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12 inflection points

(http://nocache-nocookies.digitalgott.com/gallery/20/8851_06_02_17_12_27_57.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20024

A high resolution comparison of inflections on a basic julia shape, and morphing the same shape in the Mandelbrot set (3.8E1221)

How claude came up with the simple inflection formula c=cs+(c-cs)*(c-cs), where c is the screen pixel, and cs is the inflection point, is a mystery for me.

I wonder if it would also be possible to find how the surrounding pattern is doubled?
Could the top image be rendered with plain double datatype just by transforming pixels...?

Here's some ratios of side lengths in a right angled triangle connected with sin/cos/tan:
https://en.wikipedia.org/wiki/Trigonometry#/media/File:TrigonometryTriangle.svg

So the first step was to use the ratios directly, instead of inverting tan and doing sin and cos.

Then the next step was to see that the multiplications by the length mean I can use side lengths directly, without having to normalize (divide by length) to get to sin and cos.

Then the final step was to see that the mutliplcations and additions in the code ended up looking like x^2-y^2 and x y + y x = 2 x y, which are very much like complex squaring.  It helps that I was by this point expecting something very simple to pop out, and knowing about renormalization of quadratic maps near periodic attractors (why minibrots look like the whole set, is because the period is locally similar to a single quadratic iteration).

Visually I really like the one without borders, in fact I've been thinking about cutting shapes out of the mandelbrot set. But this feels a bit like breaking sacred stuff. Inflection method seems to give a shortcut for this. Is there software for this which would run on Linux?

Not that I know of but it would be cool for this to be implemented into Kalles Fraktaler