Predictive Dice Roll: Physics & Accuracy

[Ward]

TL;DR Summary

Predictive Dice Roll in controlled environment.

Hi,

1. A multi-sided dice - 20 sides for example. Highly accurate each side equal size.
2. Dice dropped from a machine - with same starting orientation.
3. Dropped in a vacuum from same height each time.
4. Dropped onto a very flat surface - 90-degree angle.
Now given I have tried to make everything as constant and predictable.

Questions:

1. Can physics calculate the outcome without doing the test to predict the outcome.
2. What physics laws / calculations would be used?
3. How accurate would the result.
4. If it can't be calculated what are the blockers for this (why we can't calculate).

Thanks,
Ward.

 

[A.T.]

Are you familiar with this?
https://en.wikipedia.org/wiki/Chaos_theory

 

[PeroK]

Summary: Predictive Dice Roll in controlled environment.

Hi,

1. A multi-sided dice - 20 sides for example. Highly accurate each side equal size.
2. Dice dropped from a machine - with same starting orientation.
3. Dropped in a vacuum from same height each time.
4. Dropped onto a very flat surface - 90-degree angle.
Now given I have tried to make everything as constant and predictable.

Questions:
1. Can physics calculate the outcome without doing the test to predict the outcome.
2. What physics laws / calculations would be used?
3. How accurate would the result.
4. If it can't be calculated what are the blockers for this (why we can't calculate).

Thanks,
Ward.

It would make an interesting experiment. Have you searched on line to see if anyone has done anything like this?

I think you have two options. a) Drop the die with a face downwards. b) Drop the die at an angle.

Case a) should be slightly more predictable, although my guess is that the second bounce is the key. You might get a regular first bounce, but the die will go slightly off its orginal orientation. If it doesn't, then you just get a series of "straight" bounces and the die ends up in its starting position on the floor. That, deterministically, is all you could calculate. I'm not sure I see that actually happening.

As soon as you predict a change in orientation after the first impact, then that can probably only be modeled statistically. I.e. there's no way to calculate the small variations in orientation from the first impact. In fact, the definition of the initial conditions imply that it should bounce vertically upwards.

Case b) would have more significant variations in the first bounce. Although there might be a general pattern to where the die ends up, I doubt it would end up in exactly the same place each time.

If you repeat the experiment many times, then you have the issue that the die and the surface will gradually change through the repeated impacts; and, if you replace the die and surface each time, then you introduce small but significant variations.

A final point is that if you do this experiment and the results are not the same every time, then by definition (if the initial conditions are taken to be the same each time) the result is unpredictable. Although you may see a statistical pattern of sorts.

 

[Ward]

Hi A.T., PreoK,

Great answers.

I have heard of "Chaos Theory" but not really familiar.

Lets assume "Drop the die with a face downwards".
Ok so what factors are working against it being a consistent result.

In my questions premise I have "tried" to eliminate the variances.

For example does energy that is absorbed and it reflective angle very inconsistent?
(Taking a wild stab in the dark here).

Thanks,

Ward.

 

[A.T.]

In my questions premise I have "tried" to eliminate the variances.

But do you actually get consistent experimental results? Do you see any reproducible deviations from a uniform random distribution?

 

[jbriggs444]

Lets assume "Drop the die with a face downwards".

Ideally, if you drop a perfectly symmetric die with one face exactly downward and with zero rotation onto a perfectly uniform and horizontal flat surface then symmetry demands that the die will rebound exactly vertically without rotation. It will finally settle down with the same face upward that started upward.

In practice, it's like an inverted pendulum (Google it). You cannot achieve such precision.

 

[anorlunda]

Ideally, if you drop a perfectly symmetric die with one face exactly downward and with zero rotation onto a perfectly uniform and horizontal flat surface then symmetry demands that the die will rebound exactly vertically without rotation. It will finally settle down with the same face upward that started upward.

That is even more true if the drop height is small compared to the size of die face. Think of a die with 1cm faces, and you lift it only 1mm above the table, then drop. 100% of the time, the downward facing face will remain downward facing.

To approximate randomness, you have to drop from higher up, and you need to give the die some spin before letting go.

The physics are pretty simple, but it is primarily the lack of perfect knowledge of the initial conditions when we let the die go that make it hard to predict.

If you take your physics textbook with you to Las Vegas, you'll still loose.

 

[Cutter Ketch]

I think it is important to note why sufficient precision isn’t readily achievable. The thing you need to think about is the sensitivity of the system to the initial conditions.

it is plausible to assume that you could model the physics well enough that the outcome would be well predicted for an exact set of initial conditions. Such a model may have a lot of detail about the materials properties and the exact shape of the die like the roundness of the corners and such, but it’s plausible that an accurate model can be made.

The question of sufficient precision then becomes a question of the sensitivity of the “trajectory” (motion through time by whatever definition) to those initial conditions. The problem here is not the physics (I mean, an accurate model of the physics may also be hard, but by premise the physics is good) The physics is right. However perfectly correct physics may, with only slight variations in initial conditions, produce completely different end states. So in this case it is the sensitivity of the trajectory to initial conditions that determines whether or not you can predict the outcome. The trajectory is completely deterministic by the laws of physics, but very similar starting conditions completely deterministically and accurately wind up in completely different conditions.
How sensitive the trajectory is to initial conditions depends on the stability of the possible trajectories (so a six sided die would be more predictable than a 20 sided die not just because there are fewer outcomes, but also because there is a larger energy barrier to changing which face is down) and the sensitivity also depends on the available energy. Thus, if you drop a six sided die with the six face up from half a die width above the table you can safely predict it will come up 6 every time. If you drop a 20 sided die from the same height, you won’t always get the same number. (Less stable trajectories). Conversely if you drop a six sided die from a foot above the table you won’t always get the same number (more energy). These determine the sensitivity to initial conditions and so the precision you would need in the initial conditions to accurately predict the outcome. Conversely, the more precisely you control the initial conditions (as in your super accurate die dropper) the more energy (height) and the less stability (die sides) and the longer length of time into the future you can accurately predict.

Weather prediction is tremendously better today than when I was young. There are bigger computers running more complicated models, but one of the main reasons for the improvement is much more and precise data (from a proliferation of satellites, a proliferation of Doppler radar, lots more weather stations with more information) producing a much more accurate measurement of the current conditions.

Another interesting feature is that not all sets of initial conditions have the same uncertainty in final results. Trajectories which come very close to other trajectories which diverge to different outcomes are less stable. Other trajectories have less chance of diverging. You may find some height where certain initial conditions are unpredictable but other initial conditions reliably lead to a particular outcome. However, as the energy increases eventually all trajectories are unpredictable for a given degree of precision in the launch conditions.

All of that is predicated on modeling the physics well enough. I should note again that this is not necessarily easy. It depends on how elastic waves propagate through and around the die and the table, coefficients of friction, the exact shape of the die, particularly the edges and the corners, etc. The model may also be more sensitive to these parameters than the precision with which you can measure them. Here too how accurate the descriptions need to be depends on the amount of energy you put in and how far in time you need to predict.