In conversation with Shannon Trust

Compared to the general public, the education levels in UK prisons are disproportionately poor. For example, according to the Ministry of Justice, over 90% of adults entering prison between April 2021 and March 2022 had maths skills below a passing GCSE (equivalent to a grade 3/D or below). While the available statistics for the general population are not as up to date, the corresponding figure for UK adults was put at 78% by the 2011 Skills for Life survey. This finding is supported by a 2019 Ipsos Mori poll, and can be compared to even wider disparities in basic literacy skills. Nearly 70% of prisoners are admitted to prisons with entry-level maths skills, corresponding to maths skills below a GCSE level.

As well as providing functional skills, education is known to have an impact on the reintegration of formerly imprisoned people in society. However, as highlighted in research from the UCL Institute of Education in 2016, there is not always consensus on how to carry this out. While education is compulsory for those of school age in prisons, provision of adult education varies according to government priorities. Most current efforts to educate the prison population focus on employability after release, rather than personal development, but Ofsted reports have shown that this is inadequate, with this education often focusing too much learning only for qualifications, and frequently not engaging with the prisoners most in need of support.

Even during their imprisonment, prisoners who struggle with numbers and reading are put at a disadvantage: much of prison life, from arranging doctor’s appointments to replacing a broken mattress, is governed by applications, commonly referred to as ‘apps’. Prisoners lacking basic skills will not only struggle to maintain their finances, but will find accessing prison services much more difficult.

This not a new problem, and there have been many attempts to address it. In 1997, Christopher Morgan founded Shannon Trust, a prison education charity, inspired by a series of letters between Morgan and Tom Shannon, a man serving a 30-year sentence for murder in Oakwood prison near Wolverhampton. Shannon Trust was founded on the principle of a peer-led approach to learning, with prisoners tutoring those who struggled to read. Following a successful trial run in Wandsworth prison, the trust now helps thousands of learners in prisons across England, Wales and Northern Ireland, and more recently, has started to offer mentoring in numeracy skills.

Shannon Trust's logo

Shannon Trust was founded by Christopher Morgan in 1997 after he corresponded with prisoner Tom Shannon through a pen friend scheme. The letters were collected in the book Invisible Crying Tree. The royalties from the book were used to set up the trust.

Hoping to get an insight into the state of maths education in UK prisons, we sat down with Dan Chadder, a facilitator for Shannon Trust, at HMP Channings Wood in Newton Abbot, Devon. Channings Wood is a Category C men’s prison with a strong focus on rehabilitation, according to a 2022 government report. Through Dan, we also managed to speak to three prisoners three prisoners at HMP Channings Wood who participate in the numeracy skills programme. We cannot use their real names, so refer to them by the aliases Ben, Sam and Callum.

About the programme

A week before our phone call with the three participants, we caught up with Dan via Zoom.

Dan has been working with Shannon Trust as a facilitator since September 2022, having worked in the prison system for four years prior to this. After learning about the low literacy rates exhibited by prisoners, and seeing the work the trust were doing and the positive effect the mentoring sessions had, he decided to join the team. With his experience in peer-led programmes, he facilitates the mentoring sessions: matching learners to their mentors, training the mentors, and managing the programme in general. He best describes the scheme as “one-to-one learning, no tests, prisoner-to-prisoner, out of the classroom environment.”

The maths programme at Channings Wood began a couple of months after Dan joined the team in November 2022. Despite being established only recently, the programme has hit the ground running; he told us: “The maths programme has been really popular since it was introduced back in November… I’ve got a waiting list now, with people trying to sign on. I can’t keep up with the demand.”

We talked about some of the challenges facing the trust: for individual mentor–learner pairs, a huge issue is the logistics of organising sessions. Dan told us that the mentors could struggle to “find a suitable learning environment”, adding that “it’s not quiet” in the prison, so it can be difficult to find an area to concentrate.

HMP Channings Wood was built on an old US army site near Newton Abbot. Map data: \copyright2023 Google. Imagery: \copyright2023 CNES / Airbus, Getmapping plc, Infoterra Ltd & Bluesky, Maxar Technologies.

When the mentoring scheme was first being introduced, there was a worry that the trust might face “scepticism from officers” and that some might feel that the participants were “not taking part for the right reasons” or were using the scheme as an excuse to get out of their cells. Any resistance like this would make it difficult to organise the sessions, and hard to create a comfortable environment for learning.

However, according to Dan, this did not become an issue: “Prison staff have been really supportive, especially as the programme has grown and they’ve seen the positive impact it is having on prisoners.” He also told us that support from the governor of the prison has been important, as this support trickles down and confirms the legitimacy of the scheme to officers, leading to stronger support from prison staff. Dan stressed the importance of this, adding: “You need that support from the top.”

In conversation with three participants

On the day of the phone interview, Dan introduced us to the three participants: a mentor for the trust who we’re calling Callum; and two of his learners who we’ll call Ben and Sam.

When entering Channings Wood, all prisoners are assessed on their numeracy and literacy skills. The numeracy test targets topics like angles, fractions and decimals, telling the time, and arithmetic; while the literacy test assesses their reading and writing skills.

Based on his score in this initial assessment, Callum was referred to Dan as a potential mentor. He was then interviewed and references from wing staff were collected to assess his general behaviour and suitability for the role. As Dan told us: “Choosing the right mentor is based off their assessment as well as their general behaviour, attitude, and most importantly their passion and drive to make a difference to people.” Following a screening from the security department, Callum was offered the paid mentorship role within the trust.

Ben and Sam, on the other hand, were referred to Dan based on this assessment as prisoners who would benefit from mentorship under the Shannon Trust scheme. Dan assigned them to Callum based on their compatibility and on Callum’s availability.

Mentors teach maths or literacy (or often, like Callum, both) in one-to-one sessions in the wing, explaining concepts and working through solutions together. During the maths sessions Ben and Sam work through a set of five books with Callum, with each book increasing in difficulty. There are no tests; the goal of the programme is to give the participants a stronger set of maths and/or English skills which they can then continue to pursue if they choose.

Shannon Trust use books published by Schofield & Sims. Sample questions from their website include “share £550 in the ratio 𝟧 ∶ 𝟨” and “a parallelogram is drawn so that its smallest angle is one third the size of its largest angle. How big is the largest angle?” Image: Schofield & Sims


Callum has been working as a mentor with the programme since he arrived at Channings Wood in October. He works with around six or seven learners at a time, who he meets either daily or every other day. With each session lasting around 30 to 45 minutes, he usually dedicates 2½ to 3 hours per day to mentoring his peers.

On the outside his work was “very different to the teaching role I’ve taken on now.” He told us: “I was a bit doubtful at the start because I’ve never actually taught anyone in my life. I think it’s always something that I feared as a career movement.” Talking about why he became a mentor, he decided to “give it a go—I’ve got nothing to lose so I really stuck into it.” But he found a purpose in his work: “I’ve seen how it helped the learners themselves, how they developed to where they are today… it gives me a real sense of pride.” Callum mentors both literacy and numeracy sessions, but told us he has always loved numbers and maths the most, saying: “I’ve always enjoyed it. I enjoy puzzles.”


Ben is one of Callum’s learners. The pair had actually met before Ben joined the programme, as Callum told us: “I mean, he was on the same wing as me but he wasn’t a learner at the time and we just seemed to speak every single day, between going for the meals, et cetera. And it was just a casual conversation we had… he asked what I teach, and also about the maths programme which had just really started and he was quite intrigued about it.” Callum encouraged Ben to give it a go, and despite his nerves he joined with Callum as his mentor.

Four months later, and Ben is enjoying the scheme: “Callum seemed like a nice person that I could get on with… I needed some of that, someone very secure and not judgmental. He shows me little tricks that I can use on challenging puzzles—I enjoy it, you know, you would not believe how stupefied maths made me before. I’m doing better.”


Sam met Ben when he first arrived at Channings Wood. It was through Ben that he met Callum—Sam saw Callum on the wings wearing a Shannon Trust shirt, and so asked him about the mentorship scheme. With some encouragement from Callum and Ben, Sam’s sessions began the next day.

According to Dan, this referral approach is quite normal: “Most of the learners approach mentors. Since the start of November, there have been more referrals coming directly through mentors.” With the mentors being out on the wings, within the prison population, they are a great point of contact for potential participants who want to learn more about the trust. The peer-to-peer nature of this means they can ask the mentors questions informally, in a no-pressure environment, which eases in potential participants and encourages them to take part to the scheme.

Often, the learners have had bad experiences with maths at school. Ben told us: “I had been told I was stupid all my life”, and that he just “froze when it came to numbers”. Sam left school in year 10, and struggled with the initial assessment tests. One topic he is finding especially difficult at the moment is percentages and fractions. The programme also covers a range of topics ranging from times tables, telling the time, and giving change in the earlier books to finding angles, Bodmas, reflections, translations, and line graphs in the later books. Another struggle for Sam is his neurodiversity, and the stigma associated with it: “I have autism and ADHD and was bullied all my life.” According to Dan, this is common occurrence for learners in the programme, with many learners struggling with attention disorders, dyscalculia, dyslexia, or more generally experiencing “a lot of trauma attached to maths and numbers.”

Peer support

For adult education in general, there are often problems with reaching out for help—but difficulties with maths are not seen as unusual. This is the reason Callum, Sam and Ben thought the maths sessions had a higher uptake than literacy; Callum said: “I really believe it’s a stigma, because if you can’t read, people might make fun of you or you might feel embarrassed. Whereas if it comes to numbers, there’s no shame. Everybody knows that almost everybody has struggled with numbers.”

Views of other participants

“Before I became a mentor, I started out as a learner. I completed the reading and maths programme at HMP Dartmoor. When I transferred to HMP Channings Wood, I started mentoring. For me, I wanted to mentor because I was mentored. I’ve really enjoyed it. It’s given me confidence in myself, that I can make a difference to others. This job has also opened so many doors for me. Because of this job, Weston College are putting me through level 3 teacher training. I’m also getting a good reference from the governor and Shannon Trust for my release. My sentence goes so quickly now because I’m busy. It’s the best job in the jail.”

“I was nervous starting with the maths, but now I’m losing the fear. I started with the reading and that’s coming on great. I had a fear of numbers from school; even picking up the maths book was a struggle. Even the thought of numbers would tie my stomach in knots. But things are changing in a big way. I actually look forward to the maths now, which I never thought I would. I’m in the hands of a really good teacher—teachers at school never explained things like now. He makes it fun, we both laugh at our mistakes! Thank you.”

