infinite zoom of the mandelbrot?

[Tglad]

We can draw fractal zooms up to e^120 or so, heck, why not an infinite zoom? [Edit: I mean a rendered picture at infinite depth]
This first 2 pics below are infinite zooms into a manger carpet and a sierpinski triangle.
So is it possible to render an infinite zoom of the mandelbrot set? depending where you zoom it might look like this 3rd pic where the base disk that it sits on is so large as to be a straight edge.
I don't know how you would calculate such a render, or if it is possible, but it might open the doors to a 3d version... I'll explain my thinking...

I've often thought that trying to make a 3d version of the mandelbrot is like trying to make a 3d version of a hexagon... it seems plausible since there is a 3d version of triangle, square and pentagon, but for a hexagon it simply doesn't exist, end of story.
Well then I realised that there actually is a 3d version of a hexagon, it is just infinitely large since hexagons fit together with a curvature of 0 (i.e. on a plane). Another way to put it is it creates a sort of sphere with size 0 hexagons. An infinite zoom into this would look like the 4th pic, a 'hexagonahedron'. (In fact we can add two other infinite platonic solids to this list, a 'trianglahedron' (triangle tiling) which is the compliment of the 'hexagonahedron' and a 'squareahedron' which is a square tiling).

This analogy with the mandelbrot is interesting because 3d mandelbrot attempts suffer the problem that any rotation applied in the formula will break conformality when the +c part is added. But just as the hexagon never rotates in the 'hexagonahedron', certain infinite zooms of the 3d mandelbrot would not require any rotation. For example, in an infinite zoom of 1,0 of the mandelbrot the ^2 operation just becomes a scaling by 2.

Anyway, regardless of the third dimension, does anyone else think an infinite zoom of the mandelbrot set is possible?

[Sockratease]

 

[pseudogenius]

Well, this is my approach to an infinite zoom.

To view the mandelbrot set, we take all the points in a region, and iterate them, possibly infinitely.
For an infinite zoom, we have to take all the points in an infinitesimal region, and iterate infinitely.

So in other words, we take the region,

a₁ < x < a₂
b₁ < y < b₂

and to find all the points we have to iterate, we take a₁ & b₁, and add the infinitesimal increments, dx & dy, until we reach a₂ & b₂.

To view an infinitesimal part, we take a similar approach.

We take the infinitesimal region,

a - dx < x < a + dx
b - dy < y < b + dy

where (a,b) is the point which we want to infinitely zoom into.

To find all points, we instead will take (a - dx) & (b - dy), add instead (dx)² & (dy)², until we reach (a + dx) & (b + dy)

The iteration of these point should be handled by limits.

Although this depends on the infinite zoom and infinite detail of the set cancelling out to leave finite or infinite detail.

Hope this helps.

[reesej2]

My notion would be to use a nonstandard model of real numbers. In set theory, there's something called "R*", which is basically R with the addition of infinitesimal and transfinite reals. So what I would do would be take a section of the infinitesimals between, say, x and the sequence x + 1, x + 1/2, x + 1/4, etc. That gives us a region of infinitesimals to use as real and imaginary components. Now, infinitesimals are so nearby that after any finite number of iterations they are still an infinitesimal distance apart--so you'd have to iterate transfinitely many times. That's the roadblock I see--I'm not sure how to characterize transfinite iteration levels.

[kram1032]

No 3D-hexagon.. hmmm

Well:
While you can't directly create an object, that looks like a hexagon from more than one direction, you can extend certain properties to higher dimensions.
For instance, the close sphere packing

I just looked it up and the last time I looked, I heard, the closest packings where unknown above R³.... Seems like that's over: They're known up to R⁸...
http://mathworld.wolfram.com/HyperspherePacking.html
Not what you're looking for, right?

@reesej2: Are you taling about Surreal numbers (and Surcomplex for that matter)?

@ pseudogenius: nice I wonder if it's possible to do the limit of the whole set in one step with that.
Although... If it was possible, that would probably be against the beauty of the set.... Either that or it would involve complicated functions like the generalized Hypergeometric function or something...

Edit: Oh, I see. Over 3D, it's not prooven that they're optimal packings

[lkmitch]

We can draw fractal zooms up to e^120 or so, heck, why not an infinite zoom?
This first 2 pics below are infinite zooms into a manger carpet and a sierpinski triangle.
So is it possible to render an infinite zoom of the mandelbrot set? depending where you zoom it might look like this 3rd pic where the base disk that it sits on is so large as to be a straight edge.

I'm not clear on why you think that these are infinite zooms into a Menger carpet and Sierpinksi triangle.  In what sense are they infinite zooms?

To my mind, the Mandelbrot set is qualitatively different from these other two fractals in that it's not exactly self similar.  Consequently, I don't know how an infinite zoom (into a point on the boundary) could be computed, or what that even means.

[hobold]

The border of the Mandelbrot set has fractal dimension two. So one could argue that an infinite zoom either does not converge at all, or converges to an "all black" image with all visible points being on the border. I'm using the term convergence loosely here, thinking of an infinite series of ever deeper zooms toward an invariant center point.

