It depends what you mean by point and line. If you mean the usual definitions in Euclidean space, then yes there is a line between any two points. If by point you mean a location in our universe, and by line you mean a geodesic that can be traversed by a material object, then no. For example there is no such path from a location inside a black hole’s event horizon to a location outside.
I think what the OP meant was moving the end of the gun by a planck length, so the angle change of the whole gun is the minimum allowable by physics.
Thinking about it, the issue is going to be that it’s not even going to be possible to rotate the gun, so as with all these hypotheticals it’s going to end up being whichever laws we are suspending for the hypothetical will give a different solution.
My best answer in the spirit of the OP would be, I think there are targets you indeed can’t hit. Think of the planck length like squares on a chess board. Are there squares on a chessboard the queen can’t hit from a particular start square? Yep.
Oh? What makes you say that? And what does that have to do with angles?
Can you please describe what physics leads to a minimum allowable angle?
Do you know what plank length is?
If you have a meter long stick that is fixed at one end, and you move the other end by a plank length, how far does a point at .5 meters along the stick move?
Hmm… So the answer seams to be to cut the barrel in half to hit the target.
It’s the length of a piece of wood.
The Planck length, on the other hand, means something else, but not what you apparently think it means. The Planck units are not the smallest possible values of each unit. The Planck mass, for example, is about 20 micrograms. There are macroscopic animals that weigh less than that. There’s no reason to think that there is any particular physical significance to the Planck length. It’s a more natural unit than the meter or the foot, but if a smallest possible length even exists (which is highly questionable), it might be greater than or less than the Planck length.
Doesn’t the same issue arise with this shorter barrel, just for a different target?
Yes it’s crazy. I can’t really wrap my head around it. That’s why I posted this thread. Lol
It’s turtles all the way down. The target of the gaps.
The point is, we don’t actually know that spacetime is quantized, and if it is, we don’t know how, or on what scale. It might be quantized, and it might be quantized in such a way that there’s a smallest possible length, and if so, the smallest possible length might be the Planck length. Regardless of length, there probably isn’t such a thing as a smallest possible angle, and if there is, we have no idea what it would be: To the extent that it’s meaningful to speak of a “Planck angle”, it’d be about 57º, and we’re absolutely sure that it’s possible to have angles smaller than that.
Cite?
Planck units are derived from the fundamental units.
The fundamental unit of angle is the radian.
One radian is 57 (and a bit) degrees.
So the Planck angle is 57 degrees.
This makes the same amount of sense as limiting movement by a Planck length.
The Planck length is also size of the smallest volume we can probe without requiring more energy than would create a black hole. That doesn’t necessarily mean it is the smallest distance you can move. Especially as we know physics is incomplete at these scales, and the apparent limit of the Planck length is really seriously violating the bounds of our knowledge.
There are interesting questions if you want to discretise space. One point than can be made is that there isn’t a limit in a discretised space defined by the voxel size. This is a common misconception. Look at digital graphic systems. We use and tender sub-pixel locations all the time. Same with audio. The lunatic golden eared brigade still occasionally froth at the mouth claiming that digital is a staircase and cannot resolve time periods shorter than the sample rate. Neither of which is true.
Cutting to the chase there is a common underpinning to all of these discussions which is looking at them in frequency space, aka considering the Fourier transform of the domain. A swapping of space sweeps a lot of these problems away.
I hesitate to mention it, but Heisenberg’s uncertainty principle is the binding idea for the problem here. And it derives from Fourier in a manner common to all the others above. (I hesitate because it invariably adds a whole raft of possible misconceptions.)
A random foam of Planck length scale bubbles constituting space or spacetime is popular. Albeit without much in the way justification. Reasoning about exact motion on the scale of the bubbles is going to be a statistical mechanics problem. I don’t think anyone thinks of spacetime as a discrete grid of voxels of the same size. Which is what a lot of discussions about Plank length limits seems to implicitly assume. Applying General Relativity to such discrete systems is left as an exercise for the reader.