Part of the advantage of the peer-led scheme seems to be built on the relationship of trust between mentor and learners. Sam told us that sometimes he and Ben continue working on the topics outside their sessions, and “Callum will come see how we’re doing and chat to us.” He explained how Callum “goes above and beyond”, dedicating time on the weekends for more sessions and constantly encouraging them to continue working on their maths and English. Throughout our conversation, Callum repeatedly encouraged Sam and Ben, highlighting their achievements to us and working on their confidence.

Dan sees that many of the learners on the programme “have trust issues with figures of authority. Prisoner to prisoner it’s easier to develop a trusting relationship, a positive relationship, with a level of empathy and understanding that only the prisoners can have… as a facilitator I can’t have that level of bond.” The peer-to-peer mentoring is essential for this scheme to work: “The trust and rapport that mentors are able to build, being prisoners themselves, helps to break down these trust barriers.”

Outcomes of the programme

Despite these difficulties, all three participants reported that they have made enormous progress with the programme and have described genuinely enjoying doing maths. Ben told us: “Maths has always been a major fight for me,” but the one-to-one nature of the mentoring has changed his attitude completely. “Before you couldn’t pay me to do it… now I’m excited about it and I just wanna learn it. I really, really enjoy it.”

Some mentors have gone on to work towards getting teaching qualifications. Callum told us how much he values the experience as a mentor: “It just fills me with pride the fact that I’ve actually contributed to their achievements in a way where I never thought I could. You know, just some simple understanding of reading, a simple understanding of numbers, knowledge to assist someone else to make themselves better and give them great opportunities. How well they’ve come forward is really amazing.”

After finishing the programme, there are further opportunities for the learners. Dan told us: “We’re about being a gateway to further education, so any learners who are coming up to completing or are engaging well and want to start education can be referred on.” The trust works closely with the education department in the prison, helping participants progress into formal education after the programme. Ben told us he was sitting a maths test next week in the education department for the next stage. Sam told us he is looking forward to sitting a maths test in the future. Both Ben and Sam described how useful the skills they have learned from their one-to-ones will be when they leave the prison, going forward and finding a job.

This scheme targets a demographic who are often excluded from education, and gives them the confidence they need to learn the maths. “We primarily work with learners who wouldn’t engage with standard education again… they’ve been left behind at school, because they didn’t get the one-to-one support they needed,” Dan told us. In a group classroom at school, Sam felt that “there were 30 other people there laughing at you”, but now he has the confidence to work towards his exams in the future. Ben feels like the programme will open up doors for him, and hopes he can become a mentor for the maths programme himself.

Prisons also offer learning opportunities in traditional classroom settings, but not everyone performs their best in such an environment Image: US Department of Justice, Bureau of Prisons

Apart from developing skills, Dan highlighted to us that one of the key goals of the programme was about helping with personal development: building trust, making learning fun, giving people a sense of achievement and worth. “We make it clear that the mentors are not expected to be experts in maths. Essentially what we’re trying to achieve is to make learning fun—to go through the books and to make them fun. Get people engaged and enjoying their learning.” According to Callum: “I think that’s what this programme offers… it gives them greater opportunities, whether they’re in prison or whether they’re released into the outside world, this opens a lot more doors for them. Even a little bit of knowledge helps them somewhere down the line in the future and to achieve something like that is worthwhile, it’s done its job. It’s made a difference in someone’s life.”


Making a Möbius strip

So, how do you make a Möbius strip? You take a strip of paper, twist an edge and join up the edges. Tadaaahhh! End of article.

Just kidding. But this shape is indeed a Möbius strip, and it has some interesting properties. For instance, if you cut the Möbius strip down its centreline and unfold your new creation you will find that you have made a surprisingly untwisted (if slightly large) paper ring. But, as cool as Mr Möbius is, it’s his relationship with Mrs Torus that I really want to talk about. Before I delve in, I will first give a disclaimer that we are about to enter the area of topology, which is inherently a rather flexible field, in that one can stretch, bend, cut and glue topological spaces, and have them still be considered… well, the same! The ability to cut and glue things comes with a caveat: two things that have been cut apart must be glued back together in exactly the same way when we’re done stretching and bending. We’ll come back to what this means later. The ability to stretch things out in topology is what I like to call the stretchiness axiom. For instance, these four shapes are topologically equivalent:

whereas this,

is not equivalent, since there is no way to bend the shape to fill in the holes. See? Pretty tangible stuff we’re dealing with. However today our topology is going to get uncharacteristically nitpicky. So without further ado, let us see whether the torus actually is the Möbius strip.

Our torus is the surface of a doughnut in shape. So if we slice through one side of it and stretch it straight, as shown in the diagram below, we’ll get another shape we know quite well: the cylinder. Then, slicing the cylinder down the side, we can fold it out to obtain a rectangle, which is basically a square in topology language (stretchiness axiom activated). This is to say that, if we remember the top edge matches with the bottom edge (to make the cylinder), and the left edge matches with the right edge (to make the torus) then our square is topologically equivalent to the torus. By invoking again the stretchiness axiom, we can rescale this to get the unit square, of side length 1. We use arrows in these diagrams to remind us what matches with what, and with which orientation. This is important, since if we end up cutting the Möbius strip and forgetting which way it used to be attached, we could untwist and reglue it any way we want. Untwisting it would make it into a cylinder, which certainly isn’t topologically equivalent to Mr Möbius.

As we can see in the diagram below, the punchline is a square. Imagine this square to be populated with coordinate points of the form $(a,b)$, where the order of the pair influences where the point is. We call this the space of ordered pairs.

To state formally what it means to identify the arrows, we can imagine each line as an interval from $0$ to $1$. Then every point in the square has coordinate $(a,b)$ for $a$ and $b$ between $0$ and $1$. The identification then means that a point $(x,0)$ on the bottom double arrowed line is the same as the point $(x,1)$ on the top double arrowed line, and that a point $(0,x)$ on the left single arrowed line is the same as the point $(1,x)$ on the right single arrowed line. These are the constraints on the square to obtain the torus.

Now, I know what you’re thinking. This isn’t nitpicky at all. In fact, for a mathematician, I seem to be curiously notpicky. So I give it to you here. Let’s take the unit square with the left-right edge identification and the top-bottom edge identification (remembering the orientations as shown in the picture), and add an additional constraint: a point $(a,b)$ in the unit square must be identified with the point $(b,a)$. In other words, we’re restricting the space of ordered pairs to the space of unordered pairs. You may think that this constraint is more flexible, even reckless! Not caring the order of the entries in a tuple is indeed madness. However in order to forget about the order of entries in a tuple—or in this case, a pair—we must make sure that everything matches up. This final constraint may indeed put a spanner in the works with its diva-level high demands, but we put in the effort to catch a glimpse of the glory that is the configuration space. A configuration space on $n$ elements is the space of unordered $n$-tuples, and here we are looking at the configuration space on 2 elements. This kind of space is one which comes up here and there in topology, and is a simple yet fruitful object of study. Furthermore, fixing the configuration constraint here may give us a surprisingly nice result.

In case you’ve lost track of all the constraints, let’s list them:

      • for any point on the top or bottom edge, we have $(t,0)\sim(t,1)$,
      • for any point on the left or right edge, we have $(0,l)\sim (1,l)$,
      • for any point in the square, we have $(a,b)\sim(b,a)$,

where $\sim$ denotes identification. The first two of these constraints define the torus, but what shape forms when we add this extra one? The easiest way to enact the last constraint is to recognise which points match up:

We can see that the points in the triangle below the diagonal correspond exactly to those above the diagonal. As the points above the diagonal add no new information, we fold down along the diagonal to make a right-angled triangle. We don’t really need the points on the diagonal at the moment (ie points of the form $(a,a)$) so let’s delete them—although I’ve highlighted where they would have been in red so we can see what became of them should they need to be salvaged. If I ever need them again, I’ll call them $\Delta$ (for $\Delta$iagonal). But now, by identifying the left edge with both the bottom edge and the right edge (and the top edge with both too), we realise this means we need to glue the horizontal edge to the vertical edge.

The identifications we will make on the torus. The top and bottom edges are identified, the left and right edges are identified, and then all points $(a,b)$ are identified with their diagonal twin $(b,a)$.

This seems a bit impossible. One way of going about it is cutting it. The running theme here is that cutting is allowed as long as you remember to glue the cut back together again in the same place with the same orientation. Indeed, we cut the triangle down the middle and flip one of the pieces to glue the two required edges, as shown above. The pink edges are the ones we have to remember to glue back together in the correct orientation.

Gluing the last two edges together seems difficult, but, again due to the fact that we are in the flexible world of topology, we can stretch our grid out to a long strip, and flip while folding so that the two pink arrows are facing the same way. Now we have a Möbius strip; all procured from the space of unordered pairs.

Another way to get from the triangle to the Möbius strip is to imagine it’s a mozzarella cheese toastie. Since the cheese is stretchy, you can pull apart the two edges at the right-angle and stretch it out to a rectangle, and then twist and glue the edges from there to make our Mr. Möbius. This is slightly more intuitive for me, but we also lose a bit of information, since the edges (or more accurately, singular edge) of the Möbius strip aren’t as explicitly shown to be the diagonal values $\Delta$.

The method we used in the first place lets us see that including the set $\Delta$ of points $(a,a)$ corresponds to including the edges of the Möbius strip, so we can include them again if we want (although it is customary to leave them out).

The method of identifying points is known as a quotient map, and here we have quotiented the torus by the relation $\sim$ defined by $(a,b)\sim (b,a)$ to get the Möbius strip. Hereon, I’ll refer to the middle square in our torus identifications as the Möbius square.

The map to the circle

Our study of the Möbius strip is all about turning lines into circles, seeing if we can manipulate things in a fruitful way, turning circles into lines, and doing it all again. With this in mind, we ask ourselves whether it is possible to find a map from the Möbius strip to the circle.

One way to get from the set of all unordered pairs to the circle is simply to use the projection map $\pi_1$ from the pair to its first coordinate; $(x,y) \mapsto x$. However we run into a problem because $(x,y)=(y,x)$ so $(x,y)$ also maps to $y$. This requires that $\pi_1(x,y)=x=y$ which is a contradiction since by removing $\Delta$ we require that $x\neq y$. So we need another idea for our map.

I’m going to ask you to trust me here, because this map might get… weird. First, rotate the Möbius square until the orange diagonal $\Delta$ is horizontal. Also, take the circle and unroll it underneath this diamond, until it’s a straight line of the same length as $\Delta$ (also horizontal). For our map $p$, any point $(a,b)$ in the Möbius square is sent to the point at which the vertical line through $(a,b)$ intersects the line below. This looks promising; for any $(a,b)$ we have $p(a,b)=p(b,a)$, so one of our constraints is satisfied (as $p(a,b)=p(b,a)$ is stronger than the two being equivalent).