[reesej2]

kram1032: No, I don't mean surreal numbers (though those would be excellent candidates as well). I mean the ultrapower of the reals over omega (which, for the non-set-theorists here, basically means taking infinite sequences of reals instead of individual ones, and imposing an ordering based on whether "most" elements of one sequence are less than the corresponding element in the other.)

[kram1032]

Ikmitch:
Those images of Sierp Triangle and Carpet are infinite in the sense of if you zoom exactly onto that point, the series will contine for ever and you'd have a loop.

That specific meaning of an infinite (looped) zoom, however, would hardly be possible with an Mset, being "only" pseudo self similar (which is exactly, where all its beauty comes from).

Reesej2: I don't think I know those and a quick search didn't yield results... Do you have any nice link for them?

[reesej2]

Here's R*: http://en.wikipedia.org/wiki/Hyperreal_numbers . For an explanation of ultrapowers: http://en.wikipedia.org/wiki/Ultraproduct .

[kram1032]

Ah, those

The phrase "reals over omega" confused me
Didn't know hte Ultraproduct yet, though.

[David Makin]

Ikmitch:
Those images of Sierp Triangle and Carpet are infinite in the sense of if you zoom exactly onto that point, the series will contine for ever and you'd have a loop.

That specific meaning of an infinite (looped) zoom, however, would hardly be possible with an Mset, being "only" pseudo self similar (which is exactly, where all its beauty comes from).

You can get almost perfect "infinite" loops out of the M-Set, at least by using a looped zoom into some of the spirals, I did one years ago but I don't know where it went - it's inifinitely lost

[Tglad]

I'll clear up my confusing wording, by infinite zoom I meant an infinitely deep close up shot.
The shots of the menger sponge and sierpinski triangle are infinitely deep because if I were to show you the whole object by getting rid of the picture boundaries then they would be infinitely large; I can continue to render with ever larger boundaries without ever drawing the whole object.

Of course they aren't the only infinitely deep shots, there will be one for every point on the full object, in fact, for every point there is a sequence of shots which repeat; a repeating animation.
An analogy is, what is a section of the digit sequence of 1000/999 infinitely far from the decimal point? Well its '001001001001' but its also '010010010010' and '100100100100'.
But what about non-repeating sequences (like the mandelbrot), well by analogy, what is the digit sequence of pi infinitely far from the decimal?
It might depend on whether pi is 'normal', if it is 'normal' then any sequence is possible, so '01234567890123' is a correct subset of the infinitely deep sequence, in fact any set of digits at all is correct.

So since the mandelbrot set doesn't really repeat, it might simply be the case that any render can be considered an infinitely deep zoom (e.g. the first pic).
A more concrete thought is to look for a converging pattern, e.g. approaching -2,0... if it is found that minibrots appear indefinitely as you zoom in then you could say that a snapshot of an infinitely deep close up can contain a minibrot... whether its tendrils converge would affect whether such a minibrot could be rendered in detail.
Perhaps a simpler task is to find an infinite zoom of a julia set, the julia at 0.25,0 probably looks like the second pic at an infinite zoom, since the boxes show how you could zoom in (and therefore out) forever without the sequence changing.

Kram, interesting thought about the hexagon, if a hexagon is thought of as the shape of the densest squashed packing of circles, then the 3d equivalent of a hexagon would be the octahedron, since that is what you get by using the optimal 'face centred cubic' packing and squashing it... incidentally octahedrons also have 6 corners.

[pseudogenius]

Tglad, on your second picture, each of the boxes contain smaller versions of the same image, and they seem to be converging to a point.
My guess is that an infinite zoom on that point would look the same as the images in any of the boxes converging to that point.

[lkmitch]

So since the mandelbrot set doesn't really repeat, it might simply be the case that any render can be considered an infinitely deep zoom (e.g. the first pic).

A more concrete thought is to look for a converging pattern, e.g. approaching -2,0... if it is found that minibrots appear indefinitely as you zoom in then you could say that a snapshot of an infinitely deep close up can contain a minibrot... whether its tendrils converge would affect whether such a minibrot could be rendered in detail.

Perhaps a simpler task is to find an infinite zoom of a julia set, the julia at 0.25,0 probably looks like the second pic at an infinite zoom, since the boxes show how you could zoom in (and therefore out) forever without the sequence changing.

Thanks for clearing up the wording.

In a strict sense, I don't think that there are any repeating cycles of images in an infinite zoom into the boundary, and here's my reasoning:  The center of each disk and midget corresponds to z = 0 after so many iterations.  Generally speaking, those components get smaller as the period increases.  This is also reflected in the numbers of spiral arms increasing as you dig deeper into M.  Alternatively, since the dimension of the border is 2, the level of detail has to increase with depth so that the area of the zoom would become more and more filled in as you zoom.  Either way, a traditional escape-time rendering zoom sequence won't repeat because it'll be increasing filled with detail.

Now, if you just want to look at a binary inside/outside rendering, then the Feigenbaum points might provide some hope.  In particular, zooming in around c = -1.401155189091968 on the spike should reveal a self-similar cycle of disks, representing a period-doubling cascade.

The tip of the spike at c = -2 can appear to be self-similarly barren, but what happens is that the midgets become small very quickly (as you approach the tip) and more isolated (on the scales of the midgets).

I don't think the Mandelbrot set is "normal," in the sense of your pi digit analogy.