Is the Mandelbrot set sufficiently self-similar to non-rigorously visually recreate procedurally from warped, nested structures?

Question

While not strictly self similar, the Mandelbrot set shows significant similarity in its pixelated visual samplings. Are the overall shapes approximating each region well understood enough, not in a mathematically rigorous way, but visual way, to roughly reconstruct what a given bounding box will contain?

For example, a bulb in the elephant valley will have as many trunks as its order, with spires coming outward at increasing angles, each of which will contain a bulb surrounded by…and so on. Does a pattern like this occur predictably enough all the way down that zooming could be simulated on a procedurally-generated fractal, only the procedure isn't calculating whether iterating a point escapes to infinity, but a more mundane warping, joining, and nesting of familiar shapes like bulbs, trunks, and seahorses?

Are we ever "surprised" by the general shapes present in a region, or is it always determinable from the highly regular patterns visible surrounding the region? Does the structure at smaller scales repeat in a predictable way as you zoom in? If it is surprising, is it because the xth arm of a seahorse lined up with the spire from a y-order bulb, and these numbers approach infinity, or for another reason?

This is not a question about the provable structure of the fractal boundary or even so much the fractal itself, but whether a clever enough visual procedure (like the one that creates the Koch snowflake) could nest and warp known patterns to fully approximate arbitrarily zooming in on the Mandelbrot set, producing similar pixel-sampled outputs, without any actual calculation of any points' inclusion or exclusion from the set.

Either such a procedure is possible and with significantly less complexity than calculating points, or possible but the complexity quickly approaches that of calculating individual points, or it is impossible for reasons of some kind of irrational sequence of what structures appear next in nesting. Which, if any of these, are the case for why a visual, procedural prediction could or could not work?

Answer 1

(This answer is more from a fractal artist's perspective than a mathematician's; more how than why...)

The Mandelbrot set is predictable

Given a high-enough resolution image of a location, the coordinates and zoom depth can often be reverse-engineered:

• Finding the location of an image of the Mandelbrot set
• How can I reconstruct a Julia set by a given image?
• https://mathr.co.uk/web/m-location-analysis.html#Mercator-Mandelbrot

Similarly, if you have an idea of what you want an image to contain, you can figure out where to zoom.

Zoom videos are highly predictable too, after going off-center towards a baby Mandelbrot set, everything seen so far repeats again, doubled up rotationally and twice as fast (repeatedly, until infinite wrapping at the baby Mandelbrot set). This can be seen in the repeating motifs in an exponential map transformation. Probably room for some pixel copying acceleration here, though near embedded Julia sets and baby Mandelbrot sets the tiles are warped from their rectangular shape. The recently developed acceleration techniques for Mandelbrot set iteration may make this less relevant.

Julia morphing can be accelerated

Julia morphing artistic zooming technique exploits the fact that when zooming "off-center" towards a point, about half-way to the next baby Mandelbrot set is an embedded Julia set with 2-fold rotational symmetry, that looks like the previous "off-center" location wrapped around the origin twice (with some conformal distortion). By choosing the locations to go off-center, shapes can be "sculpted" out of the boundary of the Mandelbrot set.

Inflection mapping technique creates similar shapes to Julia morphing without zooming, instead of one high precision center point $c$ it stores a list of points $c_k$ about which the shape is wrapped successively, using something like $z_{k+1} = (z_k - c_k)^2 + c_k$, with $z_0$ as image coordinates.

Caveat: the dense rings of decorations that you get from Mandelbrot set zooming are missing in inflection mapping, you only get the central shape.

Caveat: you can use inflection mapping to prototype Mandelbrot set deep zooms, though in the Mandelbrot set baby Mandelbrot set locations are quantized so you can't recreate all inflection mappings exactly.

Inflection mapping software

• DinkydauSet's Explore Fractals
• my own Inflector Gadget

Example

A shape with three things in inflection mapping (no decorations):

Three (different) things in a Mandelbrot Set deep zoom (dense rings of decorations):