# Parking problem (sequential packing) simulations in two and three dimensions

## Abstract

The goal of the parking limit problem is to determine the mean fraction of a space that would be occupied by fixed objects, each of the same size, that are placed or created randomly in that space until no more can fit. The goal of the two-dimensional parking limit problem is to determine the fraction of an area that would be occupied by disks of one size; a three-dimensional parking limit problem involves creating spheres in a volume. We tried a simple numerical simulation algorithm combined with regression analysis to do the two-dimensional problem and found the parking limit to be 0.51 to 0.56 area fraction for disks on a plane, which is consistent with the results of others. We then used the three-dimensional version of this algorithm to obtain the parking fraction in a cubical region with penetrable walls. Our results can be multiplied by a geometrical factor to give the parking fraction for a cube with impenetrable walls and can be extrapolated to give the parking fraction for an infinite region. Regression equations were obtained for the effects of the ratio of the sphere radius to the side of the cubical volume and of the dimensionless number of attempts to park. We found the parking limit for an infinite volume to be 0.37 to 0.40 for spheres. © 1987.