Is there a mathematical formula that can make all pools balls....

[Dr.Arthias]

@Dr.Arthias

That video is so obviously fake.

…wanted to edit my post and explain why instead of just making a claim, but it wouldn’t let me.

We can even completely ignore the fact that the person making the break shot is employing far too little force for the amount of energy that follows…

because if you pause the video during the moment of contact, or just keep your eye fixed on almost any color you choose, you’ll notice that the direction in which some of the balls take off in is not realistic when considering their starting positions. In some instances, they are even switching places entirely (like the 3 going where the 5 should, and vice versa).

[Dr.Arthias]

I’ve been playing pool for several years; it’s one of my favorite hobbies. If the video was real, you would be able to make sense of where the balls are going. Another giveaway is that no one in this seemingly crowded bar/pool hall seems to care, not even the shooter themself, as if this was some everyday feat.

Then there’s the fact that the rack is numerically organized 1-9. No one who would have the patience to attempt this the number of times it would take to succeed on video (assuming that it’s even possible) would bother to re-rack them this way every time. The 1ball goes in front, and the 9ball goes in the middle. The rest would likely be completely random if this was real. Just further evidence that it was likely an amateur who used some fancy editing and thought the balls should be arranged in a “pretty” and orderly way.

[MikeS]

@septimus

How many degrees of freedom does a cue stroke have? I count five:

Orientation of cue stick and cue travel direction: horizontal angle, vertical angle.
Point of contact with cue ball: 2 parameters.
Cue momentum: 1 parameter.

This assumes that the cue stick velocity is in the same direction as the cue length, and that any acceleration is irrelevant. (If the acceleration affected the cue ball, wouldn’t it be an illegal double cueing?)

It also ignores shape of cue tip and amount of chalk, but I assume these are fixed and set to maximum chalk respectively. And I’ve combined cue stick speed and mass into a single momentum parameter. Is that OK?

But I’m not a real billiards player. What is the correct answer? How many parameters does it take to model a billiards stroke?

You can also look at this from perspective of what the cue ball is doing immediately after the stroke. There are definitely five parameters here, assuming that the ball doesn’t leave the table: linear velocity in the horizontal direction (2 parameters), and angular velocity (rate of rotation about each of three axes, roughly speaking.) So even if there was another parameter that you could modify about the cue stroke, you couldn’t actually make the ball do anything different.

(Note that I’m accounting for the English on the ball by allowing free rotation about all three axes. If the ball is rolling without slipping, then the angular velocity is completely determined by the linear velocity.)

[pulykamell]

@CookingWithGas

My eye must not be as good. I’m not challenging you but what are the signs that this is fake, other than that the feat itself is so difficult/dubious?

Without going Zapruder film frame-by-frame about this, just the fact that there is no reaction whatsoever to the break by anyone in the bar or the shooter himself is enough to ping the bullshit detector all the way to 10.

[Leo_Bloom]

@Trinopus

Also, such a mathematical formula would depend on perfectly spherical pool-balls, a perfectly flat felt table with a perfectly consistent coefficient of friction, no eddies or air-currents, no vibrations through the floor, etc.

Otherwise, you wouldn’t have a “formula” so much as a supercomputer simulation, like a Cray modeling the temperatures inside a hurricane.

You know, on a more approachable and well known level, the analogy of OP is understandable with what most everybody learns what a Really Big Computer is for (and historically is a fact for technological development):* simulating atom bombs, with all those billiard balls and a lot more than felt and frictions even after the “break” of fission.

*Although in fact I believe that Really Big Computers, depending on definition of big, became a government-funded (i.e., getting to “Really Big”) for cyptography in WWII.

[dracoi]

@Dr.Arthias

The answer to that question is actually yes. It fact, at most angles, it doesn’t take a lot of bounces for a pool ball to find a pocket, even without infinite motion. On a typical table under normal conditions, it’s difficult to make contact with the cushions more than 6 or 7 times, no matter how hard you hit the ball, but if friction was reduced even by just half, I think most shots (at least, those that are more than a few degrees away from being perpendicular) would probably sink in fewer than a dozen rebounds.

