mandelbrot.

Set the Mandelbrot plug-in to J-type, seed (1,-0.5), and check “Tweak z before first iteration?”. You have the option to “Add” something to z or to “Replace” z. You have the option of a simple linear expression or using a plug-in Transformation. Experiment. The default Transformation is “Function Transformation”, which has many plug-ins; you can see them as thumbnails if you click on the Browse button on the line below “Function Transformation”. Experiment. Here is the result for Replace, e1 = (0,0), e2 = (1,0), a = (1,0), and transformation “16 acotanh”.

Uncheck “Tweak z before first iteration?” and check “Tweak z after each iteration?”. Experiment. Here’s the result for Add, e1 = (-1,0), e2 = (0.2,0), a = (1,0), and transformation “27 log”.

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The transformations are of the class UserTransform. Their basic step (the Iterate function) is to take a complex value (z in this case) and return another complex value. Besides those in jlb.ulb, there are many others in Standard.ulb, dmj5.ulb, mmf.ulb, mt.ulb, om.ulb, and reb.ulb. Experiment. For example, try “Trap Shape” in mt.ulb, which has its own plug-ins. Later I’ll discuss Transforms some more.

Check the “Basic version of Switch Formula”, check M-type, and click on the Browse button to the right of Mandelbrot on the “FormulaX” line. There are many possibilities. Click on “Polynomial”.

You can change the degree of the polynomial, from 1 to 6, inclusive, and change any of the coefficients. If you set the degree to 2, a0 = 0, a1 = 0, and a2 = 2, you’re back to Mandelbrot. Experiment.

On the “Function type” line, you can also do a polynomial in some Transform of z. Try “04 asinh”. Uncheck M-type and try different seeds.

Also on the “Function type” line, you can use a sum of two transforms, each to a different power. These powers can be complex. Here’s the result for Polynomial, J-type with default seed (1,-0.5), “Two Powers”, and the default settings of T1 and T2 both “05 cos”, p = (3,0), and q = (2,0).

Experiment. It can get complicated!

Click on the Browse button on the “FormulaX” line and choose “Poly Newton”. The image will be mostly red until you check Convergent. Poly Newton is a generalized version of the Newton formulas in Standard.ufm and in Standard.ulb, where both are Convergent formulas. (Mathematically, these use iterative methods to solve a nonlinear equation, but that’s irrelevant for fractal purposes.) Uncheck M-type, set the seed to (1,0), and set a0, a1, and a2 to zero; this gives the original version.

The “Standard” versions use a “Step size” of (1,0) and “Iteration type” of Newton. (The other Iteration types in Poly Newton are different mathematically, but that’s irrelevant for fractal purposes.) Step sizes not near (1,0) may not converge, but may still give interesting results. Keep the same polynomial, set the bailout type to Divergent, Type 3 iteration, step size (-1.5,0).

Poly Newton can also use polynomials in some functions of z or a sum of two functions with different powers. Experiment.

Click on the Browse button on the “FormulaX” line and replace “Poly Newton” with “Two Formulas”. This FormulaX does a set number of iterations of the first formula, then a set number of iterations of the second formula, and repeats. Uncheck M-type for the first Mandelbrot and set “Formula 1 iterations” to 2.

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If you check “Switch formulas 1 and 2?”, or check M-type for the first Mandelbrot and uncheck M-type for the second Mandelbrot, you’ll get something different. Experiment.

On the “FormulaX” line, click on the Browser button and select “Combo FormulaX”. This combines two or three FormulaXs in a variety of ways. The default “Combos” is “f1 op f2” and the default “Op” is multiply. This means evaluate f1(z) and f2(z) and multiply them to get the new z. The operations of plus, minus, divide, and exponentiate are similar. The operation of “( )” is different—evaluate f2(z) to get z2 and the new z is f1(z2). Experiment. Some combinations won’t give anything at all useful. The default Formula 1 is Mandelbrot, and the default Formula 2 is Polynomial. In Formula1, uncheck M-type and you should see this.

Experiment. Change to “Combo UT Formula”. It’s the same as “Combo FormulaX” except for using Transforms instead of FormulaXs, and adding c at the end. Uncheck M-type.

Experiment.
On the “FormulaX” line, click on the Browser button and select “Gnarl”. Uncheck M-type. With the default values and plug-ins you’ll get this:

You’ll get different results with different values for “Step size” and for different bailout values. Change the bailout value to 1000 and increase the “Maximum Iterations” to 500.

Other colorings often give more interesting results. Experiment.

Finally, any Transform can be turned into a formula. Click on the Browse button on the “FormulaX” line and choose “Transform”. Click on the Browse button on the “Transform T” line, go to mt.ulb and choose “Crazy Spiral”.

I haven’t included the AboveBelow and Gnarly FormulaXs because I’m still working on them.

Congratulations if you made it this far while going through the examples instead of just skipping to the end.

Have a favorite formula that might make a good plug-in? Several that could be combined?

I welcome comments and suggestions.

Many Thanks for this amazing tutorial, Jim! I have already made two explorations of some of its features, with one resulting fractal that I have just used as my avatar for the UF Forum section here and another one that I will post on DA, crediting you with this tutorial. I hope to compose an Ultra-Fractal-Redux Challenge that utilizes more of the techniques presented here.
Margaret Kieckhefer (Ampelosa) https://www.deviantart.com/ampelosa

Great tutorial Jim, thanks for sharing this with us!