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My formula for Ultra Fractal - mmf3.ufm:Newton by Roots - allows you to control the Newton produced by specifying the degree and roots of the polynomial to be solved - for the traditional Newton form switch to the Julia version.

I'm looking at it right now, its very advanced matter. COMPELLING :-)

I used simple NovaMandel and NovaJulia framework to insert new equations. I repeated the framework many times to test properties of new equations. On the internet there is a large Newton's fractal from the movie Biocursion.
<a href="http://www.youtube.com/v/I0wGQaxGg_c&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/I0wGQaxGg_c&rel=1&fs=1&hd=1</a>
Its so big that it looks like a city. What I see different there is that its not limited to small neighborhood of the coordinate ZERO. Instead its wide and differs in parts. I don't expect that it is necessary to use polynomial as a function.

Sometimes I make an error when calculating the first derivative and the final result can not represent an accurate equation for Newton-Raphson method, but fractal can be drawn easily. I had some GOOD ones in that series of experiments!

Now you see, I constantly wonder what is the true, the pure and beautiful in one fractal. Or I wonder what is the science behind it, or something valuable in the background information. I can categorize shapes that way. For example - don't panic - I'd categorize your default display of fractals in the above mentioned formula file as fertile, but ugly - not very ugly, a little. When I zoom in there is an impression of overlapping shapes and pile-ups of dense colored stripes - that's one characteristic to call slightly unpleasant. Another category is - makes me almost angry - the diabolical, perverted exhibitions of yet another Mandelbrot set shape mutated and mean, lurking bellow the prominent outer features.

In contrast, NovaMandel type for (z^2-1)^2 contains Mandelbrot set and all shapes smoothly contoured around clean Mandelbrot shape. Dense areas (extreme up and down edges of a bell shape) are nicer. They insinuate almost a 3-dimensional curvature, a bell shape pointing to the left (or right depending from -+#pixel). To see more order - like Mandelbrot set is nice - in the category beautiful, but unfortunately it also fits a category - bring-me-something-new.

Rarely one complete fractal expands beyond certain limit. Some trigonometric functions can bring that. Mandelbrot and Julia sets fit into circle of radius 2. It reminds me of limited mathematics, because when all shapes stick to near coordinate zero I feel more bounded to basic principles instead of having fractal bricks for construction. I mean, fractal would appear in relation to few basic reasons over and over again. What if it is possible to add another sub shape with another function on one spot, adding a third one someplace else, or combine properties one into another one. Just a small problem there, NO ONE sees any mechanical parts of a fractal. Not a computer not a human (maybe few crazy professors).

I don't think there's a formula for UF to do that for a Newton yet but you can for a Nova.

Select mmf5.ufm:Generic Switch Formula as the main formula then plug in mmf.ulb:Switch Nova instead of "Switch Standard" then enable the "Enable Advanced Settings" and set "Switch Value" to "Exponent".

After further consideration, I realize that you don't need a transcendental function in Newton's method to get an image with infinite extent, contrary to what gamma's impression was. Even applying Newton's method to the humble Z^3-1 equation will generate an image with infinite extent, although a rather boring one.

What's cool about the Biocursion muhtoombah video is that it has variation in the detail at high magnifications, which makes me think it's more like a Nova-type fractal, rather than a straight Newton's method.

I agree you can't really create information here. I also agree they probably used a complicated polynomial.

But I think it's possible to get a lot of complexity from a very simple equation ... like z=z^+c. The necessary ingredient is feeding the result back into a nonlinear equation. That's why I suspect Muhtoombah was done with something more like the Nova equation, which is basically the relaxed Newton's method modified to include +c: x = x - k f(x)/f'(x) + c. A huge amount of complex, "interesting" (non-scale-invariant) detail can emerge from this form of iteration even with f(x)=x^3-1. I would expect a more complicated polymials would be even better.

A more complex polynomial may add more detail since it has more roots, but not necessarily the kind of detail that's interesting. For example, I recently posted a picture of Newton's method applied to sin(x). It's pretty, but totally scale-invariant, so it looks exactly the same at all magnifications as far as I can tell.

Conversely, to get a fractal with infinite extent, you don't necessarily need a very complicated function: the basic x^3-1 function will give an infinite-extent fractal under Newton's method. But again, as with sin(x), it's totally scale-invariant and kind of boring.

I'm not sure, but I think that in order to get details that change with magnification, like in the classic Mandelbrot set for z^2+c, you have to have something analogous to the "+c" part. It has at least been that way in my experience so far.

I'm looking forward to the results of your experiments....

 

excellent images, especially 10,42 and 56, i have to play with that formula

I've only had a chance to take a very quick look but these images are awesome. I'll have to check out exactly what you've done mathematically and see if I can incorporate it into my software. I'll bet it is really s ... l ..... o .........w to render these due to the complexity of the calculations.