With ZERO friction/infinite motion, the chance that one of the balls would fall into a pattern of bouncing exactly perpendicular to two opposing rails is extremely low. Slight imperfections in the cloth, rubber cushions, and the balls themselves would make that pretty much impossible, even if the ball had initially been hit perfectly straight by some kind of specially calibrated machine.

Well, yes, if we kept all the other imperfections of the materials and merely eliminated friction, they would eventually leave a perpendicular path. Heck, the rotation of the Earth would also guarantee that everything eventually finds a pocket.

But if the OP is talking about mathematical formulas, there are paths on an idealized surface that will never land in a hole regardless of force applied. (A small number of paths, but not zero). On a real surface, there are paths with so many ricochets that you cannot apply enough force. At some point excess force just knocks balls off the table or destroys them.

[BwanaBob]

@pulykamell

Are you talking about the diamond system?

Thank you for this.

[Robot_Arm]

@bob_2

Here is a 9 ball shot: https://www.youtube.com/watch?v=tf2Pv_yhvVk

In addition to the other analysis about why this is a fake, I’d ask what the circumstances were that led to them filming that table in the first place. Does this pool hall have cameras pointing at all the tables? No pool hall I’ve ever been to has done that, and why would one bother? It isn’t the setup they’d use if this was a tournament that was being broadcast. I don’t buy the coincidence that they just happened to be filming when someone just happened to sink all nine balls on a break.

@Dr.Arthias

The answer to that question is actually yes. It fact, at most angles, it doesn’t take a lot of bounces for a pool ball to find a pocket, even without infinite motion. On a typical table under normal conditions, it’s difficult to make contact with the cushions more than 6 or 7 times, no matter how hard you hit the ball, but if friction was reduced even by just half, I think most shots (at least, those that are more than a few degrees away from being perpendicular) would probably sink in fewer than a dozen rebounds.

With ZERO friction/infinite motion, the chance that one of the balls would fall into a pattern of bouncing exactly perpendicular to two opposing rails is extremely low. Slight imperfections in the cloth, rubber cushions, and the balls themselves would make that pretty much impossible, even if the ball had initially been hit perfectly straight by some kind of specially calibrated machine.

I think you’re trying to have it both ways here. On a mathematically perfect table, a ball that’s rolling exactly parallel to a side will just bounce back and forth forever. Ah, but there is no mathematically perfect table; the cushions don’t give a perfect bounce, there’ll be some amount of sidespin, etc., so the ball will eventually go in.

But there may still be an infinite path on a real table that doesn’t require such precision. The ball picks up sidespin when it bounces off a rail. the cushions may be worn in the middle such that bounces a little off from perpendicular are redirected toward the middle. The real-world factors that perturb the mathematically pure case may themselves be self-cancelling and define a real-world infinite path.

I hope that makes sense to someone other than me.

[Leo_Bloom]

@Robot_Arm

…
I hope that makes sense to someone other than me.

Now this I would use more often than “Regards.”

[don_t_ask]

@Dewey_Finn

I wonder about that video. You can’t seem much of the player or the others around the table, but I’d expect someone who did that would be jumping up and down in excitement.

Not if he had already done it a few times earlier in the night and THAT"S why they decided to film this one.

[Francis_Vaughan]

Winding back to the OP, there is a deep question here.

In a very precise way, I think it would be safe to say “No”. The idea of a “mathematical formula” is usually taken to mean a closed form formula, one where you plug the numbers in, and the answer pops out. Probably the best know example of a problem where no such formula is currently known to exist is the three body problem. Given two bodies (planet and moon for instance) we can create a formula that allows you to plug in all the parameters, and for any time you can find exactly where the planet and moon are. Add an additional body - say another moon - and you can’t do it. Whereas there may actually be such a formula, even for such a tiny increase in the complexity of the problem, no-one has ever been able to work one out. There are also problems for which we know there are no closed form solutions. A related example of something you can ask, but can prove you cannot do is to trisect an angle with only a straight-edge and a compass. Galois proved you can’t do it, and in the process opened up a whole area of ability to reason about solubility of certain problems.

