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# Dynamics of dice games

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AACIMP 2011 Summer School. Science of Global Challenges Stream. Lecture by Tomasz Kapitaniak.

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### Dynamics of dice games

1. 1. Dynamics of dice gamessCan the dice be fair by dynamics? Tomasz Kapitaniak Division of Dynamics, Technical University of Lodz
2. 2. Orzeł czy Reszka? Tail or Head? A Cara o Cruz? Pile ou Face?орeл или решкa?
3. 3. Ἀριστοτέλης, AristotélēsMarble bust of Aristotle. Roman copy after a Greek bronze original by Lysippus c. 330 BC. The alabaster mantle is modern
4. 4. DICE
5. 5. Generally, a die with a shape of convex polyhedron is fair by symmetryif and only if it is symmetric with respect to all its faces. The polyhedrawith this property are called the isohedra. Regular Tetrahedron Isosceles Tetrahedron Scalene Tetrahedron Cube Octahedron Regular Dodecahedron Octahedral Pentagonal Dodecahedron Tetragonal Pentagonal Dodecahedron Rhombic Dodecahedron
6. 6. Trapezoidal Dodecahedron Triakis Tetrahedron Regular Icosahedron Hexakis Tetrahedron Tetrakis Hexahedron Triakis OctahedronTrapezoidal Icositetrahedron Pentagonal Icositetrahedron Dyakis Dodecahedron Rhombic Triacontahedron Hexakis Octahedron Triakis Icosahedron
7. 7. Pentakis Dodecahedron Trapezoidal Hexecontahedron Pentagonal Hexecontahedron Hexakis Icosahedron Triangular Dihedron Basic Triangular Dihedron 120 sides Move points up/down - 4N sides 2N sides Basic Trigonal Trapezohedron Sides have symmetry -- 2N Sides Triangular Dihedron Trigonal Trapezohedron Move points in/out - 4N sidesAsymmetrical sides -- 2N Sides
8. 8. GEROLAMO CARDANO (1501-1576)
9. 9. Galileo Galilei (1564-1642)
10. 10. Christian Huygens (1625-1695)
11. 11. Joe KellerPersi Diaconis
12. 12. Keller’s model – free fall of the coinJoseph B. Keller, “The Probability of Heads,” The American Mathematical Monthly, 93: 191-197, 1986.
13. 13. 3D model of the coin
14. 14. Contact models
15. 15. Free fall of the coin: (a) ideal 3D, (b) imperfect 3D, (c) ideal 2D, (d) imperfect 2D.
16. 16. Trajectories of the center of the mass of different coin models
17. 17. Trajectories of the center of the mass for different initial conditions
18. 18. Basins of attraction
19. 19. Definition 1. The die throw is predictable if for almost all initialconditions x0 there exists an open set U (x0 ϵ U) which is mappedinto 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 impliesthat if B ϲ U then randomizer is predictable.Definition 2. The die throw is fair by dynamics if in theneighborhood of any initial condition leading to one of the n finalconfigurations F1,...,Fi,...,Fn, where i=1,...,n, there are sets of pointsβ(F1),...,β(Fi),...,β(Fn), which lead to all other possible configurationsand a measures of sets β(Fi) are equal.Definition 2 implies that for the infinitely small inaccuracy of theinitial conditions all final configurations are equally probable.
20. 20. How chaotic is the coin toss ?(a) (b)
21. 21. ωη 0 [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
22. 22. In the early years of the previous century there was a general convictionthat the laws of the universe were completely deterministic. Thedevelopment of the quantum mechanics, originating with the work ofsuch physicists as Max Planck, Albert Einstein and Louis de Brogliechange the Laplacian conception of the laws of nature as for thequantum 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 auniverse where random events of objective nature exist. ContradictingAlbert Einsteins famous statement it seems that God Plays dice after all.But going back to mechanical randonizers where quantum phenomenahave at most negligible effect we can say that:God does not play dice in the casinos !
23. 23. Главная / Новости науки Выпадение орла или решки можно точно предсказатьActualité : Pile ou face : pas tant de hasard
24. 24. 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 dynamicalpoint 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. _________________
25. 25. Thank youWe are not responsible for what you lose in the casino!