Turning our attention to the other constraints, we see that we need the equivalence $p(0,t)\sim p(1,t)$, as in the Möbius square. This is a bit strange, because the the line down from $(0,t)$ is nowhere near the line down from $(1,t)$. Hmm. Okay, let’s try to make progress with the other constraint; here we want $p(t,0)\sim p(1,t)$. And again, we have that the line down from $(t,0)$ is nowhere near the line down from $(t,1)$. That’s not so helpful. Then what can we do with this?

Well, I suppose if we pick some $t\in [0,1]$ and draw out the diagram, we can see that the image of $(0,t)$ and $(1,t)$ are exactly a half-line apart. The same is true for $(t, 0)$ and $(t,1)$. So perhaps let us identify not only the start and endpoints of this line representing the circle, but also the midpoint. Then the line is representative of two loops of the circle. Another way to say this is that the map is 2-periodic.

In a way, this makes sense: if we identify $(a,b)$ and $(b,a)$ to be the same then all the information we need is where the line joining them meets our orange diagonal $\Delta$, so really our map $p$ is a projection from $\Delta$ to the circle, where $\Delta$, as we found earlier, represents the edge of the Möbius strip, wrapping twice around the circle. So, since $\Delta$ is a double looped circle, it’s no wonder that the interval we’ve laid out below turns out to be one too. You can imagine that if we squeezed the Möbius strip width-ways until the width was 0 (ie the strip was a single line) then the edge of the Möbius strip would literally run the same circle twice.

With this in mind, let’s reassess. We have a map $p$ from Mr Möbius to the circle with $p(0,t)=p(1,t)$; $p(t,0)=p(t,1)$; and $p(a,b)=p(b,a)$. This loops twice around the circle. Recall that the first two identifications come from the identification of the two single arrows and the two double arrows in the Möbius square. The third identification specifies the unordering of pairs in the Möbius strip, ie reflection in $\Delta$. So the preimage (known in topology as the fibre) of each point in the circle will have two lines in the Möbius square, as shown.

This is all well and good, but why can we only seem to find 2-periodic maps; why do we have plenty of maps to a double loop of the circle, and not just the circle itself? To answer this, let’s figure out on the Möbius square where the centreline will be.

This shows that the centreline passes through all the unique points in the image. So, we can imagine that any map which doesn’t pass through the centreline has to map twice around the circle; to stay continuous, we need to stay either in the top half or the bottom half of the Möbius square. To give you an idea of why this restriction would have this effect, we are limited to maps like the one sending the top edge of the Möbius strip to the circle. As we know, following the top edge of the Möbius strip amounts to looping twice around the Möbius strip, since the strip has only one edge. Intuitively, if we imagine instead that we are mapping from the boundary of the Möbius strip to the circle that forms its centreline, we loop twice around it, getting a 2-periodic map. Meanwhile, the map from the centreline (and those paths equivalent to it, except for slight detours) maps to a once-looped circle. We can imagine that a continuous path similar to the centreline except for a slight detour is like a deformation of the centreline. Therefore, any such path will also have a once-looped map to the circle. Having found your long sought-after circle map, you may now put your feet up; our work is done!

In this article, not only have we gained the invaluable skill of making a paper Möbius strip, we have also (con)figured out how to construct it topologically from another shape: the torus. In doing so, we have discovered that the only difference between the two spaces is the specification on ordering, which is an entirely unexpected result. The circle underpinning the theory of the Möbius strip has also been catered for: the map we have found from the Möbius strip down to the once-looped circle was constructed so that we can map back up to the Möbius strip if we want to. Indeed, these maps are those from slight deformations of the Möbius strip’s centreline. This is with the knowledge that straying too far from this lands us on the Möbius square’s diagonal, and the Möbius strip’s edge, which, as we know, loops twice around the circle, and so doesn’t give us a nice two-way map anymore.

I will leave you with a question. Clearly cutting a band and giving it a half twist gives a Möbius strip, which is not topologically equivalent to it. However, an even number of half twists would match up the arrows such that the arrow orientations agree. So, is a band with an even number of half twists topologically equivalent to the untwisted band? Using this, can you conclude whether it is possible to find an $n$-periodic map by taking further half twists of the Möbius strip? I hope that the reader will be enticed to ponder further on this problem, and the many others in the literature surrounding the Möbius strip.


Penguins: the emperors of fluid dynamics

Let me guess, you like penguins? I mean, who doesn’t?! And I am guessing you like maths too? Well, what if I told you I could combine the two? Have I captured your interest?

Antarctic penguins experience some of the most extreme weather conditions on Earth, where wind speeds exceed 70mph and temperatures drop below -40°C. However, penguins have adopted an effective survival strategy. The whole colony comes together to form one large huddle to shield the birds from the wind and to conserve each penguin’s heat. While huddling, the birds are constantly waddling around, causing the entire huddle to move. But what are the underlying mechanisms driving this movement? And can we track the huddle’s movement over time?

They’re hot, then they’re cold

To answer these questions, we need to view the problem more mathematically. Intuitively, we know that the penguins are huddling together for warmth, so let’s start by thinking in terms of heat and energy. In other words, let’s consider the thermodynamics of the problem. We identify two key heat transfer processes at work.

First, the penguins are losing heat to the cold air and the fast flow of the wind means that a temperature gradient—a difference in temperature between the air and the penguins—is always sustained. You will have experienced this yourself in everyday (non-Antarctic) life. On a windy day, you feel colder, right? This is because your body loses heat to the air, and the air you just warmed up moves on and is replaced by fresh, cold air for you to lose heat to again.

Second, the penguins are receiving heat from neighbouring birds in the huddle. Once again, think about your everyday life: imagine standing in a crowded queue waiting to buy the new Zelda game. Despite the cool company you are surrounded by, you feel hotter in the crowd as you’re gaining heat from the people around you. It is the same in a penguin huddle (except they can’t play the new Zelda game: their flippers can’t hold the controller). This heating is particularly effective near the huddle centre, where temperatures can reach up to 37.5°C: higher than the average human body temperature!

Unfortunately, this temperature is actually too high and can cause the penguins to overheat. To counteract this, the birds regularly reorganise themselves in the huddle: those in the centre move to the edge and vice versa. In other words, they are all massive Katy Perry fans—by reenacting the song Hot N Cold, all members of the huddle are kept at a nice, cosy average temperature.

We can start making a model of the huddle’s movement based on these two heat transfer processes. First though, let’s add one final condition: we assume that no penguins leave or join the huddle, so the size of the huddle, ie the number of penguins, is conserved. So, there are three things in total affecting the huddle’s movement: cooling by the wind, heating by the penguins and conservation of the huddle size. These three effects become important later, so keep them in mind!

In love with the shape of $\mu$

Our aim is to see how the entire huddle moves over time. Therefore, instead of viewing the huddle as hundreds of individual penguins (a discrete model) we treat it as a continuous ‘blob’ of penguins (a continuum model).

Imagine looking down at the penguin huddle from above and tracing around its edge. The resulting curve is called the huddle boundary and we label it $\mu$. All the penguins are inside of $\mu$ so we call this the interior ‘penguin’ region $P$. The huddle is tight enough so that no wind passes through to $P$ and all wind flow is confined to the exterior ‘wind’ region $W$. Wind flowing over the top of the huddle just contributes a constant heat loss to every penguin; this will not affect how the huddle moves. Hence we can treat our problem as two-dimensional.

Top down view of the huddle

Side view of the huddle

This is an example of a free boundary problem which is exactly as it sounds: you want to find the motion of some boundary which is free to move wherever it chooses, ie its flow is not confined within walls or pipes. In this case, we want to find how the penguin huddle boundary $\mu$ moves, which would then tell us how the entire huddle moves.

We need to find the velocity $v$ of the curve $\mu$ in the outward normal direction—the direction marked by the unit normal vector $\hat{\boldsymbol{n}}$ in the ‘top down view’ diagram. Consider energy conservation: the (heat) energy lost by the penguins must be equal to that gained by the wind. From this idea, we get an equation for $v$, \begin{equation}v = \hat{\boldsymbol{n}}\cdot\boldsymbol{\nabla}T_W-\hat{\boldsymbol{n}}\cdot\boldsymbol{\nabla}T_P + A(t),\tag{⁂} \end{equation} where $\boldsymbol{\nabla}T$ is the temperature gradient in the wind/penguin regions $W/P$.

Looking at the right-hand side of this equation: the first term relates to the temperature of the wind, the second term relates to the temperature of the penguins and the third term, $A(t)$, ensures that the area enclosed by the curve $\mu$ is conserved. As if by magic, these three terms exactly represent the three huddle effects we defined earlier: cooling by the wind, heating by the penguins and conservation of huddle size, respectively!

In order to solve for $v$, we need to find the wind and penguin temperature gradients $\boldsymbol{\nabla}T_W$ and $\boldsymbol{\nabla}T_P$. The tricky part is that we need to solve three more equations in order to get these two terms. For $\boldsymbol{\nabla}T_W$, we must solve the wind flow and wind temperature equations and for $\boldsymbol{\nabla}T_P$, we need to solve the penguin temperature equation (given in the box below). Adding some suitable boundary conditions, we get the complete system of equations ready to be solved.

So, to get the normal velocity $v$ of the huddle boundary $\mu$, we have to solve three equations across two regions: the wind and penguin regions. The whole thing is starting to look a bit big and scary, so how can we deal with it?

The circle of life

I love circles! You may think they are one-sided and pointless, but they have constant radius and infinite lines of symmetry: how can you not like that?! Penguins like circles too because there is a known solution to our huddle problem when the boundary $\mu$ is a circle. The issue is, penguins do not always huddle in circles, but any random boundary shape can always be smoothed out into a circle, right?

Dealing with wind

The wind flow and wind temperature equations, respectively, are \begin{equation*} \nabla^2\phi=0,\;\;\;\;\;\;\;\; \textit{Pe}\,\boldsymbol{u}\cdot\boldsymbol{\nabla}T_W = \nabla^2 T_W, \end{equation*} where $\phi$ is defined such that its gradient $\boldsymbol{\nabla}\phi$ is the wind velocity $\boldsymbol{u}$, the Péclet number Pe is a measure of the ambient wind strength and $\nabla^2$ represents diffusion. The first equation defines the motion of the wind. The second defines the steady convection and diffusion of temperature in the wind.

The penguin temperature equation is \begin{equation*} \nabla^2T_P=-R, \end{equation*} which represents the diffusion of temperature in the penguin huddle, where $R$ is the rate at which each penguin generates heat.

Let’s take our huddle boundary. We reshape it into a circle and find the known solution. Then we just ‘un-reshape’ the curve back to the original geometry and in this ‘un-reshaping’ step, we also transform the known solution. This process is, in fact, a mathematical method known as conformal mapping which uses a theorem called the Riemann mapping theorem.

