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Time-like curves are points following a trajectory.
I realised they are different from spatial curves because their dimension can be negative.

We have to use the Einstein version of space-time (in fact Minkowski space-time), rather than Newton's for this to work. In this space-time a square distance (or period) in time is negative, whereas is it positive for a square distance in space. Consequently the proper time period of a trajectory (the time experience by the moving point) actually gets smaller as you add more iterations, if this happens quickly enough you can get negative dimensions:

This curve is the sibling of the Von Koch curve and the Weiserstrauss function. They differ by having one dimension who's square is negative, positive and zero respectively.

The smaller dimension trajectories look like they have no structure but that is because the point is travelling close to the speed of light so we can't see the bends and curves. If we animated the future trajectory depicted from the point's perspective, I think you could see a lot of twists and turns.


If the point is moving in two spatial dimensions we can get a different modification of the Von Koch curve in the two spatial dimensions this time:


As the dimension tends to 2, the shape tends to a smaller and smaller 90 degree Koch curve (solid diamond), as it tends towards -infinity the shape tends to a single vertical reflection. But it is worth stressing that this is not a shape, it is the trajectory of a point going at light-speed, in fact it is the motion of a photon, so it is a bit like the internal scattering process when light reflects off something.  

This diagram is helpful...
When you rotate a spatial axis y towards another spatial axis x, the vertical component decreases (it rotates down).
When the vertical axis is time the equivalent of rotating is changing speed. According to Newton when you change speed the vertical component doesn't change.
But in fact (and according to Einstein) when you change speed the vertical component increases.

This is precisely the reason when a Koch curve gets longer in each iteration, the Wieserstrauss function time period stays the same, and my time-like curve's proper time decreases each iteration.
This is why spatial fractals increase in dimension with roughness, but spatio-temporal fractals decrease in dimension with
roughness (using Eintein / Minkowski space), or don't change dimension with increasing roughness if using Newton's space-time.

I think it is a photon. The fractal gets closer and closer to light-speed each iteration. So the exact fractal is either light-speed or infinitely close to light speed. So we can think of it as a single photon bouncing around along the shown paths.

It is basically a Koch curve formula but the constant bend angle is replaced with a constant Lorentz boost angle in the 1+1D case. Note that in space-time you don't write 2D as temporal dimensions are different to spatial ones as you see in the diagram, so you write 1+1D, or for the real world it is 3+1D. Meaning 3D spatially and 1 time dimension.

and here's a final example, which is everywhere C(1) smooth spatially, despite having a fractional timelike dimension:


(by a+bD I mean its spacelike extents are aD and its temporal extents (measuring proper time) are bD).

What definition of dimension are you using? I don't see how you define a negative Hausdorff dimension or a negative Minkowski dimension.

Negative interval doesn't necessarily mean a negative dimension. Since a light ray has zero interval, it can be covered by a ball of radius 0, so the dimension should be 0. But you say it's -1?