The three-dimensional dynamics of the die throw

Area of Science:

• Physics
• Mechanical Engineering
• Computational Dynamics

Background:

• Understanding the physics of random number generation is crucial for various applications.
• Previous models often simplified die-throw dynamics, neglecting crucial factors like bounces and table interactions.

Purpose of the Study:

• To develop a comprehensive three-dimensional model of die throws.
• To investigate the influence of bounces, dissipation, and table motion on die-roll outcomes.
• To simulate trajectories for dice of various shapes and compare with experimental data.

Main Methods:

• Formulation of a three-dimensional dynamical model incorporating die bounces and energy dissipation.
• Simulation of die trajectories on both fixed and oscillating tables.
• Experimental validation using high-speed camera recordings of die throws.

Main Results:

• The model accurately simulates die trajectories for different shapes.
• Numerical results align with experimental observations.
• For realistic initial energies, the die shows a higher probability of landing on its initially lowest face.

Conclusions:

• The non-smooth nature of the die-throw system is a key factor in generating dynamical uncertainties.
• Small variations in initial conditions lead to significant trajectory variations, approximating a random process.
• The developed model provides a more realistic understanding of random number generation through mechanical means.