Social distancing dictates that people stay six feet apart. And, people are talking about a possible return to school. How many people could you fit in a classroom while obeying that six feet? I have to try to calculate it.
Some assumptions: 1) Let’s say it’s a 30 ft x 30 ft room. 2) We’re not concerned with desks, windows, etc. It might as well be an empty room, no obstructions to consider…I know it violates fire code but block the door if you need to.
I’ll spoiler my answer, let you arrive at your own, and see if we agree.
TLDR? I get 36.
Long version:
You could grid it the room in six feet increments and get a 5 x 5, for 25 total spaces that would be distanced appropriately. If you stand in the center of the square, you’re 3 feet from the center of each side…but the corners are farther away, and you’re wasting that space. The safe space around each person is a circle, not a square.
And do we really need to provide a full “circle of safety for ALL people?” IMO no, you wouldn’t have to put anyone six feet from a wall. You could put him in the very corner of the room. No one would be behind him or to one side of him, so that’s safe. He’d need a “quarter circle of safety,” so to speak. Others could be against the back or side walls—needing a half circle of safety. Those in the center of the room need the full circle.
So if you have four corners, totalling one full circle, area-wise. You could fit four between each corner on the perimeter. Four half circles, four sides, that uses up eight full circles, area-wise. Sixteen full circles in the center. That makes 25 circles, total. . Area=3.14 x 3 x 3=28.26 per circle. 25 circles makes 706.5 square feet used total. Since circles only touch tangentially, more than 20% of the space is wasted from the 900 (30 ft x 30 ft room) available.
4 corners, 16 sides, 16 center=36 would fit in the room. Rather than the 36 square feet gridding would allot to 25 squares, or even 28.26 square feet given by a 6 foot circle, the average per person works out to 19.625, and a lot is wasted. Which bothers me…
You need 6 feet BETWEEN the people, and your calculation doesn’t allow for the space of the people themselves. If 6 people are in a row with 6 feet between them, the distance between the first and last will be more than 30 feet.
If the people are thin enough, you MIGHT be able to squeeze 25 into that room.
True, I’m treating the people as small points. Consider infection that might be transmitted from coughing, for instance—droplets travel from someone’s mouth to someone else’s mucus membranes, not their ankles or their elbows or their hips.
“How many unit circles can I pack into a rectangle of given dimensions” is an old, old math problem. At some point for a large enough rectangle it becomes more efficient to pack the circles hexagonally, rather than squarely. But when exactly that happens is a difficult problem.
I think it complicates things to think in terms of circles or squares, so I laid out the first row starting in the corner going up (x=0,y=0 to 0,30) - that’s 6 people. The next row can be offset on a diagonal, which means x=4.2 (calculated based on the hypotenuse of the triangle=6). This row can fit 5 people.
Expand that out with another row every 4.2 feet, for a total of 8 rows, half with 6 people and half with 5.
You’re right, I screwed up. The sides of the triangle aren’t equal. With this arrangement, the y has to be at 3’, halfway between 0 and 6. That puts the rows spaced at 5.2 feet, so you can fit 6 rows, for 33 total people. And now I’m not sure if that’s the most efficient placement.
Yeah I plugged and played till I got those coordinates as well. As Chronos was saying the trend seems to be that if you have a ginormous square to fill, it’s a different deal. The only other direction I can think to go would be a soccer ball—mix of hexagons and pentagons. Deflate one, cut it carefully, fit it into a square, hmm…
Actually, with the parameters you gave, you’ll get better spacing just laying them out on a simple 6x6 grid. That will give you 36 people total, because you can fit in that last row. In practice, you couldn’t have someone sitting exactly at the edge, so you lose the last row and the offset distribution would be better.
It’s not too hard to model the students as cylinders, especially since the really hard work of solving the related circles-in-a-square packing problems has already been done by others and is conveniently available on the Internet. If you count each person, teacher and students, as a cylinder with a radius (not diameter) of 0.5 feet (little kids), I come up with a capacity of 25. If you model them as cylinders with a radius of one foot (varsity linebackers), I get a capacity of 20. And if students are allowed to wave their arms around (cylinder radius of 3 feet, say) and if finger-tip-to-fingertip distance between students still has to be 6 feet, I get a capacity of only 9. If you treat the cylinders as vanishingly small (radius of zero) I get 36, which is the answer others in this thread have gotten.
