Where is the Minibrot?

[Kalles Fraktaler]

<a href="https://www.youtube.com/v/SMLalCKkvWk&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/SMLalCKkvWk&rel=1&fs=1&hd=1</a>


Finally I managed to render this formula with perturbation.
This is the same formula that DeepZoomNet rendered in this movie
<a href="https://www.youtube.com/v/IJrzX4d-_0U&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/IJrzX4d-_0U&rel=1&fs=1&hd=1</a>
I followed the path of his zoom in the beginning of this movie.
The sparse areas looks different in my render, but otherwise it is the same.
This is a very fascinating fractal! The zoom starts centering at e42, so there should be a minibrot at e84. But instead, even though there is a clear center in the pattern, the pattern keeps coming without any clear duplication in density.
The pattern looks unique at all depths. Will there be unique patterns all the way down to infinity without hitting a minibrot?
I guess we have to wait for Kalles Fraktaler 2.10, when the current limit of e300 is removed.

[quaz0r]

interesting!  i havent explored these other formulas much, but at e22 it looks like you zoom into a disconnected part?  seems like when ive played around with different formulas, any time you go off into a disconnected part it just goes on forever without getting anywhere.. is that right?

[Kalles Fraktaler]

interesting!  i havent explored these other formulas much, but at e22 it looks like you zoom into a disconnected part?  seems like when ive played around with different formulas, any time you go off into a disconnected part it just goes on forever without getting anywhere.. is that right?

I have zoomed in disconnected parts in burning ship and variations, and eventually found minibrots where expected, or that the pattern is empty in the center. I have never seen before that the pattern keeps having a center but no minibrot.
Someone on youtube noted that the pattern doubles every double exponent. That is indeed interesting

[stardust4ever]

Very neat. I experienced this somewhat in my Third Order Quasi Burning Ship Maiden Voyage, I zoomed into an inflection point between two Minibrots and zoomed a considerable distance before encountering empty space. This can be seen from about 1:32 to 2:07 in my video, before a rhombus design with empty space inside appears:
<a href="https://www.youtube.com/v/Tz1KKxwJzjU&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Tz1KKxwJzjU&rel=1&fs=1&hd=1</a>

This can occur whenever you have two lines of reflection that intersect each other. Often these type fractal areas have a rhombus shape when real 3rd order patterns within the abs() fractals typically form hexagonal symmetry in the area surrounding the centroid / minibrot.

There are also areas within some of the abs() fractals where the set tends to be devoid of minis with dendrites that go nowhere or decays into cantor dust. The submerged side of the 2nd order burning ship and buffalo (Panzerboy's and my implementation), opposite to the masts, are such areas. I once attempted to explore this region to no avail.
<a href="https://www.youtube.com/v/d5sbv2Tl6I4&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/d5sbv2Tl6I4&rel=1&fs=1&hd=1</a>

I also discovered some nice Koch Triangles in such a region (submerged area) within the Perpendicular Burning Ship. These shapes consist entirely of cantor dust. Here is one such location:

http://stardust4ever.deviantart.com/art/Koch-Triangle-600411815

I have also witnessed a unique area within the Celtic Perpendicular (2nd order) where a mini is present that has no dendrites surrounding it at all. The fractal has dendrites leading up to it, but they actually disappear and melt away before it reaches the mini, leaving fairly smooth iteration bands surrounding it.
<a href="https://www.youtube.com/v/87TtLjFe3os&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/87TtLjFe3os&rel=1&fs=1&hd=1</a>

Panzerboy and I attempted to recreate the variant of the Buffalo fractal that HPDZ used, but we found it to be devoid of minis. IIRC, the HPDZ fractal was a hybrid quadratic formula that needed a non-zero seed value to get proper fractal detail with minis and Panzerboy was unable to figure out how to start with a nonzero seed. Forgive me but I do not recall what the necessary seed value was, but suffice to say with or without the appropriate seed, the HPDZ "Buffalo" was a different beast compared to Panzerboy's and my implementation.

EDIT: Upon watching the video, I noticed some anomalies. First observe in the opening seconds the needle is fully visible, but rather than a well defined terminus, it decays into cantor dust. Similar scenario occur when changing the seed value of the classic Mandelbrot set along the real axis, to something other than zero or C. This tells me changing the seed value along the real axis may fix it or otherwise alter the fractal. Secondly, the patterns very definitely get more complex as time goes, however as one comment on the youtube page suggests, the periodicity occurs at e42, 84, 126, and so on without narrowing of the iteration bands. Normally the iteration bands narrow and bailout grows without bound as we approach the event horizon (minibrot). Here, the patterns get more and more complex as one zooms further down the rabbit hole, but no event horizon in sight. I wonder what this beast would look like at e600 or e1200?

Keep us posted; this is an awesome development!

[Chillheimer]

hey stardust, the koch curve made out of cantor dust is awesome! love it. It kind of sums up my worldview in one picture..

[quaz0r]

IIRC, the HPDZ fractal was a hybrid quadratic formula that needed a non-zero seed value

this was my other question.  are you guys figuring out what the critical points for all these different formulas should be?  i remembered reading some posts about that like this one

http://www.fractalforums.com/index.php?topic=20898.msg81208#msg81208

[stardust4ever]

this was my other question.  are you guys figuring out what the critical points for all these different formulas should be?  i remembered reading some posts about that like this one

http://www.fractalforums.com/index.php?topic=20898.msg81208#msg81208

According to the post,

http://www.fractalforums.com/saturnandtitan/location-dependent-critical-points/msg81208/#msg81208

Critical point is to solve for the derivative of the equation, set to zero.

