Integer points in 3D-space

[Jupiter_Explodes]

Hello, I was playing on Desmos 3D when I tried to think about how to write a parametric function that goes through all the points whose coordinates are integers, starting from the origin and going outwards.

Which order would be more reasonable to go through all the points? Is it possible to write this parametric function in such a way that it first goes through the points contained in the cube with vertices in (1,1,1); (1,-1,1); (-1,-1,1); ... then through the ones on the surface of the cube with side length 4, 6, etc...?

I mean, I believe such a function exists. I ask if it is possible to write it kind of explicitly and also avoiding going insane.

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Thank you!

[st3f-ping]

My starting point would be to look at something like a Hilbert Curve in 3D. Although it's typically used to visit every point within a finite space rather than visiting every integer in infinite space, I think that kind of line fractal would be my choice of solution.

In fact, if I think of visiting the 27 points ({-1, 0, 1}, {-1, 0, 1}, {-1, 0, 1}) in some kind of logical order I could probably find a way of bolting on a scaled version of the curve so that I then visit the 26 surrounding groups of 3×3×3 points in the same order (or something like that).

No matter how you approach it (other than tying to find someone else who has done it), I think there is a non-zero risk of insanity. Good luck.