So how do we work out where the planets will go? Numerical simulation. We take tiny time steps, and work out where each planet is going at one time, and assume hope out time step is small enough that there isn’t enough change in the location of the other planets in that step to make the calculated new location wrong enough to matter. The longer you keep going to more the errors will accumulate, so care is needed. Worse, you can’t plug in a time in the future and just get the answer, you have to run the simulation all the way up to that time.

So, back to pool balls. Just working out where the pool balls go in any given break is almost this bad. There is no closed form formula that tells us. There are a bunch of formulae that govern the motions, but they are only useful until the next collision. Then we need to work out the next set of parameters and go from there. Luckily we don’t have to worry about the time steps, we can assume the balls are independent between collisions. Even if we assume a prefect pool table and perfect balls it is computationally evil. Even if we don’t care about the niceties of dropping balls into holes, if we used a numerical computer to do the calculations the system is so numerically unstable that there isn’t enough universe to hold the precision needed.

And that is just to get the answer to where the balls go for any given break. There is no guidance whatsoever as to how to answer the task. Maybe just try lots and lot and lots of different breaks, and hope you find one. The numerical instability of the problem makes such a search very very difficult. The usual techniques and heuristics for refining the search won’t work.

And none of this is a formula. We lost sight of a single formula for expressing the system dynamics a very long time ago. As discussed above, there are know closed form formula that can be used to work out how to pot a single ball. even allowing for cushion bounce. But of the balls collide again, all bets are off. No useful formula even here. It becomes a process of application of formula even here.

Given time reversibility, the problem is also identical to asking - what times, angles and speeds should I throw the balls out of the pockets at to ensure they all assemble into the triangle and the cue ball rolls up to the top of the table?

[Saint_Cad]

@pulykamell

Without going Zapruder film frame-by-frame about this, just the fact that there is no reaction whatsoever to the break by anyone in the bar or the shooter himself is enough to ping the bullshit detector all the way to 10.

If you do go all Zampruder on this and look frame by frame the editing is pretty obvious.

[HoneyBadgerDC]

I have witnessed 10 balls going in on a break. The break had more than enough energy to sink all of them.

[Backwater_Under_Duck]

Back to the OP, I think the only way to determine the right shot would be to have a mechanical racking device that can accurately place the balls exactly in the same place, properly snug against each other. That’s pretty tough and can never be achieved in a pool hall setting. Then it would require a robotic cue which could accurately deliver a specific angle, force, english and however many other variations there could be. And then, after however many different shots, you might get proper telemetry data for a repeatable incident. But only for that table, probably.

[MikeF]

@dracoi

It sounds like the question is asking whether balls with infinite motion would eventually fall into the pockets. The answer to that is no.

There are paths on a pool table in which infinite movement will not eventually put the ball into a pocket. As an example of that, a ball moving exactly perpendicular to an edge of the table would never enter a pocket, no matter how long it continues moving. It will ricochet along the same back and forth path until friction slows it down.

I don’t think that on a break you could get a ball to hit a rail at exactly a 90 degree angle, could you? And even if you could, it would sooner or later hit another ball, no?

[Chronos]

Well, theoretically, in perfect-conditions-land, if the balls were all identical and racked symmetrically, and the cue ball was shot in straight forward, then none of the balls on the centerline would ever leave the centerline, and pairs of balls off of the centerline would follow mirror-imaged paths to each other.

Of course, we’re not in perfect-conditions-land, so that would never happen.

[Shalmanese]

@MikeF

I don’t think that on a break you could get a ball to hit a rail at exactly a 90 degree angle, could you? And even if you could, it would sooner or later hit another ball, no?

There’s at least two conditions where you could get perfect 90 degree shots:

1. Place the cue ball directly in the center and shoot perfectly straight.
2. Place the cue ball to the far left/right and aim at the side of the front ball so it shoots of horizontally.