Let’s say that the transformation, or conformal map, from the circle to the penguin huddle boundary $\mu$ is given by $f(\zeta,t)$, where $\zeta$ represents the coordinate system for the circle and $t$ is time. Our whole system of equations can be combined into one single equation in terms of $f$,

\begin{equation}\frac{\operatorname{Re}\Big[f_t\overline{\zeta f_\zeta}\Big]}{|f_\zeta|} = C-H+A(t),\tag{‽} \end{equation}
where $f_\zeta$ and $f_t$ are partial derivatives of $f$ with respect to $\zeta$ and $t$, $\operatorname{Re}[🐧]$ is the real part of $🐧$ and $\overline{🐧}$ is the complex conjugate of $🐧$. The equation looks complicated but the important point is, the left hand side of (‽) is just the normal velocity $v$ expressed in terms of the transformation $f$.

Riemann mapping theorem

Consider the penguin huddle in the ‘$z$-plane’. There is always a transformation, or ‘conformal map’, from the huddle boundary $\mu$ in the $z$-plane to the unit circle in some other ‘$\zeta$-plane’, and vice versa. The conformal map can also transform the entire region exterior to the huddle boundary (the wind region) to the region exterior to the circle, and vice versa. The interior regions—the penguin region and the circle interior—are not additionally transformed.

Note: the wind flow and wind temperature equations are unchanged in the conformal mapping process—they are ‘conformally invariant’. This makes things easier when finding the solution to our penguin huddle problem.

The right hand side of (‽) represents the three huddle effects: $C$ is the wind cooling effect, $H$ is the penguin heating effect and $A(t)$ is the area conservation effect as we saw in (⁂). These three terms can now be found numerically, ie via computer approximation, which involves much more work than this simple sentence suggests, but the margin is too small to write it all down.

Instead of solving our big, scary system of equations, now all we need to do is find the transformation $f(\zeta,t)$ from the circle to the huddle boundary. Once again, we can find $f$ numerically using any suitable programming language. I will be using the best one: Matlab. Sorry Python lovers, please consult Chalkdust issue 16’s big argument for my justification.

How do you like your eggs in the… huddle

Now it’s picture time! Let’s see how a penguin huddle moves in response to a wind flowing horizontally from left to right.

First, let’s take an initially circular penguin huddle. Here is how the huddle moves:

The shape outlined in black on the far left is the starting shape of the huddle—in this case a circle. Then each subsequent plot going right is how the huddle shape is changing over equal time increments. We notice two things: the huddle moves downwind and the boundary changes shape until eventually it becomes egg-shaped.

Let’s try something more complicated. We will keep all the conditions the same but, this time, the starting shape of the huddle will be a wonky pentagon. Its evolution looks like this:

Again, the huddle moves downwind and its shape turns into an egg, the eggsact same egg that the previous huddle evolved into. In fact, regardless of the initial starting shape of the huddle boundary, the penguin huddle will always evolve into the same egg shape. Eggstraordinary!

Now, what happens if we increase the wind speed? Let’s take an initially circular huddle again:

The huddle still turns into an egg, but the egg is much narrower now, like a tear drop. Eggsciting!

And finally, what happens if the effect of penguin heating is raised? Well then the huddle evolves like this:

Now the final huddle shape is a very circular egg. Eggsceptional!

So, all huddle boundary starting shapes move downwind and turn into an egg shape. A higher wind speed gives a more streamlined egg, whereas an increased penguin heating effect gives a more circular egg. This is what we eggspect to see: penguin huddles in the real world move downwind and have similar boundary shapes as the ones we’ve shown in our pictures. Intuitively, it also makes sense for there to be some optimal huddle shape for minimising heat loss. So all these conclusions seem sunny side up to me.

A whole new world… of free boundary problems

To sum everything up, penguin huddles move and change shape during cold winds to minimise the heat loss each penguin experiences. The two key thermodynamic processes, cooling by the wind and heating by neighbouring penguins, result in a whole system of equations governing the huddle’s movement. But the method of conformal mapping simplifies everything to a single equation, which means a numerical simulation of the huddle’s movement can be made.

Similar methods can be used to solve other free boundary problems. You could model other natural phenomena, such as water waves, melting ice or dissolving rocks. Even wildfires, the polar opposite of penguins, are free boundary problems and can be simulated in a similar way to penguin huddles. You could also simulate other animal groupings, like bird flocks, fish schools or insect swarms. Free boundary problems exist throughout nature and multidisciplinary mathematical tools are needed to solve them.

One final note. Lots of rigorous mathematics was needed for us to understand penguin huddles. But penguins have been forming huddles for millions of years. How did they figure it out? Perhaps huddling is just a natural instinct for them. Yes, that must be it. Or maybe not. Maybe they’ve known all along. Thermodynamics, conformal mapping, Matlab’s superiority over Python, all of it! Was Madagascar right? Are penguins smarter than us? What else have they been hiding? Wait, who’s that behind me? … I’ve said too much. You didn’t see anything.


How to be more Pythagoras

It all began in the second lockdown of 2020. Takeaway containers littered the floor of my flat, I hadn’t exercised in months, and after what felt like an eternity shut inside, my body clock had forgotten the difference between night and day. The rate of change in my life with respect to time ($\mathrm{d}\Delta/\mathrm{d}t$) was at an all-time low, and I desperately needed to increase it.

Many people, when faced with the same daily challenges, turned to social media for their wellness advice. From waking up at sunrise for their morning yoga routine to curating the perfect spinach and kale smoothie, everyone else seemed to be live, laugh, loving their way through lockdown in an exasperatingly Instagrammable manner.

Forever the mathematician, I decided to turn to an altogether different source for my wellness tips and tricks: Pythagoras, the Greek mathematician of triangle fame. I decided to embark on a week-long journey to live in Pythagoras’ sandals, and to my amazement, discovered a remarkable overlap between the ancient philosopher’s teachings and those of modern lifestyle gurus.

History’s greatest influencer

When Pythagoras first arrived in southern Italy in around 530 BC, the distinct lack of social media didn’t stop him from gaining a large number of followers. His admirers, or fandom, eventually garnered their own name—the Pythagoreans were quite literally the Swifties to Pythagoras’ Taylor Swift. In fact, Pythagoras was so influential that even 2500-odd years later, Pythagoras’ theorem is taught in almost every classroom around the world. Everyone and their mother can tell you that:

Charli D’Amelio could only ever dream of that level of reach, especially when you consider that Pythagoras is unlikely to have thought up this famous theorem himself. Historical evidence points to earlier proofs of the theorem in ancient Babylon, India, and even by other mathematicians within Pythagoras’ sect.

The goal of many social media stars is to become something bigger than themselves; to become immortalised in memetic form for the whole world to remember. Well, not only has Pythagoras been immortalised by this small but significant theorem, but the ancient philosopher was also said to have taught his followers how to attain immortality in a very literal sense. Though I have a feeling we would be aware if he had actually succeeded in this, I decided to try it out for myself. Here are just some of the wise words of advice I followed in my week spent as a Pythagorean:

 1. Take daily morning walks

Much like fitness personality Joe Wicks, a large part of the average day in Pythagorean life was getting up early for exercise, in the form of a long walk every morning amongst the verdant hills of Croton.

Not just content with getting up early for #FitnessInspo, a Pythagorean cultist would fit in a whole host of activities before heading off to work: they would use this time to meditate, contemplate, or even curate a healthy vegetarian breakfast.

An emphasis on athletics and therapeutic dancing helped to supplement the daily exercise of a Pythagorean. Pythagoras himself is said to have been friends with legendary Greek wrestler Milo of Croton after Milo saved him from a collapsing house. According to legend, old Pythag even had a thigh made of pure gold that he proudly displayed at the Olympic games—he clearly took his exercise very seriously.

2. Embrace your own personal style

It is said that Pythagoras was a trendsetter in his day, wearing white trousers while everyone else was wearing the robes from last decade’s edition of Samian Vogue. I don’t think it is too unrealistic to say that, if he were alive today, Pythagoras would be posting videos of his morning workout routine while advertising the newest pair of white Lululemon leggings.

3. Utilise both sound and silence

From Olivia Rodrigo to your K-pop bias from Blackpink or BTS, there is a massive overlap between musicians and the cover stars of your monthly lifestyle magazines. After hearing the varying sounds of hammers striking anvils within a blacksmith’s, Pythagoras excitedly began investigating and found a link between the proportions of the hammers and the sounds that they created. It was clear to him that music abides by the laws of mathematics, prompting him to talk about the proportions and ratios of the stars and planets as the “harmony of the spheres”. Clearly, the only thing between Andrew Wiles and the cover of next month’s Cosmopolitan is a hit debut album.

On the other side of the spectrum comes the silence of meditation. Pythagoreans were said to have meditated at the beginning and end of each day, setting aside time to relax, calm, refocus and rebalance. Self-awareness and meditation are at the very base of your average influencer’s hierarchy of needs: without centring themselves at the start of the day how would a Pythagorean even start to tackle some of the biggest problems in mathematics? I’m no historian, but I’d wager that the day Pythagoras drowned his follower Hippasus for believing in irrational numbers was the day he skipped his morning meditation.

4. Never eat a bean

A widely known but heavily disputed fact about Pythagoras revolves around a particular aspect of his vegetarianism. One of the main philosophical schools of thought that Pythagoras subscribed to was that of metempsychosis, or the transmigration of souls. Belief in reincarnation is fairly widespread amongst different cultures and histories; however, it was the belief that beans are part of this grand circle of rebirth that makes Pythagoras stand out from the crowd. Yes… it is said that Pythagoras believed that beans had souls.

According to the great philosopher, a bean could have been a scorpion in another life, which could have in turn been his great uncle Edmund. This belief in metempsychosis has made Pythagoras a bit of a rock star amongst modern day vegetarians: Pythagoras didn’t eat meat, or beans, because of the morality of killing and eating a living being with a soul.

His belief was so staunch that the most famous story of his death involves him refusing to run through a bean field while being chased out of town by an angry mob. It is worth noting that all of the information we have on Pythagoras today is third hand texts that paint him as an almost mythic figure, so should be taken with a pinch of salt (so long as the salt doesn’t have a soul, that is).

5. Don’t touch a white rooster

There is some science to explain why the lifestyle techniques of the Pythagoreans are being repeated by social media celebrities a few millennia later. A consistent morning routine can create a sense of structure and order to your day, improving sleep patterns and setting you on the right path. Exercise improves your wellbeing through the release of endorphins and in the long term helps improve cognitive function and self-esteem. Even something as simple as sunlight can have a marked effect on your serotonin levels, help battle insomnia, and improve your mental health.


