Social Distancing and Sardines

It all began on Facebook a few days ago. A friend of a friend had observed that the people at the DfE needed to take remedial Maths classes. Somebody there had assumed that, if the statutory social distance was reduced from 2 metres to 1 metre then we would be able to get twice as many people in any given space. "Hurrah!", they said. That's sixteen people in a classroom - whereas we can only manage about eight (legitimately) at the moment. What will happen to the other fourteen? Erm. Maybe they'll have to be taught in a tent. After the thunderstorms that we've had over the past few days, I'd rather give that a miss, if you don't mind.

So, how much 'uninvaded person space' (UPS - teachers will get the 'joke') does someone need to keep a social distance of two metres? Well, they'd have to have a UPS with a radius of 1 metre. If everyone had this UPS, nobody would get within two metres of anybody else - think about frog spawn, if it helps. However, this runs into problems straight away as (i) people, too have an area when viewed from above - they aren't just infinitesimally small points in space; (ii) circles don't tessellate and (iii) a typical USP would be an ellipse as most people measure more from shoulder to shoulder than they do from their foremost extremity to their rearmost.

Let's go with the idea that UPS are hexagonal and that each of the six points is 1 metre away from a matching point on a smaller hexagon within it. This smaller hexagon represents the person themselves ... OK so there aren't many people who are hexagonal when viewed from above but stick with it.

Now the sums are telling us that, while a policy of two-metre social distancing (2 mSD) requires everyone to have 6.75 sqm of their own space, a policy of 1 mSD only entitles them to 1.6875 sqm - a quarter of what they got when 2 mSD was the norm.

"Eureka!", cries Boris (like Archimedes of bath fame, he speaks Greek - he was taught it at my old school where he was Head Boy in 1981 and I was a frightened 13 year-old). "I can put 32 children in every classroom again if we go down to 1mSD. People can be wedged into shops, offices, banks and supermarkets and we can make loads of money to fill up Rishi's piggy bank again!"

"Hang on", says someone with a few spare brain cells in the DfE (we live in hope), "Isn't there some legislation that entitles every pupil in a classroom to 1.8 sqm of teaching space in schools? This means that if we legislate for 1 mSD, we're actually saying it's OK for schools to be more crowded than they were before the pandemic even started. Every class will play a giant game of Sardines in every room they enter. We're saying there's less risk of infection now than when there was no COVID-19 virus around and there was no risk at all - a negative probability of being infected, i.e. something that would happen less often than never. (Negative probabilities do exist, it turns out, but only in really heavy quantum mechanics - see the Wikipedia page No, I don't understand it either.

But let's be done with this idea that 1 mSD gives us licence to double the number of people in a given space - actually it quadruples it. But only if we all walk around with giant honeycombs around our waists (which would mean we couldn't walk through any doors, or do PE, or go to the toilet for that matter), tessellate perfectly with one another as we walk the corridors and exist in classrooms whose dimensions are an exact multiple of 1.25 metres; where the teacher, the cupboards, coats and the folders get allocated no space at all, and there's no movement space (so no way of getting out to the loo without invading someone else's UPS).

Go back to the drawing board, Gavin.


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