I’m presenting here introductory remarks and thoughts about the scientific study of the movements of dice and coins and about some recent experiments .
When a coin is tossed it is usually said that the probability of a head or tail turning up is (1/2) , and when dice are thrown the probability that a certain face of a die with its
corresponding number turning up is (1/6) .The motions of dice and coins are complex and many variables , physical factors and physical conditions are to be considered and
therefore probabilities have been used as a simpler way to describe the outcome of tossing or throwing them.But more precise and exact scientific methods could also be
A large number of experiments could be made in a controlled or laboratory environment by tossing or throwing coins and dice and using high precision and high speed cameras to
film and photograph their motion , record it and show the motions or trajectories in slow motion.Data could be recorded on high speed computers and computer graphics and
simulations can be helpful in this study.The physical conditions could be changed,such as the form ,shape ,volume and constitution of dice and coins , the type of surface that the
dice or coins fall upon , etc.
After careful observation mathematical models and physical tools could be elaborated and used to study and describe the movements of dice and coins and the physical forces
acting on them (tools such as tensor calculus ,the physics of rotation and translation,etc).For example ,a coin could be considered as a circular cylinder with a small middle lateral
area (h< r,where h is the height and r is the radius of the top and bottom areas).When tossed this cylinder undergoes a combination of rotations and translations before falling on a human hand or some other surface.
And a tossed or rolled die is similar to a cube (made of a certain material) thrown and moving and then hitting a certain surface and bouncing on it before resting .The initial
conditions and the ways of tossing , rolling and throwing could be changed too.
By employing these scientific ways and methods general mathematical and physical equations could be found to describe the exact motion and the kinematics and dynamics of tossed coins and dice, which will result in understanding the physics of tossing these objects , and even make playing with dice and tossing coins scientifically predictable.
There are also non-cubical dice , but for dice to be fair they should be isohedral polyhedra
or isohedra. In these objects , each face must have the same relationship with all other faces, and each face must have the same relationship with the center of gravity.
Five of the most common isohedral dice are shown in the following image ( these are also Platonic solids):
The octahedron , dodecahedron and icosahedron have respectively eight , twelve and twenty faces.
Here are some images of dice used in games and gambling:
In the past few years a group of researchers made experiments and studies by throwing dice and tossing coins using special devices and mechanical randomizers (see the references below).
They found out that the dynamics of throwing dice and tossing coins can be described by Newtonian Mechanics and by deterministic equations of motion.The result is generally predictable in terms of dynamical system theory , provided that the initial conditions are precisely defined. But the initial conditions have to be set with a very small uncertainty which cannot be obtained in real experiments .Due to high sensitivity to initial conditions very precise devices are needed , and any uncertainty in these initial conditions leads to uncertainty and randomness in the final results and outcome.
It was deduced or conjectured that at the scale of the universe as well as in quantum mechanics ‘God plays with dice ‘, contradicting Einstein’s statement and not working well with the deterministic conception of the laws of Nature expounded by Pierre-Simon Laplace. However , at the smaller level of mechanical randomizers and gambling , they concluded that ‘God does not play with dice’ in the casinos.
In my opinion , these experiments and results are an important step towards a more accurate and rigorous study of the motions of dice and coins . Perhaps with time more experiments could be made with more precision in order to obtain scientifically predictable results for most or even all of the outcomes of throwing coins and dices . I think that
probability and statistics are the least exact parts of mathematics , and probabilistic methods ought to be used only after all possible exact and deterministic solutions have been tried .
I find it intriguing that Laplace was one of the important founders and proponents of causal determinism , and at the same time he made important contributions to the theory of
probability and statistics . But he made it clear in his statements that probabilistic and stochastic methods and processes are used due to our ignorance of all the physical data and variables needed to study and describe natural phenomena. As we become less ignorant statistical methods should be less used, and I think it is an important progress when the study of certain events and phenomena move from the domain of statistics and randomness to the domain of exact deterministic science and equations.
Albert Einstein said that God does not play with (or throw ) dice in reference to quantum mechanics. Einstein had pantheistic views ,which means he equated God with Nature and the Universe , and believed in Spinoza ‘s God . He sometimes used a spiritualistic language when referring to Nature or the Physical World. He also considered himself to be an agnostic. So his statement about God and dice could be rendered in a more scientific language such as: The natural and physical World and Reality are not fundamentally based on probability ,statistics and randomness .Quantum Physics is not a completely finished and coherent theory and relies too much on probabilistic methods. Exact , scientific and mathematical methods and equations ought to be used as much as possible to explain the Physical World.
Probabilistic and statistical models and methods are known to be used at the basis of quantum mechanics. Some scientists want to keep things unchanged , others have criticized the foundations of quantum physics and suggested certain solutions. I’m not going to get engaged in a long discussion about quantum mechanics , but i ‘d like to say that I was glad when I learned about the experiments which showed that silicon oil droplets moving on a fluid surface exhibit behaviors similar to those associated with quantum mechanical systems.There has been a regain of interest in the pilot wave theory of Louis de Broglie , and maybe this will lead to introducing more exact and deterministic approaches to the interpretation of quantum mechanics.I think that reassessing the foundations of physical theories in relation with new empirical data and experiments is necessary from time to time. In fact the theory of relativity and quantum mechanics were elaborated at the beginning of the twentieth century partly as an effort to reevaluate the theories of classical physics in the light of new scientific experiments .
Discussing probability and randomness versus causal determinism and exactness in science and their role in the physical World could evolve into a long philosophical debate. In any case I’d like to add one more remark:
Probabilities and statistics are used when we are ignorant and in the dark. When we become less ignorant ( i.e less in the dark and less blind) and have more knowledge and data , we use and should resort to accurate , correct scientific approaches and mathematical models and equations. Hence I think the most important procedure should be to apply and use exact scientific methods in the ( exact ) sciences.
Some useful references and links :
- Understanding Coin-Tossing .
- Probability, physics, and the coin toss .
- Dice Throw Dynamics Including Bouncing .
- The pilot wave dynamics of walking droplets (video link).
- A presentation and enumeration of fair dice can be seen in here.