A parametric curve covering "most of" n dimensional space

Question

I am trying to cover an n dimensional space with a curve. What I mean by that is that I would like to have a parametric and preferably smooth curve which is able to cover a significant part of n dimensional space. The goal is to sample data points from this space for a neural network. In 2d one good exmaple is: https://en.wikipedia.org/wiki/Rose_(mathematics) which can cover 2d nicely. I would like to have something like that in 6 dimensions. The curve below fills the space densely but it is not easily parametrizable. https://en.wikipedia.org/wiki/Space-filling_curve

Thank you in advance for any suggestions.

Answer 1

Because I don't know the details of your problem, this suggestion may be off-base:

Pick six positive irrational numbers $\alpha_{0}$, $\alpha_{1}$, ..., $\alpha_{5}$ whose products $\alpha_{i}\alpha_{j}$ and ratios $\alpha_{i}/\alpha_{j}$ are also irrational. Construct the points $$ x_{n} = n\alpha_{0}(1, \alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}, \alpha_{5}) \pmod{0.6}, $$ in the sense that if any coordinate goes above $0.3$, subtract $0.6$ to get back into the cube. The claim is, if the $\alpha_{i}$ are well-chosen, the resulting points efficiently sample points in your cube.

Strategies of this type are used industrially to numerically integrate functions of a large number (a least hundreds) of variables by "sampling points representatively in a cube".

The hard part is picking the $\alpha_{i}$. (In the case of "industrial applications", banks were hiring postdocs in algebraic number theory.) Because you're dealing with only six dimensions, it may be possible to pick naively. If this type of things looks as if it may suit your needs, I'd start with square roots of small primes, perhaps \begin{align*} \alpha_{0} &= \sqrt{2} - 1, \\ \alpha_{1} &= \sqrt{3} - 1, \\ \alpha_{2} &= \sqrt{5} - 2, \\ \alpha_{3} &= \sqrt{7} - 2, \\ \alpha_{4} &= \sqrt{11} - 3, \\ \alpha_{5} &= \sqrt{13} - 3. \end{align*} (Caveat: I'm not a number theorist, algebraic or analytic.)

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The geometric idea behind this scheme is to view your cube as a "$6$-dimensional torus", analogous to the screen of a video game such as Asteroids or Pac-Man: The right-hand edge is "glued to" the left-hand edge, and the bottom edge is "glued to" the top edge. In your situation, we'd make six of these "gluings", one for each dimension.

Having converted your cube into a torus, choose a "solenoid" $$ c(t) = (t, \alpha_{1}t, \alpha_{2}t, \alpha_{3}t, \alpha_{4}t, \alpha_{5}t) \pmod{0.6} $$ that "visits points of the torus as rapidly as possible", and sample along this solenoid at regular intervals.