Sixteen Ways of Looking at a Fractal

(Literary readers: yes, this is a reference to the Wallace Stevens poem, “Thirteen Ways of Looking at a Blackbird.”)

This image has 16 fractals arranged in a 4 by 4 array. All 16 have the same gradient and the same Formula tab (Julia from Standard.ufm). All of the Outside tabs use Plug-in Coloring (Gradient) from Standard.ucl, and the Distance Coloring plug-in from jlb.ulb. The differences come from the options in Distance Coloring and to a lesser degree in the gradient’s Color Density, Transfer Function, and Rotation.

For the top row, all four fractals have both horizontal and vertical symmetries. Fractals in the next two rows have only symmetry under a 180-degree rotation. Fractals in the bottom row have only the symmetry of the outside boundary.

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Here’s a simplified description of how UltraFractal makes images.

For each point (#pixel) in the plane, an initial Z is defined, most often either Z = #pixel or Z = (0,0), the “origin,” the point x=0, y=0. New Z values are generated using mathematical formulas as defined on the Formula tab. Think of these successive values of Z as marking out a path in the plane. The iterations end when the bailout criterion has been reached or when the limit on the number of iterations is reached.

The Outside tab (and sometimes the Inside tab, too) calculates a color to be assigned to each point in the plane. For Gradient-type coloring, the formula calculates a number for each point; the number is used as an index into the gradient to determine the color.

Some plug-ins use only the final Z value at each point, such as the distance of Z from the origin.

The Distance Coloring plug-in has (too) many options.

First is the Accumulation Phase where information is saved about each iteration. Information can be accumulated based on Z or various manipulations of Z, called ZZ. This phase uses plug-ins to calculate “distances,” either real or complex numbers. There are many plug-ins in jlb.ulb, each with parameters.

Distances can be accumulate using all the iterations, or just some. Each of the 16 fractals uses only iterations from the sixth onward.

In the Final Phase this accumulated information is converted to an index into the gradient, with (too) many options.

Experiment. To work on any single fractal, delete the mapping (which just moves the center), change the Solid Color from transparent to Black, and change the magnification from 0.375 to 1.5 by pressing F9 or I twice.

I can answer any questions posed in the replies.

I thought I would take a peek at your parameters but I'm afraid there is a problem with them; UF is reporting over 90 errors when I try to open the above parameters and many of your example images are not being reproduced as you intended.

The errors mostly say things like:
!Could not locate parameter 'fut' in 'GenericGradientColoring' in Standard.ucl
!Could not locate parameter 'center' in 'GenericGradientColoring' in Standard.ucl

When the image finally opens this is what is displayed:

There is something very wrong here, any ideas? I have updated my formulas but no joy.

I apologize: I had forgotten to update my formulas after doing a lot of cleaning up. I just updated them, so please try again.

Yes, that's fixed it. Thanks!