There's an article about the Rieman Zeta function (and other strange functions) https://arxiv.org/abs/1103.5274 whose parameter plane contains copies of the quadratic Mset. I don't know whether those themselves show minibrots to any depth, but if so, one might store information in a function that is inherently correlated to primes - quite interesting.

Main purpose of my https://fractalforums.org/fractal-mathematics-and-new-theories/28/holomorphic-mandelbrot-extensions/1475/ thread was to answer this question, but no-one has ever come up with something like M-set shape stacking.

AFAIK this is true: For holomorphic iterated functions of n variables, only n=1 produces this effect and there only for \( z^k+c,\
 k>1 \). For non-holomorphic 2D fractals (\( z \leftarrow f(z,\bar{z}) \)) I think some form of path encoding is present (simplest case: Mandelbar \( z \leftarrow \bar{z}^2+c \)), but I think this is usuall "trivial".

What do I mean with "trivial"? If you look as an example at a general (complicated) rational function of 1 variable  and start zooming (skip thinking about critical orbits etc) you'll find Mandelbrot minis. They have slightly different decorations than in the true set but you can do zoom and shape stacking as in the original M-set and get almost identical results, differing only in decorations. I call that "trivial". The 3d order polynomial in Kalles Fractaler (called redshifter) is like that.

Even functions  like \( \frac{z^2+c}{z^2+1} \) have Mandelbrot minis if visualized with pretty much any method, though nothing ever escapes. And you can zoom on those and do all the shapestacking tricks you know for the regular M-set and they'll work. See attached example, which is obtained by zooming in an apparent M-set mini, finding Julia etc. But nothing escapes in this image, so how it knows about M-set is not known to me and maybe even God doesn't understand why.

Non-trivial fractals that show shapes unlike M-set exist (general real 2D in KF for example) but none of them have shape stacking like M-set.
And >1 complex variable fractals are different (some) and have no M-set minis. But they also are not zoomable: nothing really new is discoverd by zooming.

Zoomable means there is something to be gained from being added to KF for zooming very deep. In M-set you can use shapestacking, which I'd prefer to call "shape weaving" as it's more like make a sock from a tread. Randomly making knots in a piece of wool string won't do. But if you know how, you can make a sock. Or a hat.

M-set is like that. Randomly zooming will show the self-similarity but you'll never discover a new shape, just variants. Shape weaving does give you results which have a structure that was designed by the user. A knot of the knitting needle here is a zoom. Only in KF with NR zooming you can do this really, unless you have more patience than I. (In my M-weaves I have some other auto tools like zooming on Misiurewicz points as well.) We can make sweaters, socks, and even octopus pants using wool and knitting, but how to do so is not so easy. Similarly various M-set knitting techniques have been developed over the decades, most complicated probably Dinkydau's "evolution" method.  Simplest "spectacular" technique I find is the "tree-building" (prerequisite for "evolution") which involves more than just knowing that "zoom path is encoded", it also reveals aspects of how it's encoded, esp. how bifurcations can be used as a construction tool.

@pauldelbrot Thanks for that. Yes, thinking of \( \infty \) as just another point which may or may not attract reduces my surprise at finding the mini Mandelbrots in such formulas as \( (z^2+c)/(z^2+1) \) (c=pixel) which is a simple one showing minibrots using pretty much any nontrivial orbit-->colorindex method.

I understand everything you wrote  but it somehow doesn't explain it to me. Maybe I'm asking too much, or maybe it means I don't really understand it. For me, I would say I understand it if I can do like in the Mandelbrot set, write down equations for an attractor and get a formula for the shape of the main cardioid and the primary bulb (circle) and write down equations for higher order minis which can't be solved analytically but they determine the shape.

Never mind what I don't understand, pauldelbrot do you agree \( z^k+c\ \ k>1 \) is the only iterated function which features shape-stacking? I think for generic 1 complex variable functions the answer is that usually the derivative gets to be zero without second derivative being zero unless you conspire to make that happen, which results in regions that see approximate quadratic behaviour (Mandelbrot set) and linear regions which is unstable stuff. You see mutilated minis as some parts of it are affected by "other terms".

Do you know why we can't repeat this using 2 or more complex variables and get different shapes from Mandelbrot set which can also be used to weave images in various forms? There seems to be something magical about complex numbers, but what? Quantum mechanics for example needs complex numbers and could not work without it (Penrose's books reiterate this surprise as the magical properties of complex numbers).

"Escaping" is really just a special case of "converging", in this case to a point attractor that happens to superattract and oh, it's also at infinity.

Good insight, thanks for sharing!

On minibrots: Well, shape-stacking will only occur around the site of a bifurcation cascade, as it's the period-doubling (or tripling or more) around it that makes the stacking possible.

not only!

This also works for the bulbs, not just minibrots. So for example, you can zoom into a bulb of a bulb of a bulb etc and it will store the unique location in the attached tendrils in a sequence, without ever visiting a minibrot.

Didn't notice that until Tavis pointed it out.
So there's two kinds of shapestacking Juliastacking and Bulbstacking.