Not all Pythagorean practices have analogues in the modern day, but here is a selection of some of Pythagoras’ rules and beliefs that might well be one of the next big viral trends:

During my week as Pythagoras my takeaway containers were quickly replaced by the remnants of Mediterranean fruits and vegetables, my body had finally got moving again, and the lockdown fog clouding my mind slowly began to thin. Through attempting to fit into Pythagoras’ gleaming white trousers for the week I was unknowingly becoming the ultimate wellbeing guru. So, there we have it, from morning walks and river talks to rhythm and vegetarianism, we could all stand to be a little bit more Pythagoras.


On the cover:

Our whole world is constantly changing, and this can be modelled by differential equations: equations that link together quantities with the rate at which they are changing. If you’re in a hot room and your friend opens the window in the middle of winter, the room will cool down. The colder it is outside, the faster the room cools down.

Mathematically we could say the change in temperature in the room ($T$) over time ($t$) is proportional to the difference between the indoor and outdoor temperature,
\[ \frac{\mathrm{d}T}{\mathrm{d} t} = \alpha (T-T_{\text{outside}}).\]
That’s a differential equation in time. But this isn’t the full story! If you stand closer to the window, you’re going to feel colder: heat spreads in space as well as in time.

Differential equations that combine changes over time and space (or just multiple variables in general) are known as partial differential equations, or PDEs. It just so happens that one of the coolest (or hottest?) PDEs is the one that describes how heat spreads out in time and space ($x$):
\[ \frac{\partial T}{\partial t} = D \frac{\partial^2 T}{\partial x^2}. \]
The interesting thing is that this model also works for anything that likes to spread out, not just heat, and in as many dimensions as you want. Bacteria on surfaces, chemicals in water, people on the Metro… all spreading out (technically assuming some sort of randomness) can be captured in this, the diffusion equation.

But if we’re describing bacteria spreading out, in addition to diffusion, we might also want to include reproduction. How quickly do bacteria reproduce? A simple model is growth proportional to the current population size, $u(x,t)$! That gives us, for some parameters $D$ and $\alpha$,
\[ \frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2} + \alpha u.\]
But why stop there? We could have different sorts of reproduction rules, and even extra species! How about some predator–prey dynamics of rabbits, $u$, being chased by foxes, $v$, in two-dimensional space?
\frac{\partial u}{\partial t} &= D_u \nabla^2 u+au-buv,\\
\frac{\partial v}{\partial t} &= D_v \nabla^2 v+cuv-dv.
These sorts of systems—‘diffusion + something’ —are called reaction–diffusion systems, and they really are everywhere in mathematical modelling of physics, biology and chemistry.

Not only that, the solutions are totally not boring. They can pulsate, move around, and collide in complicated ways! But for us to appreciate this, we need to be able to see it.

And that is actually quite a challenge. Because the solutions are functions of space (often 2D) and time, they are hard to draw on paper. But nature has already solved this problem! If we are modelling two competing populations of plants in a desert (say a green plant and a white plant), we could climb a big rock, look down from above and see where each plant is growing. Darker spots means more of the green plant, and lighter spots means more white plant. All we have to do to see the solution is then wait. And wait. And wait. And so long as we’re not eaten by a snake, we’ll see the whole solution in time.

Aerial view of a Gapped bush plateau in the W National Park, Niger. Image: Nicolas Barbier, CC BY-SA 3.0.

It is reasonable not to want to risk death by desert snake, so instead we can just plot a solution in 2D by getting a computer to colour in a rectangle and then change the colouring over a few seconds…

…so long as we can get the computer to (a) plot, (b) make videos, and (c) actually solve the system! Because the other downside is that solutions to these systems are hard, and sometimes impossible, to find. We can do a pretty good job with numerical approximations using a computer, though, but even approximations can be awkward to set up and require a lot of tuning on top of technical programming knowledge. And what if all you want to do is understand what changing a single parameter does?

This is where our new website,, comes in. We made it for our own interest and we think you might like it too. Think Desmos for PDEs, except it makes fun pictures that the cover art for this issue is based on. And it does so instantaneously, right before your eyes!

Here’s a solution to the diffusion equation:

We started with heat in the centre of the square and then let the heat diffuse outwards. We’ve chosen to make red hot and blue cold, but you can pick the colour scheme that works best for you. But the other images are more exciting—what about them? Read on!

You might have heard about recent trends in scientific computing where we give graphics cards work to do, instead of the standard processing chip (the CPU). Your phone or your computer displays boomer memes or fascinating Chalkdust articles on the screen by telling each pixel on the screen what colour to be. It’s a big grid. And we can numerically approximate a PDE solution as well by treating the space we’re solving on—for example, our desert floor—as a big, but really fine, grid. On this grid, the derivatives become differences (it’s replacing $\mathrm{d} y/\mathrm{d} x$ by $\mathrm{\Delta} y/\mathrm{\Delta} x$) and the maths becomes simple.

The nice new stuff is that before, you’d have to get your CPU to solve the (even simple) maths on its own grid, and then display it on the screen. This is slow. Today, you can get the graphics card to do the maths itself, directly displaying the solution on the screen. It’s lightning fast!

This is what does. We use Javascript, a WebGL library and a low-level language called GLSL so that you can write your equations in your browser, on your phone or computer, and then see the solution evolve in time. It works instantly. You can even prod it with your finger to change the current solution.

OK, but maybe you think we are interested in this just because it’s our research (it is) or because we teach this stuff to undergraduates (we do). But really it’s because the solutions are really pretty! Just look at these!

You might not know that the first run at looking at these solutions was done by codebreaker Alan Turing—yes, this is Alan Turing’s Other Work! Let’s take a look at some fun examples, which we invite you to play with on

Artificial life

The Gray–Scott system, given by
\frac{\partial u}{\partial t}&=\nabla^2 u+u^2v – (a+b)u,\\
\frac{\partial v}{\partial t}&=D\nabla^2v-u^2v + a(1-v),
has been heavily studied as a model of pattern formation, and even as a model of ‘artificial life’, analogous to Conway’s game of life. Many websites and research papers classify the different things it can do. As the values $a$ and $b$ vary, this system can exhibit waves, oscillations, stationary and moving spots, holes, or chaotic oscillations. Here’s one behaviour when $D=2$:

As with non-artificial life, the model gets weirder and more fascinating the more you study it!

Beating hearts

The cover is inspired by the FitzHugh–Nagumo model, pictured at the front of this article,
\frac{\partial u}{\partial t}&=\nabla^2 u +u-u^3-v,\\
\frac{\partial v}{\partial t}&=D\nabla^2v+ \varepsilon_v(u-a_v v-a_z).
Originally, it was a simple model of voltage in the squid giant axon (not to be confused with the giant squid axon). It has since been used to model electrical activity in the heart, as well as formation of patterns in contexts including the folds and wrinkles of the brain. The model can do several things such as: relax to a uniform state, oscillate uniformly, form fixed patterns, and form moving patterns (typically spiral waves and/or chaos). In the example you can find on, we’ve set the model to allow both uniform oscillations and stationary patterns, so you get a competition between the two, with one behaviour winning if you tweak the parameters slightly. This scenario is not just mathematically interesting: scar tissue in heart muscles has been shown to prevent waves of excitation from moving through the heart. These patterns are pretty but deadly!

Your face as a PDE

Models can contain functions of just space, $S(x,y)$. Maybe $S$ is the location of nutrient-rich soil for plants or dangerous terrain for animals. But why not let $S$ represent a picture? And why not let that picture be your face? We adapt the so-called Schnakenberg model,
\frac{\partial u}{\partial t} &= D_u \nabla^2 u + (1-S)-u+u^2v,\\
\frac{\partial v}{\partial t} &= D_v \nabla^2 v + 1-u^2 v,
where the brighter your picture is, the higher the value of $S$. Behold the glorious scorpion picture!

Visit on your phone, take a selfie, and see your face as a PDE pattern!

What can you find?

So we invite you to have a play! Head over to and see what cool pictures and effects you get when you adjust the parameters on different models. If you’re techy, it’s all open source so you can see how we did it on GitHub. We’d love to see what you can do!


Do the shuffle: finding π in your playlists

Being a mathematician makes applying your vocation in apparently non-mathematical situations something of a reflex. Very often, this can be to the annoyance of others, including your long-suffering friends, who may label you with words such as ‘party-pooper’ or ‘killjoy’. However, in some cases, mathematics can make you the life and soul of the party. The following—the mystery of the repeating Starship song—is one such example.

At the party in question, the host had the foresight to put a playlist on shuffle to keep us pacified. But he was soon to regret this decision, as proceedings were interrupted by an unwelcome guest, like Banquo’s ghost at the feast—namely, Starship’s 1985 classic We Built This City. Don’t get me wrong: the song is a surefire singalong firestarter. But as with most things from the 80s, its glossy studio sheen is liable to wear off easily. When the song came on for the first time, we were relatively enthusiastic—but upon the second time in 10 minutes, we began to ask ourselves searching questions, such as—who is Marconi? Why was he playing the mamba? And more pressingly, why had we heard the same song twice in close succession? Were we merely victims of chance? Was it a sign from God? Perhaps some arena rock-based apparitions were at play? None of us knew how to interpret it. An eerie silence fell upon the room as our mortified host pressed skip.

I think I see the problem

Thankfully, however, I managed to rescue us from our music-based turmoil with my combinatorial prowess. Quickly, like a cowboy from the wild west whipping out their pistol, I broke out my comically large pad of paper. Although all of my friends (bar one) were either inebriated or otherwise distracted, I was undeterred in attempting to come up with a valid explanation that could satisfy the remaining two of us. Given a playlist of $n$ songs, how long will it be before a song is played twice? I set to work. Little did I know that the ensuing mathematical odyssey would take me from Harry Styles to hashing functions, from Ra-Ra-Rasputin to Ramanujan, from popular music to population dynamics and back again.

Don’t you remember?

So, to start off: the chance of the first song being unheard is exactly 1 (provided you are not suffering from debilitating deja vu). For the second song, the chance it is different from the first one is exactly $1-1/n$ for a playlist with $n$ songs, and by the time we reach the $k$th song, our odds become $1-k/n$: exactly the proportion of $n$ songs we haven’t heard yet. Then, if we have a run of different songs 1 through $k$, the probability of the $(k+1)$th song being one we’ve heard before (perhaps our beloved Starship) is $k/n$. So the probability that the first repeat is at the $(k+1)$th song, which we’ll call $P_{k+1}$, is \begin{align*} P_{k+1} &= \frac{n}{n}\cdot\frac{n-1}{n}\cdot\cdots\cdot\frac{n-k+1}{n} \cdot \frac{k}{n}\\ &= \frac{n!}{(n-k)! n^{k}} \cdot \frac{k}{n}. \end{align*} Now—observe that the gap between the repeated songs, at positions $k + 1$ and $j \leq k$, is equal to $(k + 1)/2$ on average. Hence, the number of songs we should expect to hear before some song repeats itself is equal to \begin{align*}{E}[\text{length between first repeat}] &= \sum_{k = 1}^{n} P_{k+1} \cdot \frac{k+1}{2}\\ &= \frac{1}{2}\left(\sum_{k = 1}^{n} P_{k+1} + \sum_{k = 1}^{n} P_{k+1}\cdot k\right). \end{align*}

Shuffle up: With a random playlist containing $n = 8$ songs, we may have a repeat after 6 songs, with the two plays 3 songs apart.

