0, 1, 2, 3, 4... - linear scale.
1, 2, 4, 8, 16... - logarithmic scale.
0, 1/2, 1, 2, - the hell is this one ?
Because everybody is like "this is totally legit, but idk how it's called or what's the general rule ".

More generally:
0; n^0;
0; n^-1; n^0; n^1;
0; n^-2; n^-1; n^-1/2; n^0; n^1/2; n^1; n^2;
0; n^-3; n^-2; n^-3/2; n^-1; n^-2/3; n^-1/2; n^-1/3; n^0; n^1/3 n^1/2; n^2/3; n^1; n^3/2; n^2; n^3; (not sure for this one)
Add negative numbers on the left for full scale.

And so on

If you scale your X;Y coordinates like this, 1/x looks like a straight line between 0; and ;0 and x^2 looks like a -sinx rotated 45 degrees around origin.

Also, it's very symmetrical: if you multiply numbers equidistant from 1 or -1, you get 1; if you multiply the exponents of numbers equidistant from n or -n, you get 1, if you multiply numbers 1/2 of scale away from each other (full scale includes negative numbers) you get -1 !

So, who knows how that's called ?

I'm not sure what the scale would be called, but it sounds very related to stereographic projection and the Riemann sphere, which are associated with the tangent function.

Let's say f(x) = tan(x * pi/4). Then



This satisfies the equation f(1 + x) * f(1 - x) = 1, except where that's undefined.

But... it sounds like that's not quite everything you're looking for.