Two Physicists Map Where You Should Park Your Car According to Math

The answer required the use of multiple mathematical techniques.

Physicists Paul Krapivsky (Boston University) and Sidney Redner (Santa Fe Institute) decided to use mathematics to answer an age-old question: where is the best place to park your car. The criteria? To find the parking space that minimizes time spent in the lot.

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Intelligent decisions

"Mathematics allows you to make intelligent decisions," Redner said. "It allows you to approach a complex world with some insights."

In their new study, the physicists map three simple parking strategies onto an idealized, single row parking lot. They call their strategies; meek, optimistic, and prudent.

Meek refers to drivers who grab the first space available. Optimistic indicates the drivers that gamble on finding a space right next to the entrance. Meanwhile, prudent refers to when the drivers take the middle path. 

The authors used multiple techniques to compute the relative merits of each strategy. For starters, the meek strategy mirrored a dynamic seen in the microtubules that provide scaffolding within living cells. They, therefore, used an equation that describes a microtubule's length to calculate the chain of "meek" cars at the far end of the lot.

"Sometimes there are connections between things that seem to have no connection," Redner said. "In this case, the connection to microtubule dynamics made the problem solvable."

The optimistic strategy was described by a differential equation, and the prudent strategy was represented by a simulation that allowed the physicists to compute, on average, the average density of spots, and the amount of backtracking required.

Prudent strategy wins

In the end, the prudent strategy won, followed closely by the optimistic strategy. 

Redner, however, acknowledges that his approach sacrifices real-world applicability in exchange for mathematical insight. "If you really want to be an engineer, you have to take into account how fast people are driving, the actual designs of the parking lot and spaces—all these things," he remarked.

"Once you start being completely realistic, [every parking situation is different] and you lose the possibility of explaining anything."

Still, for Redner, the exercise was about the joy of thinking analytically about everyday situations.

"We're living in a crowded society, and we always encounter crowding phenomena in parking lots, traffic patterns, you name it," he said. "If you can look at it with the right eyes, you can account for something."

The research is published in this week's Journal of Statistical Mechanics.

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