Geometric Social Distancing

I have another blog on which I post what I call “Quibans” – Questions Inspired By A News Story.  Those tasks are aimed at Yr 12/13 Core Maths classes.  While this post is inspired by a news story, it isn’t really Core Maths appropriate, so I’m putting it here instead.  It might be of interest to Yr 10 and Yr 11 Higher Tier GCSE classes, or as some trig revision at the start of a Yr 12 A-level course.

The article has been edited to remove material that is irrelevant (more on this in a future post); nothing has been altered or misrepresented.

Here is the article, taken from the Daily Telegraph:

Parallelogram, hexagon or pentagon? The safest way for six people to picnic revealed 

A basic knowledge of geometry could come in handy for people who wish to meet when the coronavirus lockdown rules are relaxed next week 

29 May 2020 • 6:51pm  

Groups should picnic in a parallelogram, hexagon or pentagon and if they want to meet together while still maintaining social distancing and good hygiene, experts have said.

On Monday, lockdown restriction will ease so that six people can gather together for the first time since measures were imposed on March 23rd. 

Maths specialist Bobby Seagull who presents the BBC Two series, Monkman & Seagull's Genius Adventures, has suggested that the most efficient way to keep people the correct distance apart is to form a series of triangles, with each person sitting at an apex. When six people do this it forms a parallelogram. 

Mr Seagull said: “My solution is not perfect but means anyone could leave without upsetting others' social distancing. we think that this problem is trickier than at first sight.” 

However mathematicians at Oxford University believe that sitting in a hexagon or pentagon would be a more efficient solution, taking up less space and allowing minimum distance between people. 

Jason Lotay, Professor of Pure Mathematics at Oxford University said that putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 

Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m.  

Alternatively, if one person is placed in the middle so that the other five form a petagram then the maximum distance between any two people is just 3.80m.However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.   

“Of course, things are much better for the person in the middle, who is 2m from everyone, but worse for everyone else, who now have an average of 2.86m to anyone else,” added Prof Lotay. 

“I would probably say that efficiency is given by the total area required and minimising the maximum distance between any two people, in which case the pentagram model is better.   

“However, as I say, it means that one person gets preferential treatment, and it is worse for everyone else.”

There are so many issues with this, which would make it fun to do with a class!

First of all, the article is a bit of fluff, it’s not meant to be taken seriously and the errors are really not a big deal. It may appear as if what follows is being ridiculously pedantic (and of course it is!), but the bullet points at the end make it clear that none of this is really serious.  

Let’s look at some of the figures involved.

Which of these two configurations looks like it has the smaller area?

Which one looks as if the “average distance that one person has to another” is smaller?

To me it appears as if the pentagon has the smaller area and the smaller average distance between the people.

The article says:

However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.  

Ignoring the typo of “pentagram” (which isn’t the same as a pentagon or, as it’s written elsewhere in the article “petagram”), this suggests that my intuition is partly wrong, in that the area of the pentagon is smaller but the average distance for the pentagon is bigger.  Hmm.  Let’s check that out.

The Hexagon

Here’s the regular hexagon, with 2m along each side.  There are lots of ways to find the area.  We could use Pythagoras to find the height of one of the triangles:

The height is 

The area is therefore √3 and the area of the hexagon is 6√3

(An alternative would be to use the area formula   which isn’t as bad an idea as it looks, because this is  

 (we know the value of sin 60 degrees) and this is clearly also the square root of 3.)

… putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 

Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m. 

6√3 = 10.39 (2dp), so we have worked out the area required and this agrees with the article.

The maximum distance between any two people is indeed 4m.  (F and C on the diagram are 4 metres apart, for example.)

How do we work out the ‘average distance’ between people? There is nothing special or different about any of the positions, so if we work out the average distance that A is from all the others, this will be the same as the average distance of B from everyone else, etc.

AB is easy – this is 2m.

AD = 4m and AF = 2m.

AE could be worked out using the cosine rule, but we could look back at the previous diagram and see that AE is the height of two of the triangles, so it is 2√3.

AC is the same as AE.

That gives the average distance of the other 5 points from A as:

This is 2.98564 …  and not 2.29 as in the article.

It is also greater than the values given in the article for the pentagon.

The Pentagon

I think the pentagon values are accurate.  (A class could be asked to work these out, using whatever method they want.)

Here, briefly, are my workings:

Area of a triangle = 
 so the area of the pentagon is (to 2dp)

Now we work out the lengths on the diagram:

Distance AF = 2.

We can work out AB using the cosine rule in the earlier diagram:

AE = AB = 2.3511…

To work out AC, I used the triangle ACF and the cosine rule. 

AD = AC = 3.8042…

This time we need to be careful when working out the average distance, because F is an average distance of 2m away from everyone else, whereas A (and B, C, D, E) are an average of

(This is a useful reminder to use the unrounded versions of the answers …)

To work out the average distance for all of the six people in this arrangement, I did

These figures agree with the article.

The Parallelogram

Bobby Seagull tentatively suggested a parallelogram seating arrangement based on equilateral triangles:

The area and the average distances aren’t included in the article, so our class could work those out!

Here is my brief working:

Area is 4 lots of the triangle we worked out earlier, so it’s 4√3

The distance AD is the height of two of the triangles, so this is the same as AE was on the hexagon: 2√3

AD is equal to CF and BE.

AB = AC = BC = BD = CD = CE = DE = DF = EF = 2

That just leaves AF to work out. I used the cosine rule to get √28 = 2√7

AF is the furthest distance in this configuration, which is 5.29m (2dp).

The average distance is, I think, 2.78m (2dp) which is better than the hexagon but worse than the pentagon.

Real-life implications

• There are clearly a few mathematical simplifications going on here, but there are some other issues too.
• People are being treated as points (I think that is reasonable in that the ‘danger area’ is presumably the mouth/nose which, when looking vertically downwards, are not particularly large.)
• How good are people at estimating 2m? I would guess that no-one is exactly accurate, so much of this is moot anyway.
• Given that the measurements won’t be perfect, it is a bit odd to give answers to 2dp, which suggests that they are accurate to the nearest cm !
• A practical drawback with the parallelogram is that C and D are the only people who can see everyone else (B and F can’t see each other because D is in the way).
• Bobby points out that the parallelogram is good because anyone can leave of their own volition. This is true for the hexagon as well. In the pentagon arrangement, for the one in the middle to be able to escape, a minimum of two other people (adjacent to each other) have to move.
• Finally, the article suggests that the person in the middle of the pentagon has an unfair advantage because they are closer to everyone else.  It will be an uncomfortable conversation for them, though, because they will need to twist their head like an owl to be able to look at everyone!

Postscript 1:

The Daily Mail has an article that starts in a very similar way. Not sure whether they both got it from the same source or whether someone has been copying their homework!  They include the error in the calculation too.

https://www.dailymail.co.uk/news/article-8372009/The-way-picnic-Scientists-say-groups-sit-hexagon-pentagon-parallelogram.html

Postscript 2:

I have written above that the article is clearly a bit of fun.  The comments under both the Telegraph and the Mail articles from members of the public suggest that not many of their readers realised this …