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2 new videos of zooms closer and closer to the surface of the Mandelbulb O0 :evil1:

http://www.youtube.com/watch?v=fxXjAUJRG10

http://www.youtube.com/watch?v=h0WOf7wOMRY

I don't have the patience for the set-up time for nice flight-paths :( At least not on this P4HT !!!
So here's my (comparatively boring) attempt at getting close to the Mandelbulb:

http://www.youtube.com/watch?v=NJ-UqNs2reI (http://www.youtube.com/watch?v=NJ-UqNs2reI)

@bib:
You may notice an issue with the exponential interpolation - just as it gets really close it seems to accelerate.
I think it's accelerating from a dolly parameter of "e" (yes the mathematical constant) down to 1.
So to avoid this you should use a minimum dolly scale of "e" i.e. >=2.71828182845904523536 or so :) (reduce the camera distance accordingly - easiest is use 3 for the minimum dolly scale and divide the camera distance by 3).

Mandelbulb red trip is awesome. Like to see more iterations and detail though, even at the cost of rendering time ;)

Really looks like snakes curled together in the deep dolly zoom.

Indeed, I am quite happy with this red trip. Actually it calculated much faster than I expected (less than 1 day in 1024*768) so I might fine-tune the trajectory and launch a higher quality version.

I noticed this behavior and gave up with this parameter, and I give it a value of 100 then I just adjust the camera distance, and sometimes a smal variations of the dolly to adjust the view. I would like to try a "true dolly zoom" (e.g the foreground does not move while the background seems to move forwards or backwards).

those videos are fantastic.

David, yours has the advantage, for me, to be closer to the 2d mandelbrot set deep zoom, i.e. you still more or less know where you are (or just have the feeling of it) even at the deepest point. it is maybe simpler but it's clear.

bib, is it a software/algorithm limitation or an artistic choice to have the details of the bulb getting more precise only after we are close enough to see that it's still "whipped cream". no offense here, but i'd love to see the bulb in its full precision all the way through the trip.
maybe, if it's technical, and if the camera path is know before hand, having the precision render pre-calculated along the way (i don't know nothing about software and limitations, so it's just a remark from a enthousiast follower :D )

I like David's zoom for its precision but I find it lacks of some perspective (a bit like zooming in a giant 2D image)

I lower precision on purpose to optimise calculation time, but I agree, I should not do it in the beginning where the render time per frame is still reasonable. Let's say it's an "artistic" choice;) it's not a matter of software limitation.

After some exploration of the Mandelbulb, I am a bit disapointed. Let me try to explain, and see if I'm right or wrong.

In the 2D M-Set, no matter where you zoom in at the border, you will always find a minibrot (if you know where to search it's easier). It means that no matter where you zoom in (e.g in the seahorse valley), you can still zoom in a minibrot's elephant valley for example, thus combining both regions' patterns in an intricate way.

I'm not sure if it's the same in the Mandelbulb. I *feel* that self-similarity is "stronger", meaning that once you have started to zoom in a particular area (they don't have "official" names like in 2D), for example the "crater" or "tower", or "bulb", or "cave", you have to stick with the patterns you find. For example if you start to zoom at the crest of a tower, you will hardly find minibulbs like if you had zoomed at first in a minibulb.

Don't know if I am clear enough. Any opinions?

That wouldn't be a "dolly zoom" - dollying is when the camera is moved (in/out), what you're suggesting would involve a change in the image plane distance.

This is exactly what I meant : I want to find a nice minibulb and then move the camera backwards while increasing the magnification (maybe changing the image plane distance as well, I'll have to test), so that the minibulb keeps the same size while the background changes.
Example :
http://www.youtube.com/watch?v=iv41W6iyyGs

You may be right about "minibrots" but to be sure of that more exploration needs to be done in those caverns where some of your animations go.
Remember where the minibrots are in 2D and then consider the locations of the 3D equivalents and how much surrounding "tendrils" etc. there will be :)
Also as to variation the same thing really applies, if you look at complex z^8+c you have to zoom in quite far (a lot further than z^2+c) to get recogniseable variation in the details and the same is true of the triplex version.
In short IMHO we just haven't investigated the objects in detail enough yet to give up on finding variety.

To be honest, apart from the areas of "whipped cream" the triplex z^8+c is exactly what I would have expected (and wanted) the 3D equivalent of complex z^8+c to look like :)

Actually I think you may be correct, you should be able to get that effect just changing the dolly and the magnification simultaeneously.

I did a quick test. This will probably be very difficult to find the right combination of parameters, linear/exponential interpolation, linear/curve variations at keyframes, etc...
Look how the bulbs in the foreground seem to go to the back and the bulbs in the back move to the front.
http://www.youtube.com/watch?v=iy3VRoTtSQ4

That issue will go away when you get the update which includes fogging to aid in depth perception ;)
Also with your current version if you change your light sources to point sources and use the "Squared Fade" and/or "Linear Fade" parameters correctly then you can at least get a fade to darkness effect to aid in depth perception :)

Note that using the squared fade usually means you have to set the light strengths a lot higher than normal.

hmmm.... what's the basic formula for both kinds of zoom?
The solution would be to "force" same optical distance to the central point for both zooms and then solve that equation to find the functional relationship between both kinds of zoom....

