## Escape Formula Variation: RectangularPolar

Not claiming there is any real mathematical genius behind this idea, but it seemed interesting enough to share. I was wondering what would happen if you build your next Z by plugging the magnitude of Z in as your real coordinate, and the angle of Z as your imaginary coordinate. I.e., abs(Z)+i*angle(Z), or in xaos abs(Z)+{0;1}*asin(im(Z)/re(Z)). This is it:

I found that the image is more visual stimulating if you dampen the real component by a factor of 0.5:

Here is the Julia taken from the colorful section on the left — Julia seed -1.2+0i.

## Mandelbrot n=2.01

Something to explorer with the mandelbrot fractals is altering the n parameter in Z1 = Z^n. A slight tweak from the standard n=2 gives some interesting texture and pattern:

Something I would like to explore more is mandelbrots with an imaginary n (e.g., 3+2i) but my version of Fraqtive only supports real values of n.

## Fractal with Barycentric Coordinates Variations

One can generate a few slight variations on the appearance of the previous fractal by altering the scaling factoring of the circle radius a little:

It is interesting also that this fractal draws in such a way that you can paint alternate cells without painted cells touching each other: I.e., they only touch at the vertices.

## Fractal with Barycentric Coordinates

This fractal idea did not originate with me, but I wrote some racket code to do the midpoint calculation using barycentric coordinates. This fractal draws a circle at the midpoint of a triangle, then subdivides the triangle and repeats:

Here is the same fractal to four iterations:

To get the midpoints, I could simple pass in the coordinates of the last triangle ABC, and then use “0.5” barycentric coordinates:

``````        [P1 (barycentric->complex 0.5 0.5 0 A B C)]
[P2 (barycentric->complex 0.5 0.0 0.5 A B C)]
[P3 (barycentric->complex 0 0.5 0.5 A B C)]``````

Here is the full code:

ftp://lavender.qlfiles.net/Racket/bc-fractal.7z