Dynamics of dice games
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1) The document discusses whether dice rolls and other mechanical randomizers can truly produce random outcomes from a dynamics perspective. 2) It analyzes the equations of motion for different dice shapes and coin tossing, showing that outcomes are theoretically predictable if initial conditions can be reproduced precisely. 3) However, in reality small uncertainties in initial conditions mean mechanical randomizers can approximate random processes, even if they are deterministic based on their underlying dynamics.
![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](https://image.slidesharecdn.com/hazard-new-110909150427-phpapp01/85/Dynamics-of-dice-games-30-320.jpg)
Dynamics of dice games
- 1. Dynamics of dice gamess Can the dice be fair by dynamics? Tomasz Kapitaniak Division of Dynamics, Technical University of Lodz
- 3. Orzeł czy Reszka? Tail or Head? A Cara o Cruz? Pile ou Face? орeл или решкa?
- 4. Ἀριστοτέλης, Aristotélēs Marble bust of Aristotle. Roman copy after a Greek bronze original by Lysippus c. 330 BC. The alabaster mantle is modern
- 5. DICE
- 6. 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
- 7. Trapezoidal Dodecahedron Triakis Tetrahedron Regular Icosahedron Hexakis Tetrahedron Tetrakis Hexahedron Triakis Octahedron Trapezoidal Icositetrahedron Pentagonal Icositetrahedron Dyakis Dodecahedron Rhombic Triacontahedron Hexakis Octahedron Triakis Icosahedron
- 8. 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 sides Asymmetrical sides -- 2N Sides
- 14. Keller’s model – free fall of the coin Joseph B. Keller, “The Probability of Heads,” The American Mathematical Monthly, 93: 191-197, 1986.
- 15. 3D model of the coin
- 16. Contact models
- 17. Free fall of the coin: (a) ideal 3D, (b) imperfect 3D, (c) ideal 2D, (d) imperfect 2D.
- 18. Trajectories of the center of the mass of different coin models
- 19. Trajectories of the center of the mass for different initial conditions
- 23. 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.
- 25. How chaotic is the coin toss ? (a) (b)
- 30. 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
- 31. 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 !
- 33. Главная / Новости науки Выпадение орла или решки можно точно предсказать Actualité : Pile ou face : pas tant de hasard
- 35. 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 of mechanical randomizers (coin tossing, die throw and roulette). The authors derive the equations of motion, also describing collisions and body contacts. It is shown and emphasized that, from the dynamical point of view, outcomes are predictable, i.e. if an experienced player can reproduce initial conditions with a small finite uncertainty, there is a good chance that the desired final state will be obtained. Finally, readers learn why mechanical randomizers can approximate random processes and benefit from a discussion of the nature of randomness in mechanical systems. In summary, the book not only provides a general analysis of random effects in mechanical (engineering) systems, but addresses deep questions concerning the nature of randomness, and gives potentially useful tips for gamblers and the gaming industry. _________________
- 37. Thank you We are not responsible for what you lose in the casino!