Circle Packing and the Math of Social Distancing

What is the maximum number of people to be allowed in a certain area to practice social distancing during COVID-19? Scientists recommend that people keep at least two meters away from each other, then we can imagine each person to be at the center of a circle of radius one meter, where the circles cannot overlap. Thus, solving the aforementioned social distancing problem amounts to fitting as many non-overlapping equal-sized circles into a confined floor plan as possible. This is the so-called circle packing problem in geometry, which concerns the best (densest) way to arrange non-overlapping circles on a plane. We will calculate the density of square packing as well as hexagon packing of circles, and go over Thue's theorem, which says that hexagon packing is the optimal packing of circles.