Then again, I wouldn't say the stacking happens at the bifurcation points. The process of "stacking" happens all the time when you zoom. Bifurcation is just where the stacking becomes visible and repeats doubled.
We shouldn't confuse this with the shapestacking that results in complex juliasets, using shortcuts in the juliasets to make them grow more and more complex (like in Kalles Julia animal series back at ff.com)


Gerrit, I like the term shape-weaving. It definitely makes sense of you actively use "stacking" to create desired shapes.
But I still think we need a general term for the fact that basically every pattern in the mandelbrot-set is caused by the decision where you zoom to and that your zoom path is stored around bifurcation, similar to dna.
Is there a commonly used term for this already? That something I wanted to find out the 3rd question in first post too.


Maybe the answer to my first question (is information storage a feature of all escape time fractals?) is already hidden in your posts, but there's too much math-talkl I don't really get.
Would you guys mind to boil it down for me? 

>>"There seems to be something magical about complex numbers, but what?"

Complex numbers provide a concise way to describe all smooth mappings that don't produce whipped cream. Suppose we are looking at z->f(z)+c iterations (z->f(z,c) doesn't seem to add much). "Whipping" the cream means stretching some directions more than others. We can consider the Jacobian (J) at a point, i.e. d(z_new)/d(z_old), which measures the local amount of stretching/shearing/scaling/rotating at each point. For 2D this is a 2 by 2 matrix with the real and imaginary axes playing the role of the x and y axes.

What do these Jacobians look like? For z->z, the identity, J is the identity. From this we build up. For z->z*z, we multiply the complex plane by z. Following the rules of complex numbers, we get J=[2r -2i; 2i 2r], with r and i being the reals. This matrix applies a rotation and a scaling, it has no shearing or stretching. What about z->z*z*z? We apply a multiplication twice, which applies this rotation + scale twice; still no whipped cream. For polynomial functions the overall Jacobian adds the Jacobians from the linear (a simple scaling), quadratic, cubic, etc terms. Adding rotation + scaling matrixes is still a rotation + scale. The + c term simply adds the identity. As polynomials are universal function approximators, so long as the function is "analytic" (i.e. not piecewise defined) and doesn't use conjugates any function will be safe from whipped cream.

The dose and the poison. If the function isn't smooth everywhere, i.e. z^2.5, branch points will break up the fractal and any zoom of. If the function uses reflection, the C term will cause a minor whipped cream in places because an identity + reflecting rotation will distort angles, but for most iterations the +C term will add very little distortion. This opens up a lot of "shape weaving/stacking" possibilities such as the burning ship which can be deeply explored. There are many ways to piece together reflections of the complex plane through the origin making non-trivial deep-zoomable fractals. However, if the distortion is in the f(z) term we will get distortion stacking which builds up very quickly and makes deep zooms hard.

Higher dimensions are barren. Conformal mappings are only interesting in 2 dimensions. In 3 or more dimensions they are limited to Mobius transformations: rotation, translation, scaling and 1/z inversions. It's true that complex z*z only rotates, scales and translates on a local scale but it still is more "interesting" on a global scale. In 3 or more dimensions, such as the quaternions, even z*z causes, on a local scale, shearing/stretching and the whipped cream we see. Mandelbulb lives with whipped cream and the mandelbox combines a burningship-esque reflection trick with Mobius transformations. No shape weaving in the mandelbox, most "exploration" is shallow zooms as well as parameters, lighting, etc. So having multiple variables isn't helpful in the land of analytic function deep zooms, unless there is a clever slicing trick I am missing or something.

Julia shape weaving plays with the root. Passing near a minibrot to get an embedded Julia means passing near a root of quadratic or higher order. Roots crush the constellation of iterating points downward, allowing the +C term to matter as well as folding it up; both are very important for shape stacking. If we restrict to the smooth analytic functions without branch points, we have polynomials as well as essential singularities to play with. "But essential singularities aren't roots!". They still can have regions that act as roots, and if we pretend f(0)=0 we can get interesting pictures in some cases. Here is an example of shape weaving using an embedded Julia's edge as well as it's spiral for z->z*exp(1/z)+c:

Which is what I tried with this thread, but still have no answer to the initial questions, so here we go again:

1. Is this information storage principle true for all escape time fractals?

2: Is there any other "class of fractals" known to behace in a similar way? does this happen in 3d in the amazing box for example?

3rd: how common is that knowledge of information storage? does anyone in the mathematic/scientific community put any importance into the fact that the mset (and other fractals) are capable of storing information?
To me that opens a huge world of new important research. (In short, if fractals can store information just like DNA, this ties fractals with evolution and adds to  a big fractal cosmology picture)

Grossly simplified:
1. No. 2. No. 3. No-one cares.

Caveat: Information storage is too vague to say much about IMO. A zoom into a blank picture stores as much info as the M-set: just use digits of coordinates to encode whatever you want. Shape stacking is a different thing altogether. Don't know if it can be even defined precisely.