Hey, this is starting to sound familiar…

There has to be a repeat somewhere, so that first sum, the one with just probabilities in it, is equal to 1. For the second sum, we can use the probability we calculated earlier; so \begin{align*} {E}[\text{length between first repeat}] &= \frac{1}{2}\left(1 + \sum_{k = 1}^{n} \frac{n!}{(n-k)! n^{k}} \cdot \frac{k}{n} \cdot k \right)\\ &= \frac{1}{2}\left(1 + \sum_{k = 1}^{n} \frac{n! k^2}{(n-k)! n^{k+1}} \right). \end{align*} Letting $j = n-k$, and with a little bit of algebra (which, in true mathematical style, is left as an exercise for the reader), this can be rearranged to \begin{align*} {E}[\text{length between first repeat}] &= \frac{1}{2}\left(1 + \frac{n!}{n^n}\sum_{j = 0}^{n-1} \frac{n^j}{j!} \right). \end{align*}

We just want to dance here

Here, I became stuck. And though at the time I attributed my mathematical oversight to a DWI (dividing while intoxicated), I realised that the problem was rather more beautiful than I thought, requiring extraordinary subtlety and elegance. As such, it took an extraordinary mathematician, no less than Ramanujan himself, to put it to rights. We define the eponymous Ramanujan Q-function to be
$$Q(n) = \frac{n!}{n^n} \sum_{j = 0}^{n-1} \frac{n^j}{j!},$$ which corresponds to the truncated series, and the accompanying, rather predictably named, Ramanujan R-function to be $$R(n) = \frac{n!}{n^n} \sum_{j = n}^{\infty} \frac{n^j}{j!},$$ which corresponds to the tail.

The $Q$ and $R$-series for $n=8$

Now, our original problem is to understand $${E}[\text{length between first repeat}] = \tfrac{1}{2}(1+Q(n)).$$ While Ramanujan may not have been preoccupied with the mathematics of playlists, the $Q$ and $R$ functions emerge naturally in the context of number theory—in particular, for enumerating numbers with a bounded number of prime factors. The prime number theorem says that the number of primes below $N$ is asymptotically equal to $N / \log N$. Ramanujan generalised this, showing that the number of integers below $N$ which have at most $k$ prime factors is asymptotically equal to $$\pi_k(N) \sim \frac{N}{\log N} \sum_{j = 0}^{k}\frac{(\log\log N)^j}{j!}.$$ In the case that $k = \log\log N$, this reduces to our friend, the $Q$-function which we wish to study. At the party, thinking on my (by this point rather unsteady) feet, I had begun to contemplate the complete series from $j = 0$ to $\infty$: namely, $Q(n) + R(n)$. Surely, I thought aloud, the tail of the series should be insignificant in comparison to the partial sum? Upon going home the morning after and calculating this on the back of a napkin, I realised that I was not only wrong, but consistently wrong. Because, as it happens, the infinite series is almost exactly twice the value of the finite one! In particular, $Q(n) \sim R(n)$: namely, the partial series $Q(n)$ is equal to almost exactly half the complete infinite series, $Q(n) + R(n)$. But why?

Pick your Poisson

Although Ramanujan used high-precision tools to see this, a more heuristic, statistical analysis gives just as much insight. The Poisson distribution, which is used to model everything from football to the internet (and occasionally, both in one sitting), is defined as follows: a random variable $X$ is Poisson with mean $n$ (or $X \approx \text{Poi}(n)$) if for all $j \geq 0$, $$\text{Prob}[X = j] = \mathrm{e}^{-n} \cdot \frac{n^j}{j!}.$$ The astute reader will notice that this is identical to the terms of the $Q$- and $R$-series up to a factor of $\mathrm{e}^{-n} n^n / n!$. So to see that $Q(n) \sim R(n)$, it would suffice to show that, for $X \approx \text{Poi}(n)$, $$\text{Prob}[X < n] \sim \text{Prob}[X \geq n].$$ To prove this, we will need some elementary facts about the Poisson distribution:

  • If $X \approx \text{Poi}(n)$ and $Y \approx \text{Poi}(m)$ are independent, then $X + Y \approx \text{Poi}(n + m)$ (ie if two Poisson processes have expected numbers of $n$ and $m$ events per unit of time, the combination of both processes ought to be a Poisson process, and expect $n + m$ events per unit of time).
  • If $X \approx \text{Poi}(n)$, then $\text{E}[X] = \text{V}[X] = n$, where E and V are the expectation and variance of the distribution (ie the uncertainty in the number of events that will occur increases linearly as the rate of the events increases).

Now, with this under our belt, we can apply the central limit theorem—the fairy godmother of statistics. The central limit theorem tells us that, if we have $n$ independent random variables (let’s call them $X_1, X_2,$ and so on), which all have the same distribution with mean $\mu$ and variance $\sigma^2$, then as $n\rightarrow\infty$, $$\frac{1}{\sqrt{n}}\sum_{j = 1}^{n} (X_j-\mu) \rightarrow \mathcal{N}(0, \sigma^2),$$ where $\mathcal{N}(0, \sigma^2)$ is the normal distribution with mean 0 and standard deviation $\sigma$.

A Poisson distribution with $n = 8$ and a normal distribution with $\mu=\sigma^2=8$

This means that if we sample a large number of times from the same random variable, the distribution of the sum of the results will start to look like a bell curve—regardless of how the variable was distributed in the first place. If we use Poisson random variables, so $X_j \approx \text{Poi}(1)$ for $1 \leq j \leq n$ and $X \approx \text{Poi}(n)$, we have $$\frac{1}{\sqrt{n}}(X-n) = \frac{1}{\sqrt{n}}\sum_{j = 1}^{n} (X_j-1) \rightarrow \mathcal{N}(0, 1)$$ as $n \rightarrow \infty$. Now, we are almost there: since $n \rightarrow \infty$, if $X \approx \text{Poi}(n)$ and $Y \approx \mathcal{N}(0, 1)$ then the fact that the normal distribution is symmetric means that $$\text{Prob}[X < n] \sim \text{Prob}[Y < 0] = \text{Prob}[Y \geq 0] \sim \text{Prob}[X \geq n],$$ which is precisely what we needed to show. So, after all this, what is the average number of songs between the first repeat? Well, observe that: $$Q(n) + R(n) = \frac{n!}{n^n} \left(\sum_{j = 0}^{n-1} \frac{n^j}{j!} + \sum_{j = n}^{\infty} \frac{n^j}{j!}\right) = \frac{n!}{n^n}\sum_{j = 0}^{\infty} \frac{n^j}{j!} = \frac{\mathrm{e}^n \cdot n!}{n^n}.$$ To figure out the latter expression, we need heavier weaponry. Since factorials are usually fairly unwieldy to deal with—especially as $n$ becomes large—we use an approximation, which gets better as the factorial gets larger, known as Stirling’s approximation: $$n! \sim \sqrt{2\pi n} \cdot \frac{n^n}{\mathrm{e}^n}$$ from whence our pesky $\pi$ emerges. From this, and the fact that $Q(n) \sim R(n)$, we have $$Q(n) \sim \frac{1}{2} \cdot \frac{\mathrm{e}^n\cdot n!}{n^n} \sim \sqrt{\frac{\pi n}{2}}$$ In fact, we are remarkably close to the real value: using more advanced analytic techniques, Ramanujan managed to show that $$Q(n) = \sqrt{\frac{\pi n}{2}}-\frac{1}{3} + \frac{1}{12}\sqrt{\frac{\pi}{2n}}-\frac{4}{135n} + \cdots$$ where our approximation follows from simply neglecting the subsequent terms. So: in answer to our original question, the average length between repeats from a sample of $n$ songs is equal to $${E}[\text{length between first repeat}] = \frac{1}{2}(1+Q(n)) \sim \frac{1}{2}\sqrt{\frac{\pi n}{2}}$$ as we approach an infinite playlist. (Here’s looking at you, Nick and Norah!)

Rooooock aaaannnd roolllllll

Starship: surprisingly good builders

Knowing this, we can circle back to the original problem: was it fate, or mere luck of the draw? Well, given a playlist of around $50$ songs (the only $50$ our host was familiar with), at a conservative 3 minutes per song, we end up with the average length of the first gap being approximately equal to: $$\frac{1}{2}\sqrt{\frac{\pi \cdot 50}{2}} \cdot \text{3 minutes} = \text{13 minutes 17 seconds}.$$ Therefore, we can conclude that the ghost of Starship was probably not haunting us on that auspicious night (although one can never be completely sure). So, beyond the realm of party-based musical quandaries, how useful is this? One example is when we consider birthdays, instead of songs on a playlist. If we have a queue forming one by one, what will the average size of the first gap between people with the same birthday be? Simply plugging in $n = 365$ to our formula, on average, we will need to wait for $Q(n) = 23.9$ to come into our queue before we see a repeat, with a gap of $(1 + Q(n))/2 = 12.5$ people between them. This is remarkably small, illustrating what is commonly known as the birthday paradox—the paradox of how very few samples are needed before a repeat will tend to occur, provided they are taken at random. Equally interestingly, we can use this analysis in reverse. Say we are listening to a given playlist, and we want to work out how many songs are on it. Then, all we have to do is simply keep waiting to listen out for the number of songs between the first repeat. Rearranging, we obtain: $$n \sim \frac{8 {E}[\text{length between first repeat}]^2}{\pi}.$$

You’ll never know how many times I pressed ‘Random article’ before I found this.

So, if we wanted, for example, to work out the size of a behemoth website like Wikipedia—a website with millions upon millions of articles—all we need to do is repeatedly click ‘Random article’, and voila: we can find out how large our website really is while only having to look at less than $Q(n)/n~\sim~0.025\%$ of the total articles. And this technique becomes very useful in biology to estimate the population of a species efficiently. In fieldwork, keeping the surrounding environment pristine is paramount—so taking as few samples as possible is vital. With our method in hand, all we would need to do is start sampling at random, marking our animals sequentially, and then releasing them; using our method, in a population of 10,000 animals, we would only need to find around 125 to understand the size of the population. As soon as we come across an animal we have seen before, we can work out the gap between the first repeat, and hence estimate the population of the species, while risking minimal disturbance to the surrounding environment.

