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Greetings!

I would like to share with you my recent Mandelbrot set related work on the mathematical entities underlying the Buddhabrot, which I have called the Apeiropolis and anthropobrot multisets (i.e.: bags).

Long story short, there are an infinitude of pareidolic Buddhabrot-like figures, anthropobrots; and they are the result of deep ectocopial (outside the Mandelbrot set) points--which approximate the outline of the Mandelbrot set--being rotated and shifted by the Fatou-Julia function.

I have written a short paper on this, which is primarily about detailing the terminological framework through which I made my discovery, along with defining the aforementioned multisets:
http://vixra.org/abs/1604.0392

My website (scroll down below the thumbnails) also has the reference source code from the article, along with compiled win32 and win64 versions, if you want to try your hand at generating them:
http://apeirography.com/en/projects/anthropobrots/

I also attach a few pictures of low numbered anthropobrots for those who do not wish to click through to the website.

Thank you in advance for your time and comments!

Sincerely,

Apeirographer

As a small addendum, here is a smooth animation transitioning between the first few anthropobrots:
https://twitter.com/apeirographer/status/727311712081334272

- Apeirographer

interesting work  :beer:

Thank you for the alesome encouragement, quaz0r!  Should I be reassured or concerned by its brevity?

Are people understanding the definitions alright, once they get past the weirdness of Salov's iteral notation and the Strachney brackets for bags/multisets?

The Apeiropolis multiset *is* the mathematical entity beneath the Buddhabrot... in fact, it is easy to think of it as a sort of perfect (Platonic) ideal of the Buddhabrot.  However, it is also the repository of multiply infinite numbers of anthropobrots, and so I have dubbed it the Endless/Infinite City.

It is interesting to note, that while certain (usually higher iteration limit) versions of the Buddhabrot can yield images with similar but non-superficially different features (such as dotted arcs or spiral-like blotches left by a single very deep point's lengthy and productive orbit); a given anthropobrot's differences will depend primarily on the precision of the visualization, which, in turn, depends on the depth of the governing seeds, making anthropobrots more visually static on account of being more strictly defined.

In other words, the Fatou-Julia anthropobrot (#2) or the Gardi anthropobrot (#17) will always have the same broad features (arising from the specific set of rotations and shifts applied to the M-set outline thereuntil)... only their "definition" (sharpness) will vary.

- István

I'm thinking of trying to put together a short video (possibly to enter into the annual competition) that gives a more pragmatic introduction to anthropobrots, and perhaps also visually demonstrates their relation to the Buddhabrot.  Will have to see whether I can make the deadline though...

I should also mention that I'd be happy to help out anyone who might want to throw together their own implementation of an anthropobrot generator.  So if you are interested, please don't hesitate to ask for help or advice on anything that is unclear.

Apeirographer

Wow!
Very demanding job, and nice to see the paper. Congrats!

I had no time up to now to understanding even the buddhabrot; but recently I experimented with parameter perturbation of bifurcation fractals and got something buddhabrot-like. (Don't publishet yet here, but will do.)
Let me share some sinusoidal perturbed logistic maps:

(http://s32.postimg.org/ahmub3xqd/Bifu_fr0_freq_1_E_4_ampl_0_1_const_0_45_zoom_hy.png)

(http://s32.postimg.org/8qv31iidh/Bifurcation_fr0_pert01_hyp.png)



And a 2D section of a 4D Mandelbrot (quadrobrot or what is its name?):

(http://s32.postimg.org/nevx5wio5/4_Dmandelbrot_0001.png)


Well, seems to we can do anything, just get a nice fractal! ;)
(OK, I know its not true, sometimes it requires a lot experimenting...)

4D Mandelbrot? Now that's getting really interesting. Can we get more of those?

Its name is Tetrabrot, I played with it before a little.
I just mentioned it because I saw your animation of your transformations, and remembered I saw something similar when I pushed a plane across Tetrabrot:
https://youtu.be/_fpfoNF4QVI (https://youtu.be/_fpfoNF4QVI)
and:
https://youtu.be/Y-b9N5AxdYA (https://youtu.be/Y-b9N5AxdYA)

It is not as witty as Mandelbulb... I had own trials with more-than-2D Mandelbrot, I called these Orangeman:
http://www.fractalforums.com/3d-fractal-generation/new-3d-mandelbrot-formula-%28-orangeman-%29/ (http://www.fractalforums.com/3d-fractal-generation/new-3d-mandelbrot-formula-%28-orangeman-%29/)

I found 2 more images about cross sections of Tetrabrot (sorry, if this is a big OFF here (?)):
(http://s33.postimg.org/8no3cea9r/4_D_Mandelbrot_2_D_metszet.png)

(http://s33.postimg.org/rk2ynpqwv/4_Dmandelbrot_0002.png)

Very nice work! Thank you for posting it here.   O0

Yeah, especially the latter is very nice.

OFF:
One more of my old pictures: a cross section of 8th order Mandelbulb:
(http://s33.postimg.org/e512vqmv3/8_rendu_Mandelbulb0005.png)