If you look at the Mandelbrot set, a lot of the results look like vortices.
2D vortices already are highly complex - even worse, in this case: vortices, made from vortices!

Now imagine a 3D vortex. And that with fractal repeated fractal detail. 3D vortices, made of 3D vortices.
That's just riddiculous.

There is a reason why the Navier Strokes formulae aren't yet solved after all...

Of course that doesn't have to be true but it might be the case that you have best chances to find 3D Minibrots by starting off the xy-plane, where the Mandelbrotset is found, zoom into a Minibrot there and then look at the surrounding 3D structure. In theory, that should also be a 3D Minibrot.

Also you could try to find the center of a Minibrot and then use the radius from the origin as fixed value and just change the angular position of the camera in order to find other off-xy-plane-Minibrots :)

Also a nice idea would probably be to look at cut-planes through the origin, being rotated around one of the xy-plane's axes.

Usual candidates for that would be angles like 15°, 30°, 45°, 60°, 75°, 90°, k*pi/n rad....

and if you find something promising (not nessecarily Minibrots but shapes that are similar copies of the whole plane you look at), you basically would look what happens when you slowly rotate the plane in either direction - when does the structure at that part go away?...

You could then use those min-, max-angles to cut a part out of the set that includes the whole smaller version.
That in theory should be one (kind of) Minibulb.

Most likely, however, unlike the Mset, those Minibulbs could look quite different to the overall shape. They're based off the plane their centers come from.
True Minibulbs "in respect to the whole set" would be harder to find and so to say a minority of those sets, found by the way I desciribed...

Do you think, that would work? :)

in this type of zoom i think camera FOV and distance sholud be in relation given:



where is camera FOV angle, is object width, is camera distance from the object, then

Yes that should work. A few weeks ago, I tried to do it in the 2nd order Mbulb, but results were not satisfying as you can imagine from the 2 pictures below, so I kept the cut-plane. I will try to do the same with higher powers.

(http://www.fractalforums.com/gallery/1/492_16_11_09_11_24_00.jpg)
(http://www.fractalforums.com/gallery/1/492_05_12_09_9_37_12.jpg)

Thanks gaston for the maths.

Dave, can you make the script ? ;) :D

the 2nd order probably is too asymmetric for this kind of stuff... :)
Already curiously waiting for higher order Msets :)

I just thought of a good analogy with respect to viewing the minibrots.

Take the complex z^8+c Mandelbrot and we can easily see the minibrots because we are 3D creatures and have a view of the entire plane.
Now imagine you are a 2D creature living somewhere on the plane "outside" the Mandelbrot and the Mandelbrot is rendered in the plane (say with colour for depth cueing) and you have the job of looking around to find and view the minibrots.
Essentially the task we face to find and view the 3D minibrots is the same.

that's why 2D slices of different kinds could come in handy :)

if the 2D-creatures could cut lines through the 2D M-set, they could under circumstances find out where such Minibrots are.

However, it would become totally mindblowing if they had to search through the 4D-set....... :)

I agree with you both and I found kram's analysis and explanations very relevant and convincing. And I realized just by looking again at the 2D z^8+c that the surface of the Mandelbulb is a totally crazy 3D maze as you approach the surface and follow the curves, the hills and valleys turning in all directions in 3D in a crazy dance!

So as suggested by kram, using a cut-plane, I focused on a "simple" and very symmetric minibulb: near the top of the main spike. But when I slightly move the cut plane, as you can imagine, it's just a mess, there are plenty of threads everywhere and it's not nice looking.

So I tried to use a spherical clip to show only the central part of the minibrot (not all the filaments), but I face 2 difficulties: 1 - finding the exact center coordinates of a minibulb is impossible to me, and 2 - I have frozen Dave's formula in UF just when I unchecked the cut-plane to see the final result. Sorry Dave, I did not save the upr and I can't replicate, I can't even remember how I did to do a mini-spherical clip on a minibulb.

So far, the best image I got is not much different from the blue image above, with a z^8+c instead of the z^2+c. I don't know what to do next. Maybe I should find how I did the mini-spherical clip, but anyway, I feel that it will not be what I'm looking for.

maybe it helps to go inside the minibulb and look from there.
It might be easier to recognize certain details as the mess of dentrites around the minibulb wont show up that way...

Also try potential Minibulbs from different cut-planes :)
Maybe for some strange reason a different Minibulb will have less mess around it...

Yes, it's true that the holy grail would be 1000x better still - that's why we're still searching. I see what you mean, the Mandelbulb is (almost, but not quite) like an organic version of the Menger Sponge, where patterns are pretty similar after zooming in.

But I still think we have a few surprises left, particularly on the inside of the bulb which has been hardly explored yet.