What's the smallest country that can fit everyone standing six feet apart?
Today’s xkcd got me wondering how much land would we need if everyone stood six feet apart.
Methodology
We are all but spherical cows
If we want everyone to keep a minimum distance of \(r\) from each other, the problem boils down to efficiently packing \(n\) circles of radius \(r\) in the smallest area possible. This can be calculated as follows:
Area of a circle with radius \(r\),
\[A_r = \pi r^2\]Optimal circle packing density1,
\[\eta = \frac{\pi\sqrt{3}}{6}\]Area required for \(n\) people,
\[\begin{align} A_n & = n \cdot \frac{A_r}{\eta} \\ & = n \cdot \frac{\pi r^2}{\frac{\pi \sqrt{3}}{6}} \\ & = 2 \sqrt{3} \cdot n \cdot r^2 \end{align}\]Crunching the numbers
Plugging in the current world population2 and the recommended distance of 6 feet3:
\[n = \text{7,773,861,168}\] \[r = 6\text{ft}\]we get
\[\begin{align} A_n & = 9.695×10^{11} \text{ft}^2 \\ & = 34,775 \text{mi}^2 \\ \end{align}\]And the answer is…
According to Wikipedia4, this is just shy of the land area of Portugal 🇵🇹. We could also squeeze into Hungary 🇭🇺 if some of us were willing to swim.
Corollaries
How many people can San Francisco fit standing 6 feet apart?
Dividing the area of San Francisco by \(2 \sqrt{3} \cdot 36 \text{ft}^2\), gives us around 10.5 million people.
Are there any countries where people couldn’t stand 6 feet apart?
1 square mile of land can have 223,563 people standing 6 feet apart. This is much higher than the most densely populated countries and territories in the world5.