Well, for those interested in black holes and fractal geometries... they probably thought that I had something like this in mind:
https://www.researchgate.net/figure/Frac...g3_4041576
However... not exactly.
The problem with that kind of spherical fractal geometry is that points at different levels of
granularity will
overlap.
That is, if you imagine that same sphere increasing its size 2 times,
without changing the "size" of the fractal pattern on it (which is mathematically equivalent to increasing the fractal granularity by a factor of 2 while
not increasing the sphere's size), then all the points at the "corners" of the fractal pattern (the seed) will overlap at those two granularities.
What I actually had in mind was (probably, as I just skimmed the title of the paper) closer to this:
Variable Granularity Space Filling Curve for Indexing Multidimensional Data
https://www.researchgate.net/publication...ional_Data
In the case of the black hole model I'm working with, this 'variable granularity' would take the form of (the only one I could find, but there may, jut possibly, be more):
a) Assume that space has a cubic structure, i.e. points (actually,
knots would be a more correct term) of space are located at the centers of the sides of a cube. Since cube has 6 sides, that means that 6 points will form the seed of the fractal pattern.
[the reason a) is assumed is because we're trying to
pack those points onto the surface of an
expanding sphere (black hole), and this particular fractal pattern, described below, starts with 6 points as its seed]
b) Start at any point on the sphere (point #1), and follow these rules to determine where the next point is located (remember, all movements are done on the
spherical surface):
b.1) Turn
right and move by 90 degrees (point #2),
b.2) Turn
left and move by 90 degrees (point #3),
b.3) Turn
right and move by 90 degrees (point #4),
b.4) Turn
left and move by 90 degrees (point #5)
b.5) Turn
right and move by 90 degrees (point #6)
b.6) And this is where the trick that makes this (variable-granularity) fractal work comes into focus. Because, if you tried to follow the right-left pattern of turning and moving by 90 degrees, you'd end up at the point #1... and that is
not what we want here. We want to avoid any and all overlapping of points as we
pack more and more of them (as the sphere expands), and that is why we are going to...
Turn
left and move by
45 degrees,
b.7) Turn
right and move by
45 degrees,
... etc...
Keep repeating the right-left pattern of movement by 45 degrees until your final move is about to take you back to a point you've already visited. At that moment, half the angle size, and keep repeating the movements again until you encounter a previously visited point (again), then half the angle once again, and just keep on moving all over the sphere, repeating this fractal pattern... for all eternity (or however large the sphere gets)...
without overlapping any two points at any moment, and with all the points of 3D space being nicely
packed onto a 2D surface... close to each others in
both frames of reference, but... not exactly
next to each other on the sphere. More like... slightly "garbled up" (that is,
interleaved)... Hey, just like on a hologram!
If I got this whole
packing process right (and it becomes truly impossible to picture it in one's imagination beyond the first 6 points, at least without any help from a computer simulation), you should end up with the whole "interior" space (all the points that used to be "inside" the sphere) perfectly
packed on its surface.
Of interesting note here is the fact that
connections (paths) between the points play no role in the
packing process, since they can be abstracted away as belonging to a higher dimensional "space" (or, ultimately, to an
infinitely divisible, or a
dimensionless one).
The only thing of importance in an
N-dimensional space are its
points (or
knots), and
not the actual
connections between them.
P.S.
It's quite possible I've made a mistake or two while explaining the
packing process, but you've undoubtedly got the general idea of how it should work.