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Wednesday, July 15, 2020

The Math of Social Distancing Is a Lesson in Geometry | Mathematics - Quanta Magazine

How to safely reopen offices, schools and other public spaces while keeping people six feet apart comes down to a question mathematicians have been studying for centuries, says Patrick Honner, teaches mathematics and computer science in New York City.

Photo: BIG MOUTH for Quanta Magazine
Sphere packing might seem like a topic only a mathematician could love. Who else could get excited about finding the most efficient way to arrange circles in the plane, or spheres in space?

But right now, millions of people all over the world are thinking about this very problem.

Determining how to safely reopen buildings and public spaces under social distancing is in part an exercise in geometry: If each person must keep six feet away from everyone else, then figuring out how many people can sit in a classroom or a dining room is a question about packing non-overlapping circles into floor plans.

Of course there’s a lot more to confronting COVID than just this geometry problem. But circle and sphere packing plays a part, just as it does in modeling crystal structures in chemistry and abstract message spaces in information theory. It’s a simple-sounding problem that’s occupied some of history’s greatest mathematicians, and exciting research is still happening today, particularly in higher dimensions...

Proving this wasn’t easy: Famous mathematicians like Joseph Louis Lagrange and Carl Friedrich Gauss started the work in the late 18th and early 19th century, but the problem wasn’t completely solved until the 1940s, when all the possible arrangements — both regular and irregular — were rigorously dealt with. That it took so long to handle the problem in two dimensions, where things are relatively easy to visualize, is a warning of what’s to come in higher dimensions.
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Source: Quanta Magazine