The formula I come up with is (b+s/2)/(w+s) where b is the radius of the body-modeling cylinder (0.5 or 1 or 3 feet), s is the between-person safety distance (6 feet), and w is the width of the square room (30 feet). Remember the number you calculate and look in a table like this one, in the column labeled “radius.” Start at a value of N larger than you expect, and work your way up until you come to the first value of “radius” that is equal to or larger than your number. The corresponding value of N in the first column of the table is your maximum capacity. Packings in this range have been proven to be optimal. For numbers over 30 (except 36), the best we can say is that the listed packings are the best known, but not proven to be optimal.
If you want to see what the packings look like, see the images here, but keep in mind the room as described in the problem (30 feet) is smaller than the square depicted (which is proportionally 36 feet). That’s because students, at or near the center of their circles, are allowed to touch the wall.
All of this assumes, of course, that no student is allowed to levitate or dangle from the ceiling. I would see that as less of a three-dimensional packing problem and more a matter of teacher’s lack of control over the classroom.
I’ve told my students many times that if any of them are capable of working at super-speed, they have my permission to do so (apparently, I look like Dash’s teacher). I suppose that would likewise give permission to any students capable of levitation.
Yes, all well and good, but what is the damn answer?
Not so fast, the teacher at the front of the room needs her/his row to be 6 feet away from the first row of students.
Plus, they really do need desks and chairs. They can’t stand up perfectly straight for an entire class, although those on Red Bull might be willing to try. But then they would have to move through other students, staying 6 feet away, to go to the bathroom.
All this figuring and planning will be a boon to architects. At least they can work from home.
The OP was interested in the mathematical question. In order to even have a mathematical question, you have to have a model for it, and constructing the model is where all of those real-world considerations come in. And of course, the real world is more complicated yet than you make it out to be, too: Like, why 6 feet for everyone? Maybe you’d get better safety with only 5 feet between students, but 8 feet between the students and teacher. Maybe you can put plexiglass partitions in the room to decrease the distance, and is that more or less practical than spacing the students apart, and so on. And of course, if you make the model too complicated, then solving the resulting math problem becomes less interesting (still of interest to those actually setting up classroom layouts, of course, but not interesting to people approaching it purely as a mathematical game on a message board).
My thought process was that the disease is spread by coughing and sneezing. Less commonly, a person gets the virus on the hands, rubs the eyes, and gets it. So really, the face is the concern and that’s a very small area, a few square inches.
If in fact I am seated in a circle of radius three feet, my desk easily fits inside that and doesn’t add to the space needed. If I lift my arm and point to the student seated in front of me, the distance between my nose and that student’s nose wouldn’t change because of it.
Why six feet? For the purposes of the question, I chose that because that’s what officials have recommended, marked off in stores, etc. IIRC they chose it to lessen the chances of getting the disease from airborne particles. Maybe that will change but maybe not. But in order to comply, I went with that.
So if you’re not spending a prolonged time, just walking past someone else’s desk, it doesnt appear to fit the CDC’s definition.
In any case, my question was basically a math problem. We can’t block the door because of fire codes, for instance. Teachers often write on the board so students can’t block that space or be positioned where they can’t see it. I once had a teacher from Paris who said in her day (1930s?) switching from math class to science class meant the teachers changed rooms, not the students. That’s problematic if my science is chemistry and yours is biology etc. but maybe…?
I can believe architecture of schools will change because of COVID-19. Have you seen that now some are designing schools with active shooters in mind?
There has been a rising concern around mass shootings at schools in the US, with the attacks at Sandy Hook Elementary School, where 26 children and adults were killed, and Marjory Stoneman Douglas High School, where 17 were killed, recorded as among the deadliest.
So the article doesn’t cite Columbine but rather the more recent ones. I’m sure the concerns evolve every time it happens again, so they double down on the tech.
Bullet stopping window film…is that really a thing? IIRC Mythbusters did something with bullet stopping glass and said yeah, to a point. 22 caliber, ok. Get out an AK47 though…so I hope people don’t think we can design a building that will provide perfect protection all by itself.
Knowing that 49 circles is the smallest square number whose most dense packing within a square is not an arrangement in square array (having worked on that problem in the past), and that 36 in a square array perfectly fit the dimensions of the OP, the answer is 36.
Seems like too many interpret the 6 foot coronavirus recommendation too literally, unless the virus always dies or falls to the ground before 6 feet. If there is one sick person surrounded by a circle of many tightly packed healthy people 6 feet away, I doubt if none would be exposed. One infected person being 6 feet from exactly one healthy person does not result in the same risk of transmission as one infected person placed in the middle of a large array of healthy people 6 feet apart.
You’re right, I screwed up. The sides of the triangle aren’t equal. With this arrangement, the y has to be at 3’, halfway between 0 and 6. That puts the rows spaced at 5.2 feet, so you can fit 6 rows, for 33 total people. And now I’m not sure if that’s the most efficient placement.