Dynamics of dice gamess Can the dice be fair by dynamics?

Tomasz Kapitaniak

Division of Dynamics, Technical University of Lodz

Orzeł czy Reszka?

Tail or Head?

A Cara o Cruz?

Pile ou Face?

орeл или решкa?

Ἀριστοτέλης, Aristotélēs

Marble bust of Aristotle. Roman copy after a Greek bronze original by Lysippus c. 330 BC. The alabaster mantle is modern

DICE

Generally, a die with a shape of convex polyhedron is fair by symmetry if and only if it is symmetric with respect to all its faces. The polyhedra with this property are called the isohedra.

Regular Tetrahedron

Isosceles Tetrahedron

Scalene Tetrahedron

Cube

Octahedron

Regular Dodecahedron

Octahedral Pentagonal Dodecahedron

Tetragonal Pentagonal Dodecahedron

Rhombic Dodecahedron

Trapezoidal Dodecahedron

Triakis Tetrahedron

Regular Icosahedron

Hexakis Tetrahedron

Tetrakis Hexahedron

Triakis Octahedron

Trapezoidal Icositetrahedron

Pentagonal Icositetrahedron

Dyakis Dodecahedron

Rhombic Triacontahedron

Hexakis Octahedron

Triakis Icosahedron

Pentakis Dodecahedron

Trapezoidal Hexecontahedron

Pentagonal Hexecontahedron

Hexakis Icosahedron

120 sides

Triangular Dihedron

Move points up/down - 4N sides

Basic Triangular Dihedron

2N sides

Trigonal Trapezohedron

Asymmetrical sides -- 2N Sides

Basic Trigonal Trapezohedron

Sides have symmetry -- 2N Sides

Triangular Dihedron Move points in/out - 4N sides

GEROLAMO CARDANO (1501-1576)

Galileo Galilei (1564-1642)

Christian Huygens (1625-1695)

Joe Keller

Persi Diaconis

Keller’s model – free fall of the coin

Joseph B. Keller, “The Probability of Heads,” The American Mathematical Monthly, 93: 191-197, 1986.

3D model of the coin

Contact models

Free fall of the coin: (a) ideal 3D, (b) imperfect 3D, (c) ideal 2D, (d) imperfect 2D.

Trajectories of the center of the mass of different coin models

Trajectories of the center of the mass for different initial conditions

Basins of attraction

Definition 1. The die throw is predictable if for almost all initial conditions x0 there exists an open set U (x0 ϵ U) which is mapped into the given final configuration. Assume that the initial condition x0 is set with the inaccuracy є. Consider a ball B centered at x0 with a radius є. Definition 1 implies that if B ϲ U then randomizer is predictable. Definition 2. The die throw is fair by dynamics if in the neighborhood of any initial condition leading to one of the n final configurations F1,...,Fi,...,Fn, where i=1,...,n, there are sets of points β(F1),...,β(Fi),...,β(Fn), which lead to all other possible configurations and a measures of sets β(Fi) are equal. Definition 2 implies that for the infinitely small inaccuracy of the initial conditions all final configurations are equally probable.

How chaotic is the coin toss ? (a) (b)

0ηω [rad/s] tetrahedron cube octahedron icosehedron 0 0.393 0.217 0.212 0.117 10 0.341 0.142 0.133 0.098 20 0.282 0.101 0.081 0.043 30 0.201 0.085 0.068 0.038 40 0.092 0.063 0.029 0.018 50 0.073 0.022 0.024 0.012 100 0.052 0.013 0.015 0.004 200 0.009 0.008 0.007 0.002 300 0.005 0.005 0.003 0.001 1000 0.003 0.002 0.001 0.000

In the early years of the previous century there was a general conviction that the laws of the universe were completely deterministic. The development of the quantum mechanics, originating with the work of such physicists as Max Planck, Albert Einstein and Louis de Broglie change the Laplacian conception of the laws of nature as for the quantum phenomena the stochastic description is not just a handy trick, but an absolute necessity imposed by their intrinsically random nature. Currently the vast majority of the scientists supports the vision of a universe where random events of objective nature exist. Contradicting Albert Einstein's famous statement it seems that God Plays dice after all. But going back to mechanical randonizers where quantum phenomena have at most negligible effect we can say that: God does not play dice in the casinos !

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Actualité : Pile ou face : pas tant de hasard

Dynamics of Gambling: Origins of Randomness in Mechanical Systems;Lecture Notes in Physics, Vol. 792, Springer 2010 – 48.00 Euro only !! _________________ This monograph presents a concise discussion of the dynamics ofmechanical randomizers (coin tossing, die throw and roulette). Theauthors derive the equations of motion, also describing collisions andbody contacts. It is shown and emphasized that, from the dynamical point of view, outcomes are predictable, i.e. if an experienced player canreproduce initial conditions with a small finite uncertainty, there is agood chance that the desired final state will be obtained. Finally, readerslearn why mechanical randomizers can approximate random processesand benefit from a discussion of the nature of randomness in mechanicalsystems. In summary, the book not only provides a general analysis ofrandom effects in mechanical (engineering) systems, but addresses deepquestions concerning the nature of randomness, and gives potentiallyuseful tips for gamblers and the gaming industry. _________________

Thank you We are not responsible for what you lose in the casino!