Vicar – Multi-dimensional image processing software

If it has one property that can change over time it's no longer a dot, but a line moving in the time dimension

the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

A space consisting uniquely of one point is thus zero dimensional since you don't need any number to figure out how to locate such point; by definition such space contains only one point.

Dimensionality is a property of a space, not of an object.

If you draw a curve on a piece of paper, is the curve 1d or 2d?

If depends on the space you choose to ask this question in! And for that question, there are two interesting spaces: the paper (where you view each point as having a position in the paper and two curves with a different shape would be very different curves) and the curve itself (where points along the curve are identified by the "linear" distance from one of the edges). There are practical examples, e.g. when you have an accident in a highway chances are it's easier to report your location using the km/mile position as reported on signs placed at regular intervals along the road (rather than your longitude/latitude).

W.r.t the "void": I'm not sure exactly what do you mean. I can think of two meanings of the word: the parts of a space that are empty, i.e. devoid of some "things"; or a space without any state at all, a pure "nothingness" without any property nor anything in it that could have any property.

If you're thinking about the first meaning of "void", then consider what we said earlier about dimensionality being a property of a space not of a location within that space. If you view each point of this void independently, then each of such positions is a zero dimensional point. But you can also say that there is a spherical region of empty space (and call it a void) and such space would be a 3d space (consisting of void).

To flesh out the sibling comment with formality: the periodic table of categories [0] gives the zero-dimensional objects as "trivial". This isn't to say that they're empty, or just points, even though those are both reasonable interpretations. It means that there's not really anything predictable or deducible. The number of deductions is zero, but also, the dimension of the largest possible proof is zero, the dimension of the largest possible diagram is zero, etc. There's not just nothing there, but nothing constructible.

Nonetheless the trivial objects exist, which is enough to allow us to draw arrows to/from them, and those arrows are not zero-dimensional. So these trivial objects show up as trivial endpoints in situations where there's only one of something.

[0] https://ncatlab.org/nlab/show/periodic+table