The problem of time before first repeats also has great significance in the context of cybersecurity—in particular, the study of hash collisions. A hash function is a method by which we take a piece of information—like a file or password—and encode it into a piece of data of a fixed size, known as a hash. More specifically, a hash function is a computationally complex map $h: \mathcal{B} \rightarrow \mathcal{H}$, where $\mathcal{B}$ is the set of binary strings, and $\mathcal{H}$ is some finite set of hashes. There are many practical uses for this—for example, for keeping passwords secure. Keeping an exact record of all user passwords is at best a terrible idea and at worst an unforgivable risk in terms of data security—what if someone was able to access the passwords directly, or there was a data breach? Instead, for a user $U$ with password $p \in \mathcal{B}$, the pair $(U, h(p))$ is stored in our database. Then, when the user tries to login again with password $p’ \in \mathcal{B}$, the values of the hashes, $h(p)$ and $h(p’)$, are compared. If they agree, we deduce that $p = p’$, and we let our user in.

The major shortfall of this is that hash functions are almost never injective—ie there always exist distinct $p, p’ \in \mathcal{B}$ for which $h(p) = h(p’)$, a phenomenon known as a hash collision. At best, our method is probabilistic: we can only have a certain level of confidence that our user is the genuine article, but never absolute certainty. This achilles heel which occurs by necessity in most hash functions—after all, there are far more passwords than hashes—is a major flaw, especially when malicious users are capable of exploiting it to modify or delete existing records. And while we can have faith that finding two values with the same hash is unlikely—and computationally very difficult to pull off—nevertheless, probability always wins in the end: we only have to sample on average $Q(n) \sim \sqrt{\pi n/2}$ passwords out of a space of $n$ hashes before two will result in the same hash. And even if they don’t correspond to the password we want, finding any hash collision at all is sufficient to compromise the entire system.

Direct hit: a hash collision in action.

This attack, known as the birthday attack, can allow a large system to be compromised surprisingly quickly, with nothing more than absolute randomness. But before readers start writing frantic letters to Dirichlet about the security of their passwords, they can be safe in the knowledge that modern hash functions are designed to resist such collisions: the value of $Q(n)$, known as the birthday bound, for the most common type of hash, SHA-256, with (you guessed it) $n = 2^{256}$, is $4.26 \times 10^{38}$, which would still take $9.66 \times 10^{14}$ times the length of the universe to compute at a rate of 1 million hashes per second. Phew. So while, to quote the venerable Starship, “the odds are against us”, chance, as they say, is a very fine thing indeed.


A day in the life: statisticians

Statistically speaking, most Chalkdust readers hope to find a job involving lots of maths when they grow up. (Statistically speaking, so do most Chalkdust editors.) There are so many different ways to do maths, it can be overwhelming to think about the different options. Luckily, Chalkdust is here to help. This is the second in our Day in the Life series, where we ask mathematicians to tell us about a typical day in their line of work. This time, we’re digging into the data-day lives of four people who work with statistics:

  • Helen Parsons, a medical statistician at Warwick’s Clinical Trials Unit.
  • Rory McLaurin, a higher statistical officer at the Office for National Statistics.
  • Vilda Markeviciute, a machine learning specialist at Expedia.
  • Evangelos Sariyanidi, a principal data scientist at the Children’s Hospital of Philadelphia.

All four of them use statistics and data to make the world better, from fixing shoulder injuries to helping people to book their holidays. Their jobs involve maths, coding, data analysis, and lots and lots of coffee. If that sounds like your thing, read on—you might be inspired.

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Mathematics and pregnancy

Mathematics has many practical applications throughout our everyday lives, but people always seem to be surprised by its relevancy to biology and medicine. Decades, if not centuries, of work in different areas of applied mathematics have allowed us to understand how fluid flows in different situations and how materials deform under varying loads. We are part of a growing interdisciplinary research field interested in how we can apply this mechanistic understanding to the processes in the human body to better understand what goes wrong in the case of diseases and illnesses, and how we can improve the situation using interventions such as surgery or medication.

In particular we work on understanding the placenta, an organ which develops during pregnancy to supply a baby with oxygen and nutrients. Pregnancy, and the various aspects associated with it, have long been problematic to research. The standard medical research routes, such as clinical trials, are rife with ethical and practical considerations when pregnancy is involved due to concerns around the health of the foetus and mother. Both the placenta and the foetus dramatically change throughout the course of pregnancy, and our understanding of the development of both is limited. Although pregnancy can be studied in animals, the human placenta and foetus are fundamentally different to those in rabbits and rats, the animals most used in testing.

So how can we proceed? There are a number of research groups globally working on developing new techniques to investigate pregnancy. We are going to give you a taster of the mathematics that our research network is using to develop theoretical models to better understand the human placenta.

What is the placenta?

The placenta is an organ which develops during pregnancy alongside the foetus within the uterus or womb. It enables nutrients and gases such as oxygen to be transferred from the mother’s blood into the blood supply of the foetus throughout pregnancy. Once the baby takes their first breath, the placenta is no longer required and is expelled from the mother’s body as part of the afterbirth. The placenta shares both the mother’s and the baby’s DNA, and if you’ve ever wondered about how a mother and a baby can have different blood groups—it’s down to the fact the mother’s and baby’s blood circulations are distinct flow systems within the placenta.

Within the placenta, the mother’s and foetus’s blood come close enough to allow the transfer of vital nutrients across a membrane, but do not come into contact. The foetal blood flows into and out of the placenta via the umbilical cord. Once within the placenta, the foetal blood flows through a series of increasingly small capillaries within tissue protrusions known as villi. Maternal blood is channelled from the walls of the uterus into the placenta before flowing through the spaces formed between the villi, known as the intervillous space, and exiting the placenta through veins in the base of the placenta. If the flow conditions within the intervillous space impairs the transfer of nutrients and gases into the foetal blood flow then this can potentially harm the foetus.

Understanding blood flow within the placenta is easier said than done. Experiments using placentas, either during pregnancy or after birth, are difficult to carry out due to ethical considerations and the fact that placentas can be damaged during birth. Every placenta, like every baby and woman, is unique and the exact content of blood varies from person to person. However, imaging techniques are improving and it is now possible to reconstruct 3D images of what the microscale blood vessels in a placenta may look like using placentas obtained after birth. With these 3D images, we have a better prospect than ever before of being able to computationally simulate blood flow in the placenta.

Fully developed human placenta with a close-up schematic of the intervillous space

Mathematics and blood flow

Blood is a difficult fluid to model, in part due to its complex composition. Blood contains both a fluid component, known as blood plasma, and a variety of different cells, the most common of which are red blood cells which carry oxygen around the body. The blood plasma makes up over half of the volume of blood and is over 90% water, with the other 10% being solutes such as sugars, salts, and proteins. So it’s fair to say that the majority of blood is water. Water and other similar fluids have been widely studied in a variety of different applications of mathematics, including meteorology, aeronautics, and many other industrial applications. We have a plethora of mathematical tools at our disposal when thinking about fluids such as water. Can’t we just use these tools? Well… it depends.

Let’s think about an example to illustrate the problem. If we take a tank of water and a tank of water with plastic beads in, we’re not going to see a fundamental difference in the flow behaviour between these two tanks, assuming the plastic beads are evenly dispersed throughout the water. Similarly if we pour both tanks into large drainage pipes, we also will not see any marked differences in the flows:

However, if we were to pour both fluids into pipes with diameters closer in size to the beads, then there would be differences. The water with beads will not flow the same way as the water alone—the beads may partially or fully block the vessels, and at the very least we expect a reduction in flow:

There’s an equivalent problem with modelling blood. Red blood cells have a diameter of approximately 8μm. The largest blood vessel in the human body is the aorta with a diameter of about 3 cm. Although there will be small differences in the flow of blood through an aorta versus the flow of plasma or a pure fluid, these differences will be minor compared to the effects of variation between individual aortas and other sources of error introduced in the experimental process. In the case of these large vessels, modelling blood using the same approaches as for fluids such as water is a fair assumption. However, in the placenta blood vessels can have diameters down to approximately 6μm. At this scale we need different approaches to those we would use with fluids such as water.

So what can we do? One property we know any fluid, including blood, must satisfy is that of mass conservation. This is essentially that a flow must obey the rule `what goes in, must come out’. If we tip one litre of fluid into an empty bucket, then we know there will be one litre of fluid in that bucket. Similarly, if we empty a bucket into a measuring jug and find that we have one litre of fluid, we know that the bucket must have initially contained one litre of fluid.

We can apply the same argument to flow through a vessel or a pipe. However, we must take into account the velocity of the fluid in and out of the vessel, and the size of the vessel. We do this by introducing the notion of volume flux, often denoted using $q$. Volume flux is a way of quantifying fluid flow and is defined as the rate of volume flow across a given area, in our case the cross-section of a vessel. If we take the volume of fluid in a vessel to be $V$, then the rate of change of $V$ in a vessel must be equal to the difference in the volume flux in and out of that vessel:

This can be written as
We can extend this concept to more complex geometries. In particular, mass conservation is useful when considering the junctions between vessels. Let’s take the example of a network consisting of one larger vessel splitting into two smaller vessels:

If we again take $V$ to be the total fluid within this network, the rate of change of fluid within these vessels is given by
Note that $V$ is an arbitrary volume based on the length scale of the network which we can label as $\mathrm{d} x$. If we take the limit $\mathrm{d} x \to 0$, so our network is reduced to flow in and out of a single point at the centre of the junction, we see that $V$ becomes infinitesimally small, allowing us to reduce our equation to
\[ q_\text{in}=q_a + q_b. \tag{1} \]
This tells us what the flow is around a junction between vessels. But what about the flow within a single vessel? The most common method to evaluate fluid flow is using the Navier–Stokes equations, the series of 3D partial differential equations for fluid motion, given the density and viscosity of the fluid. The existence and smoothness of solutions to the Navier–Stokes equations is one of the seven millennium prize problems and is currently unsolved. However, we do have solutions to the Navier–Stokes equations in certain physical situations. One of these is the Hagen–Poiseuille equation which states that the pressure change, $p_B-p_A$, over a long cylindrical pipe is proportional to the rate of change of volume of fluid passing through the pipe. This is often stated as
\[ p_B-p_A = \frac{8 \mu L q}{\pi r^4}, \tag{2}\]
where $r$ is the radius of the pipe, $\mu$ is the viscosity of the fluid, and $L$ is the length of the pipe:

The Hagen–Poiseuille equation provides a good approximation for the flow of many fluids in pipes, which can be seen as analogous to blood vessels. But both the Navier–Stokes equations and the Hagen–Poiseuille equation make the assumption that the viscosity of the fluid, $μ$, or the friction between molecules within the fluid, is constant. This is known as a Newtonian fluid, and examples include water and air. Due to the composition of blood, the viscosity is not constant and past work has shown the viscosity of blood is a function of the proportion of red blood cells in the flow and the size of the vessels. As blood is a non-Newtonian fluid, we cannot use the Hagen–Poiseuille equation to describe the blood flow, but we can use it to inform our approach: we can take $μ$ to be dependent on parameters such as the size of the vessel and the amount of red blood cells in the vessel and still use equations of the same form as the Hagen–Poiseuille equation with variable viscosity. This is not a perfect solution to our modelling problem and would not be guaranteed to give the absolute values for the flow properties in all possible cases, but it allows us to build a model able to investigate complex blood flow.

