[3blue1brown]

The original pattern is good for going "out-to-in", in the sense of starting with a finite area, a square, and filling the whole thing.

That same pattern cannot apply to go "in-to-out", as in starting with a unit square and trying to go to all of space.

You might think you could have the Pseudo-Hilbert curve pattern fill 4 unit square, then 16 unit square, then 64 unit square, etc. However, no proper limit curve would exist in this case, since each specific value on the curve tends to diverge to infinity.

[pcmaffey]

So then it's not really able to define infinite space so much as infinitesimal space. Going from 1 to 0.

On a side note, the Hilbert Curve pattern quite resembles the folds of a brain. Which makes me wonder about the attributes such a pattern would lend our brains:

1) the ability to hold fixed points in space relative to each other while increasing information density between those points, and

2) our ability to stand on the edge of space (reality), and look / measure inward (1 to zero), without seeing infinity looming behind us.