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

This Letter presents a numerical study across parameter space to calculate the aspect ratio (ratio of length to diameter) of a fair “three-sided coin”: a cylinder that when tossed, has equal probabilities of landing heads, tails, or sideways. The results are cast in the context of previous analytical studies, and the various mechanisms that govern the dynamics of coin tossing are compared and contrasted. After more than $7\times {10}^8$ tosses of coins of various aspect ratios, this study finds the critical aspect ratio to be slightly less than (but not exactly equal to) $\sqrt{3}/2\approx 0.866$.

Figure 1

Schematic of the discretization used in this study. We use the definition of aspect ratio $\varphi =L/D$. In this model, the diameter of each Lennard-Jones sphere is $d=2^{1/6}\sigma$, which is also the center-to-center distance (along a straight line, not the circumference) of adjacent spheres. (a) Discretization of an ideal cylinder into spheres. (b) View along the cylinder axis. Here, $h$ is the height of the cylinder axis above a horizontal plane when the cylinder is in equilibrium, and is used as a measure of cylinder roughness.

Figure 2

Results of coin tosses for various aspect ratios. We have used the definition of aspect ratio as the ratio of the length to diameter of the cylinder, i.e., $\varphi =L/D$.