I read a newspaper article about the police breaking up a party of 20 people in a room of 20 square meters, because they could not keep corona restrictions of a minimum distance of 1 meter between each participant.
This got me thinking. As a pure math problem, how many people could you actually fit in a 20 square meters room under those restrictions?
My first instinct is to treat this as a circle packing problem where each person is replaced with 1/2 metre circle (area about 0.8m^2) describing their “personal space” imagining the person as a point at the centre of that circle. According to that wikipedia article, the maximum packing density of a room is about 0.9, so a 20m^2 room would have about 18m^2 of usable space which would ideally fit 22.5 people if they actually achieve maximal packing density.
We can model the more realistic situation where a person actually needs some non-zero footprint to stand on by expanding the “personal space” circle. If we gave each person a 0.3m circle to stand on, the circle then has radius 0.65m, area about 1.3, and you end up fitting about 13.8 people in.
But, we also need to take into account the fact that people only need to be separated from each other, not the walls of the room, so a chunk of our personal space circle can extend outside the walls (for those people who are pressed up against the wall) making this not quite a circle packing problem. This is equivalent to expanding the walls by half a metre in each direction. How much this expands the area depends on the shape of the room.
Of course my smart-alek solution would be to return to the “people require no area to stand in” assumption, then assume the room is one light year long by however narrow it needs to be to have total area of 20m^2, and then fit 9.461e+15 people in it. 
Thanks for the detailed answer. I must admit that I did think about a circular room, but not the long, thin one. That would actually make the problem almost meaningless.
Not really. We live in the actual world. Forget the mathematicians. 20 people standing in a hallway 1 meter apart (shouldn’t it be 2 meters?) is a 1 meter by 20 meter hallway. People standing as closely packed as reasonable would be a 4 x 5 pattern and it is still 20 sq meters.
But a real room would most likely have furniture which would compress things and require additional room to maintain spacing. The police were right.
I have been involved in figuring maximum capacity for a room for safety reasons. You have to allow for chairs, tables, etc are they fixed or movable, etc.
Regardless of the maths, 20 people in a small room for any length of time, even if they were all masked (unlikely considering many would be drinking), is clearly a high risk.
They should all be required to do time on unpaid work at a hospital.
Are they clowns?
Also, you could fit more people if you think in 3D. If you have some folks hanging from chandeliers (or set up some scaffolding) you can get more people in the room. I’m not sure the exclusion zone is spherical If a 1.75m tall person is standing on my shoulders, our faces are > 1 m apart. Though I’m not sure how much droplets float vs settle downward…
Brian
Of course, theory is never established as correct until there is a confirmatory experiment…
“Assume a spherical cow…”
Here’s a funny …
When I first glanced at the preview pic of your vid I thought the woman at the left was fiddling with her mobile phone. Then I noticed Groucho & Harpo and had to reparse the whole pic.
Do the people have to alive and intact? Human slurry allows you to fill the entire volume pretty efficiently.
Keeping the 6’ ~= 2m spacing is difficult w slurry.
OK, so keep the noses intact, and arrange them in the slurry in a face-centered cubic structure.
As leahcim pointed out, it wouldn’t have to be a cubic structure, but rather could be stacked as oranges.
I said face-centered cubic. Which is in fact one of the two regular ways spheres can stack.
Ah, my bad. Just googled it 
Well, there is the story of the Black Hole of Calcutta, where 64 people were supposedly stuffed into a room 14’ x 18’:
…the surviving British soldiers, Indian sepoys, and Indian civilians were imprisoned overnight in conditions so cramped that many people died from suffocation and, and that 123 of 146 prisoners of war imprisoned there died. Modern historians believe that 64 prisoners were sent into the Hole, and that 43 died there.
Of course, they were not provided a meter space each in that situation, but the OP brought this story to mind.
Something else which came to mind is phonebooth stuffing:
I am reminded of an ad I saw decades ago for self-sticking floor tile. It opened with a bunch of people milling around in a small space, each holding a tile. The camera drops to a guy sitting cross-legged on the floor also holding one. With a few people passing in front of him he extolls the virtues of the product and demonstrates how easy it is to install by stripping off the backing and placing it.
Stripping another he says, “Why, by myself I could tile this 10x10 foot room in about an hour. With sixteen friends—” and someone steps on the tile, planting it all cockeyed. “…It could take forever.”
This should be approximately 1 person per square meter.