Equation (1) describes the volume flux of blood between vessels and equation (2) tells us what we need to determine the flux through each vessel. By combining both these equations in a network of vessels, we can determine the flux through the network, given the flow in and out of the network. However for biological tissues such as the placenta, this can result in keeping track of many hundreds and thousands of vessels and junctions between vessels. As mathematicians we tackle this by taking inspiration from network theory. We define a matrix, $\boldsymbol{\mathsf{A}}$, known as the incidence matrix, which describes the relationship between each vessel and each junction. Each element of $\boldsymbol{\mathsf{A}}$ is defined as

\[A_{ij} = \begin{cases} 1 & \text{if vessel $j$ flows into junction $i$,} \\ -1 & \text{if vessel $j$ flows out of junction $i$,} \\ 0 & \text{otherwise.}\end{cases}\]

Using the incidence matrix notation we can reduce the hundreds of copies of the equations to a series of matrix equations which are solved using numerical methods within programming software such as Matlab or Python. Work on how to describe blood viscosity and other flow properties required as input for this model, in terms of the cell content of blood and vessel geometry, has been going on since the 1980s. However, this work has not been specific to the placenta: vessels in different organs will have different geometric properties, and there are still many open problems related to describing blood flow in the placenta.

Simulation of cellular blood flow modelled as a suspension of red blood cells in plasma across a microvascular network during development

Mathematics and…

We are of the opinion that the best mathematical modelling is never done in isolation. Some of the most impactful research in recent years has come about from mathematicians collaborating with researchers in other disciplines. The equations we’ve looked at and other equations in fluid dynamics can give us an insight into the function of the placenta but only with input from other disciplines to estimate the missing parameters such as viscosity and vessel geometry.

If you think about the number of babies that are born every day, there are a lot of placentas being disposed of! There are a lot of different measurements we could obtain from these placentas which could help with mathematical modelling. Unfortunately, placentas can become damaged during birth, so not every one donated to science is usable, and relatively few maternity wards are attached to research centres so most don’t have the facilities to conduct experiments with placentas. So, despite the fact that human placentas should be easier to obtain than, for example, human hearts or lungs, there is still a disparity in the amount of direct medical and biological research being done with the placenta compared to other organs.

Other ways exist to get valuable information to improve our mathematical modelling. One method is to reconstruct realistic microscale blood vascular geometries using plastics or polymers, allowing us to vary the composition of the fluid and obtain measurements of volume flux against which we can parametrise our mathematical model. This is known as microfluidics. Microfluidic chips can be constructed—for example by using 3D printing or by milling vessels within solid sections of plastic—using information on the vessel geometry obtained from imaging sections of organs. We are then able to flow solutions through these microscale geometries to understand the flow within the microscale blood vessels in our body. We say solutions here, simply because this approach gives us a large degree of freedom in terms of what fluid is used. Water and glycerol combinations can be used to consider Newtonian flow with different densities and viscosities. Real blood can also be used, but this comes with problems in terms of the cell variation between samples and length of time each sample can be used for. The use of artificial capsules and droplets shows promise by offering control of the flow properties to experimentalists and allowing us to replicate blood properties.

We can further explore recreating blood vessel geometries to inform mathematical modelling using high performance computing. Here we can use methodologies such as finite element, lattice Boltzmann and immersed boundary methods to simulate cell interactions on the microscale level within the vessels. These methods are highly computationally expensive and require specialist experience to use when considering non-Newtonian fluids (eg blood) within complex blood vessel geometries (eg the placenta); but they can provide a large amount of information for mathematical modellers and, as in the case of microfluidics, a great deal of control over the experimental system.

There is still a great deal of work to be done to advance our understanding of the placenta, and by extension pregnancy and women’s health. In 2021, the UK government published a report recognising that women receive poorer healthcare than men. There are many reasons as to why this could be: prevalence of different diseases, access to healthcare or even implicit bias. But one reason is consistent in the many reviews on gender bias in medicine: we simply know less about female-assigned bodies compared to male-assigned bodies. There are many open problems in medicine which need to be addressed to counter gender and minority bias in medicine.

One of mathematics’s greatest strengths is its ability to investigate problems involving many parameters and variables with minimal financial and time costs, and this is increasingly becoming an asset to medical research. We believe that mathematical modelling will have a crucial role in advancing medicine in the coming decades and we hope we’ve convinced you that placental research is an exciting area to be in.


Significant figures: David Singmaster (1938–2023)

Some years ago, at the MathsJam annual gathering, David Singmaster ambled up to one of the other delegates. “Nice T-shirt,” he said. The design showed the small robot from Star Wars, loyal companion to C-3PO and widely adored for his playful beeps and boops. But instead of his usual cylindrical body, in its place was a Rubik’s cube. Printed underneath was the name: R2-D2.

“Yes!” said the delegate, clearly delighted that someone had noticed, “But it’s even cleverer than you might think. See, the robot’s name is R2-D2, but that’s also a move in the notation we use for solving the cube!”

David chatted with them for a while, during which time the truth slowly emerged. The notation in question is called the Singmaster notation, invented by David in late 1978 during his process of devising a method of solving (or restoring) the cube. So the delight increased, to meet the man who’d brought a method of solving the cube to a world of enthusiasts.

It’s via the cube that most people will have encountered David, but if you met him in person it rapidly became clear that his interests extended far beyond that one mathematical object. I don’t remember when I first met him, but I remember spending some time with him at the Gathering 4 Gardner in 2008. He had a new (to him, and to me) toy that we call ‘the ring on a chain’, and he was both practising how to do it, and delightedly showing it to anyone in range, sometimes getting it right.

David at Gathering 4 Gardner 13 (G4G13) in 2018

This was the theme… David had a delight and a passion for sharing. A visit to his house was an adventure, and once you were there it was extraordinarily difficult to leave! “Just one more thing” was his catchphrase, as he would first show, then watch you struggle with, then struggle in turn to remember how to solve a mechanical puzzle of one sort or another. His tongue would poke out, first on one side, then the other, then back again, all the while as he talked about the history of the object at hand.

He delighted in the struggle, and each time he succeeded in solving a puzzle again his joy was evident, and you were invited to share it with him.

There was much more to him than just puzzles. As an undergraduate, he solved a prize problem in his number theory course, and another problem that eventually led to two academic papers. He taught in Beirut, then lived in Cyprus, and later while working as a photographer for an underwater archaeological expedition off the coast of Sicily, he accidentally discovered the oldest known warship wreck in existence. He then worked in Pisa for a year, before finally settling in London.

David is known for Singmaster’s conjecture. In the pattern we know as Pascal’s triangle, every number greater than one will appear somewhere, and each will appear only finitely many times, but David noticed that none of them seemed to turn up very often. He conjectured that there is some fixed finite bound on how often a number can appear, a conjecture that remains unresolved. So far, it has been found that the number 3003 appears eight times… no other number is known to turn up as often, and while there are many related questions, almost all remain open.

John Railing, John Conway, Richard Guy and David at Gathering 4 Gardner 12 (G4G12) in 2016

David’s knowledge of the history of puzzles (among many other things) was extensive, and so was his network of connections. He was friends with John Conway, Richard Guy, Martin Gardner, and many others, and he was right at home with them, as he was with anyone else who loved puzzles, games, maths, and toys. He was kind and welcoming to all who would engage.

David was also quick and sharp. On more than one occasion, at various gatherings such as MathsJam or the Gathering 4 Gardner, a puzzle would be proposed and he would give an answer within seconds. Sometimes it was because it was similar to a puzzle he’d seen before—and it felt like he had seen them all! But sometimes it was simply because he had solved it there, on the spot. This could make him challenging company, but if you were willing to go along for the ride it was immensely rewarding, for he was free with his knowledge, and his enthusiasm was undeniable and irresistibly infectious.

Just part of David’s collection

He was also an avid collector. His book collection has something in the vicinity of 10,000 books, including works on recreational maths, the history of maths, plus a range of cartoons, humour, and language. His collection of mechanical puzzles is extensive, including (of course) many, many examples of, and variants on, the Rubik’s cube. Having written the first book outlining a method of restoring the cube, and devising the notation that is almost universally used, people often sent him early versions of new modifications. An extension to the house was constructed to hold the collection—an extension that has been amply filled.

Dreihasenfenster (‘window of three hares’) in the cloister’s inner courtyard of Paderborn Cathedral

Perhaps slightly less expectedly, David was also interested in the ‘motif of the three hares’, a design in which there are three hares following each other in a circle. They each have the requisite two ears, but each shares an ear with another, so there are only three ears in total. There is some evidence that this was presented as a puzzle in ancient times, but it has also appeared in many unexpected places as a motif of some sort. It is unclear whether it was purely decorative, or intended to convey a message and carry meaning. David showed me a splendid coffee table book, which he had contributed to, about a journey along the Silk Road tracking down places where the three hares had been found. This went some way to explaining why his ever-present bag of toys had the three hares appliquéd on it.

David was full of surprises like that, but when you came to know him it was somehow less surprising. His interests were as unbounded as his youthful enthusiasm.

To paraphrase Rob Eastaway: David will be greatly missed by everyone who knew him, but his spirit will live on whenever someone casually picks up a Rubik’s cube, smiles, and starts to play with it.


Ode to Newtonian mechanics

My mistress shapes ellipses round the Sun;
        Their perihelion does not precess.
If neutron stars combine, no waves are shun,
        If stars be dense, black holes may not progress.
I’ve read that matter makes space-time be curved,
        But no such curvature, just force I see;
The field equations Einstein to us served;
        Yet she prefers $\mathrm{d}(m v)$ by $\mathrm{d}t$.
I love to talk of her, yet well I know
        That relativity be general;
I grant I never saw dark matter though;
        My mistress is far less ephemeral.

        And yet, by heav’n, I think my love as rare,
        For Newton and Einstein I shan’t compare.