Thickness of a three-sided coin: A molecular dynamics study

Area of Science:

• Physics of granular materials
• Fluid dynamics
• Statistical mechanics

Background:

• The behavior of tossed objects, like coins, is typically analyzed assuming two outcomes (heads or tails).
• A 'three-sided coin' (cylinder) offers a unique case with three potential landing states: heads, tails, or sideways.
• Understanding the critical aspect ratio is key to achieving a fair, three-outcome toss.

Purpose of the Study:

• To numerically calculate the precise aspect ratio of a cylinder that results in equal probabilities for heads, tails, or sideways landings.
• To compare numerical findings with existing analytical models of coin-tossing dynamics.
• To investigate the physical mechanisms influencing the outcomes of tossing cylindrical objects.

Main Methods:

• Extensive numerical simulations of coin tosses were performed across a wide range of aspect ratios.
• Over 7x10^8 individual tosses were simulated to ensure statistical significance.
• The aspect ratio, defined as the ratio of length to diameter, was systematically varied.

Main Results:

• The critical aspect ratio for a fair three-sided coin toss was determined to be slightly less than sqrt(3)/2 (approximately 0.866).
• Numerical results were consistent with and extended previous analytical predictions.
• The study identified key dynamic mechanisms governing the toss outcomes.

Conclusions:

• A cylinder with an aspect ratio near 0.866 can be considered a fair three-sided coin.
• Numerical simulations provide a robust method for exploring complex physical phenomena like coin tossing.
• This research contributes to the understanding of object dynamics and probability in physical systems.