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Fire and Ice



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=15711

Doesn't really look like fire or ice, but I didn't really know what to make of this one. This is a new formula I've been playing around with. It has mini brots, tricorns, and ships, and has elements of all three throughout. It's the same as the formula for the metabrot fractal, but using the burning ship as a base.

Cool. That looks like an interesting function. Will you reveal it?

Thanks. Unfortunately, I don't remember the formula I used for this. I was playing around with a lot of different formulas. I know it was of the form:
J = z
J = J^2 + J
J = J^2 + J
z_new = J + c

The real metabrot formula uses J = J^2 + z instead, so this one was actually a mistake. There were one or two abs functions in there, but I don't remember where. An expanded version of the formula is:
z_new = (z^2+z)^2+(z^2+z)

To render it, you need to compute all 3 critical points (I used online derivative calculators and polynomial root finders), iterate it with each critical point as the starting value, and color it by either taking the maximum iteration count yielded  by each critical point (coloring black if a pixel stays bounded for all three starting points), or taking the minimum iteration count and coloring black if the pixel stays bounded for any starting point.

Some interesting variations on the power 2 burning ship I've been playing around with are taking the absolute value only on even iterations or odd iterations (even iterations, starting at 1, gives a fractal with a mix of mini ships, tricorns and mandelbrots, like this one). These variations look really promising as well.

I rediscovered the formula I used for this image. It's:

J = z
J = J^2 +z
J = J^2 + z
z_new = abs(J) + c

Where abs is the absolute value of the real and imaginary values independently.

The critical points for this formula are:
X1=(-0.11965507329885812-i0.4289368132975893)
X2=(-0.11965507329885812+i0.4289368132975893)
X3=(-1.2606898534022837+i0)

Edit: To render it, use just the first critical point (X1) as the starting value. If you combine all 3 critical points, like I suggested before, the result won't look like my image.

Features just like my image can be found at the coordinates (-.222709,-.764333), at zoom level 11459.7. Assuming the coordinates are the same between fragmentarium and other programs. There are a lot of really interesting features in this fractal if you look around. Some parts bear an interesting resemblance to Kalisets.