Mathematically Distant

Here’s another article that tries to show some calculations but ends up being distant from the genuine mathematics.  And the source data is, in some ways, even funnier.

This blog starts with a filleted version of the article.  (The link is provided so you can see the whole thing, but none of the mathematical info has been removed from my version.)

https://www.theguardian.com/business/2020/jun/05/shops-reopen-england-social-distancing-rules-coronavirus

Plans to reopen shops in England 'in chaos' over social distancing rules

Study finds 10 sq metres of space is required per shopper to stick to government’s 2-metre guidelines

Fri 5 Jun 2020 14.19

High street retailers must give each shopper a minimum of 10 sq metres of space if they are to maintain social distancing rules, a new study has found, prompting businesses to complain that a lack of official guidance has left reopening plans in chaos.

With 10 days to go until most non-essential shops reopen in England, a research paper by Manchester Metropolitan and Cardiff univeristies has concluded that larger shops must allow substantially more than a 2-metre radius for each person so that they can move freely.

The study finds that people need 10 sq metres in smaller shops, 11 sq metres in larger shops, and 12 sq metres in outdoor spaces where they move more rapidly. That is up to three times as much space as the 2-metre radius required when people are stationary.

The report’s conclusions are based on the speed and unpredictability of people’s movements, which means they are much more likely to get near each other when shopping than when queuing or sitting at a desk – and therefore require a bigger “halo”.

Chris Turner, the chief executive of British BIDs – the representative group for Business Improvement Districts – said the report showed that maintaining a 2-metre distance was “hugely problematic”.

“There’s a real sense that the government is making it up as it goes along,” he said.

What’s the issue?

Have a look at the calculations on the diagrams.  Here is the first one.  What is going on here?

Each person is in a circle, radius 1 metre.  So the area is π m²

Suggesting 4-sq-metres is therefore a bit strange.

That is, however, the area of a square with each person at its centre.  The square would be of side 2m.  Maybe they assumed it was a square?  Maybe they have rounded up (despite pi being closer to 3 than to 4)?

OK – now let’s look at the next diagram:

There’s no explanation here as to why each person now needs to be in a circle with a radius of 1.75 metres.  This gives, apparently, an area of 11-sq-metres. 

Well a circle of radius 1.75m has an area of about 9.6m², which doesn’t round up to give 11, so that theory was wrong.  If this were a square for each person then it would have sides of 3.5m, which dives an area of 12.25m², which isn’t 11 either.

So are there two different errors here?  Or something else entirely?

The original report

I followed the link to the original report (https://www.highstreetstaskforce.org.uk/resources/details/?id=bc16b6bc-0ebb-4b7b-8df8-d8aa6b9a3a9f ) and found that some of it makes sense, some has been reported badly, and at least one part of it is hilariously mathematically ridiculous.

Starting with the sensible part:

The research paper points out that when people move around a shop they can’t stay exactly 2m away from everyone else.  We see this at traffic lights.  If you are four or five cars back when the lights turn green, you can’t immediately start moving forwards.  The first car moves, then the second driver reacts to that movement, then the third car responds a little later, and it is possible that the lights have turned red again before the tenth car has even started moving.  In a shop, if one person stops to browse, or moves towards a display, those around are all affected and need to respond.  To account for this the authors of the paper suggest a bigger circle is needed.  And they go for a circle of radius 1.75m (more on this later).

So that explains the bigger radius.  But what about the two dodgy areas?

The Areas

The paper points out that circles don’t tessellate, so they use a ‘square tessellation’. When arranged in this square formation, the total area divided by the total number of people gives an area of 4m² per person.

[Two diagrams (fig 4 and fig 6) from the paper.]

On the same diagrams I have shown the tessellations:

Later in the paper they mention the hexagonal tessellation.  Let’s try that out with a circle of radius 1.75m

The hexagon goes around the edge of the circle, so the ‘height’ of the shaded triangle is 1.75m.  The base is therefore 2 x 1.75 x tan30.  When we work out the area of this triangle and multiply by 6 (to find the area of the hexagon), we get an answer of 10.6088112 m².  Aha – that rounds off to 11 – and is presumably where that figure came from!

What a shame that the article doesn’t explain this, and what a shame their diagrams are so unclear.

But, I mentioned there are ridiculous things in the report as well.

The first maths error

Here is a quote from the report:

In square or rectangular tessellations, the density of the circles is 0.7854 (Williams, 1979, p. 49). In other words, 78.5% of the space can be utilised.

Based on a square tessellation, in fixed space each person will require a space of π(r)²/0.7894m² (3.9797m² when r = 1m) for the square or rectangular tessellation.

What is going on here?  The diagram shows the scenario, where a circle of radius 1 is inside a square, which clearly has sides of length 2.  The area of the circle is π, and the area of the square is 4, so the fraction of the area of the square that is taken up by circle is π ÷ 4 = 0.7853981634…

The book they are quoting uses a rounded version of this value (0.7854) , which is where the figure of 78.5% of the space being filled comes from.

Rather than just saying the area of the square is clearly 4, the report then tries to divide the area of the circle by 0.7854 to get the area of the square.  This would give 3.999990646, which is close to 4, but is a ridiculous way to do it!

Unfortunately, having written the correct rounded value of 0.7854 in the previous sentence, they then miscopy that in their calculations and divide the area of the circle by 0.7894 instead, which gives 0.979722135, and they have correctly rounded this off to 3.9797. 

A nice bit of maths (with another error)

Having done some bizarre things when working out the area of a square of side 2m, there is then some nice maths to decide that a circle of radius 1.75m is required. 

They use a study of walking speeds in different environments (roughly 1.5m/s), and a different study that looked at how long it takes someone to stop walking (0.5 seconds), worked out that this is therefore a stopping distance of 0.75m and that in order to stay 2m apart, that stopping distance of 0.75m should be added on to each person’s circle.  The original radius of 1m, plus the stopping distance of 0.75m gives 1.75m.

This assumes an instantaneous deceleration and that everyone walks and reacts in the same way. I think these are reasonable assumptions to make.  Unfortunately, the formula they use to calculate this extra amount is noted as this:

We set the value of x, the radius of the inner circle giving freedom of movement, as walking speed / stopping time.

It should be speed multiplied by stopping time.

This then gives them different areas required for different sizes of shop, because people walk at different speeds in smaller shops compared to larger ones and therefore the stopping distance is different.

What now with this?

Unfortunately, this is another reason not to trust the maths you see being used in the newspaper.  Rather disappointingly, there were basic errors in the news article (which didn’t explain the whole square and hexagon thing), and in the university report it was based on, which included multiple ridiculous things.

Perhaps worth using with a class to point out the folly of applying a formula when you don’t need to, and the danger of rounding off too early?

Given how much of this is just twaddle, it seems appropriate to end with the last line of the original article:
“There’s a real sense that the government is making it up as it goes along,” he said.