HPDZ Buffalo is Z1 = |Z0|^2 - |Z0| + C

While I am unsure of how to handle absolute value when doing derivatives (my calculus is a bit rusty), we can take the derivative of

f(Z) = Z^2 - Z + C

which yields

f'(Z) = 2Z - 1 + 0

Now we solve for Z

2Z - 1 = 0

2Z = 1

Z = 1/2

Based on the results, I would attempt to start the iterations with a seed value of Z0 = 1/2 rather than zero.

However there are abs() functions in the original equation, which I assume like the Burning Ship, takes the absolute value of both the imaginary and real components of Z prior to iterating. In the real domain, |Z|^2 = Z^2, but -|Z| = Z in the event Z is negative. The solution of this derivative would be Z = -1/2 which checks out because Z becomes negative, so it would be of academic interest to try both values as initial seeds.

Z0 = 1/2 or Z0 = -1/2

EDIT: I just did the logic inside my skull. The first iteration will have the exact same result whether the initial value Z0 = 1/2 or Z0 = -1/2 is used, ie both are valid solutions. IIRC, this should eliminate the "cantor dust" area near the western terminus of the needle as well.


@Kalles, why not try to include both formula variants, with seeds set to 0 and ±0.5?

[stardust4ever]

Further musings on this unique fractal. I believe I have solved the minibrot riddle. To hit the mini requires infinite zoom depth:

Second order fractals halve the depth each periodicity and third order fractals third the depth with each periodicity.

For second order, a diversion found at zoom depth N will double at N + N/2,
quadruple at N + N/2 + N/2^2, 8-fold at N + N/2 + N/2^2 + N/2^3, and so on...

For third order, a diversion found at zoom depth N will triple at N + N/3,
9-fold at N + N/3 + N/3^2, 27-fold at N + N/3 + N/3^2 + N/3^3, and so on...

For any Power P, a diversion found at zoom depth N will result in a P-fold formation at N + N/P,
P^2 formation at N + N/P + N/P^2, P^3 formation at N + N/P + N/P^2 + N/P^3, and so on...

Minibrot of depth M appears at the limit of

M = N/P^0 + N/P^1 + N/P^2 + N/P^3 + ... + N/P^∞

for Mandelbrot of power P with divergent zoom path at depth N.

For second power, limit M = 2N
For third power, limit M = 3N/2
For fourth power, limit M = 4N/3

Notice the pattern. For any power P, limit M = NP/(P-1)

M = M*1/(1-1) is not allowed, because we are dividing by zero.

However, the limit as P decreases approaching one, M = NP/(P-1) is infinity!

Proof, for first power fractal P=1:
M = N/P^0 + N/P^1 + N/P^2 + N/P^3 + ... + N/P^∞
M = N/1^0 + N/1^1 + N/1^2 + N/1^3 + ... + N/1^∞
M = N + N + N + N ...
M = ∞

Therefore a hypothetical first power equation will never hit a minibrot, because said hypothetical minibrot exists only at infinite zoom depth!

HPDZ's Buffalo implementation is neither second order nor first order, with Z = |Z|^2-|Z|+C being quadratic in nature. There is clearly a Z term present with both second power and first power exponent. So naturally, it would have some hybrid traits of both a second power and first power fractal. This could provide a clue as to why you can zoom indefinitely into the centroid without hitting a mini. The scaling of periodicity for a first order fractal is one whole rather than one half, one third, one fourth, or so on! Therefore the fractal iteration bands will never reach critical mass and one may zoom into the centroid forever without a formal ending to the zoom sequence!! HPDZ's Buffalo therefore has no minis as all zoom paths are infinite!!!

[stardust4ever]

Sorry for the bump. I am working on a video of my own. I chose an inflection point somewhere far away from the needle zone, at relatively light depth (about e9). Fractal is really weird with the way periodicity works. It takes two or three periodicity cycles before the pattern repeats ad infinitum. There is an unmistakable compass like formation that is part of the repetition. I use this as a metric sort of like mile markers in my video. Shortly beyond e100 I take another detour. The circle like compass pattern gets zoomed into eventually creating a dual football type shape around e216. This pattern duplicates again around e324 creating a four football like points in a rectangle shape at about 60 degree angle. At e432, a snowflake pattern appears with six football points. Finally, an additional snowflake like pattern occurs at e540. The zoom sequence cuts off here, at 7.14e541 (2^1800). This will yield a 15 minute video if rendered at constant zoom rate 2 zoom frames per second. When the sequence is finished, I'll post a more thorough analysis of my findings. It will take less than a week to finish at the current rate, and as we all know, it gets faster as the zoom depth decreases so should be done well before then. Iteration depth increase appears to be fairly linear though this depends upon the zoom path chosen. Take care...

BTW, when I use the term "football" I am referring to American style gridiron football which oblong shape represented by two minor arcs, not the perfectly circular soccer ball.

[Kalles Fraktaler]

Looking forward to that movie!

Yann just published another movie of this perculiar fractal, that is jumping between inflection points to show the linear development.
He has used either real or imag only for high bailout, which gives some structure to the empty areas which emphasize the development of the density
Very interesting I think
<a href="https://www.youtube.com/v/7We86a8uKIo&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/7We86a8uKIo&rel=1&fs=1&hd=1</a>

[stardust4ever]

That is indeed fascinating. Here's a preview of my location, starting with the ten point compass:


As you can see it takes a couple periods before the pattern truly repeats itself. the 2e430 and 2e650 locations appear identical; ditto for 2e540 and 2e760. The pattern appears to get rotated with each period.

Also amazing I got such round numbers for the zoom depths! Each period is exactly 1e110 smaller than the last...

EDIT: Got around to posting it...