The Maths of Doctor Who #4.2 – “There’s an awful lot of one, but there’s an infinity of the other.”

“Saving the day through a heartfelt sing song and the illogical powers of an emotional leaf felt like a distinct cop-out.”

– Mark Snow, IGN.1

“I caught the sound of a man airing the preposterous notion that the sum of all primes approaches infinity.”

– A complaint to BBC Radio 4 regarding an episode of More or Less.2

Infinity makes people cross. The very idea itself asks us to the imagine the most impossible of notions, that something can goes on forever. Fans of Doctor Who are likely to say that it is a show that will ‘go on forever’, but what they probably more accurately mean is that ‘Doctor Who will go on for the rest of my life and many, many years after’. If Doctor Who were to run for 100,000 years, then yes that might seem like a very, very long time indeed, but 100,000 years is just the teeniest tiniest drop in the ocean when compared to the all-encompassing enormity of infinity.

How then can we even begin to comprehend something like that using only our finite and comparatively tiny lived experiences? And when presented with rational arguments and logical conclusions on the consequences of such an idea, why do we intuitively decide to reject these answers as preposterous and absurd? Perhaps it is because it’s unlike anything else we know. Perhaps it has something to do we how we feel about infinity instead. This blog I hope will unravel the mysteries and shed some light on our understanding of infinity itself. But why have I actually brought up the idea of infinity? Because of The Rings of Akhaten (2013), of course.

The Rings of Akhaten and The Notion of Infinity

Judging by the Readers’ Poll conducted by Doctor Who Magazine in 20143, it would seem that The Rings of Akhaten is one of the most unpopular episodes among fandom at-large. This idea is examined in considerably more depth within William Shaw’s entry in The Black Archive range on The Rings of Akhaten, but Shaw has observed that “nearly all of the [contemporary reviews] were quite positive, or only mildly negative. The only strongly negative review from the time was in Doctor Who Magazine. Something about this episode seemed to hit differently with hardcore Doctor Who fans”4. One of the main points of critique is the episode’s climax, which not only involves the notion of infinity but also what Mark Snow of IGN has described as “the illogical powers of an emotional leaf”. So first off, let’s quickly recap the dialogue from that climatic moment:

CLARA: Well, I brought something for you. This. The most important leaf in human history. The most important leaf in human history. It’s full of stories, full of history. And full of a future that never got lived. Days that should have been that never were. Passed on to me. This leaf isn’t just the past, it’s a whole future that never happened. There are billions and millions of unlived days for every day we live. An infinity. All the days that never came. And these are all my mum’s.5

Upon Clara’s successful resolution to the problem, the Doctor then decides to come back to the fore and act like he knew this was the answer all the time:

DOCTOR: Well, come on then. Eat up. Are you full? I expect so, because there’s quite a difference, isn’t there, between what was and what should have been. There’s an awful lot of one, but there’s an infinity of the other. And infinity’s too much, even for your appetite.

This dialogue suggests a rather intriguing question: are we to accept that the Doctor’s past memories offered here are merely finite, whereas the lost future of Clara’s mum, symbolised by “the most important leaf in human history”6, represents an infinity of unlived days? What perhaps makes this even more unintuitive and radical a conclusion to the story here is that we generally perceive the Doctor as this immortal hero, who stars in a show that we like to think will go on forever. Yet here, in this particular moment, it actually pales in comparison to the seemingly finite, and tragically cut-short, lifespan of Clara’s mother. Allow me then an exciting digression into the ideas and consequences of infinity itself, in the hope that I will be able to answer this question more comprehensively. I do hope it’s not too much for your appetite.

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Pictured: Clara, the Doctor, and the Most Important Leaf In Human History.

“Hey, do you mind if I tell you a story?” – The Early Days of Infinity

The earliest recorded mention of infinity is widely regarded to be from the philosopher Anaximander (c.610-546 BC) who used the word ‘apeiron’, which more literally means “unbounded” or “indefinite”, though many philosophers such as Heidegger and Derrida have debated the translation of this term. However, the Ancient Greeks were seemingly terrified by the notion, a fear which has since been termed as a “horror of the infinite”7. This is especially notable in the works of mathematician Euclid who, by using a clever recursive argument, first proved the existence of infinitely many prime numbers. Yet he deliberately avoids the word ‘infinite’ altogether. Instead, his proof in Book IX of The Elements translates into English as “Prime numbers are more than any assigned multitude of prime numbers”8. The Ancient Greeks may have been among the first to entertain the notion of infinity, but they certainly refused to take it seriously.

Perhaps the most well-known use of the infinite in philosophical pop culture is Zeno’s Paradox about a hypothetical race between the hero Achilles and a tortoise. The tortoise is given a head start on Achilles and the race begins. By the time Achilles has reached the starting position of the tortoise, the tortoise has moved some distance ahead. Once Achilles has reached that position, the tortoise will have moved ahead some more, albeit a shorter distance. This process continues ad infinitum and so the argument here is that Achilles will never overtake the tortoise to win the race. Intuitively though, we know that in real life a man would easily overtake a tortoise in a running race and herein lies the paradox. What makes this paradox of infinity somewhat inadequate is that it does not explicitly recognise that an infinite sequence of events can still lead to a finite result. This is the entirely logical result of summing up sequences of numbers that converge towards a particular value and in mathematics we call that a ‘limit’.

Infinity, however, was still slow to catch on in the minds of mathematicians and doesn’t get its iconic ‘figure-of-eight’ symbol (∞) until 1665 when John Wallis first described an infinitesimal as the fraction 1/∞9. His idea caught on with the likes of Newton and Leibniz who would go onto independently discover calculus (formerly known as infinitesimal calculus) during the latter half of the 17th Century. By this point, it seems that infinity was here to stay.

“I’ve seen bigger.” – Are Some Infinities Bigger Than Others?

So far then, we’ve talked about the origins of infinity and things getting infinitely smaller, but what about when things get infinitely bigger? In fairness, mathematicians didn’t tackle that one head on until towards the end of the 19th Century. The principle figure behind this area of work was Georg Cantor, who introduced the radical new notion of cardinality, which essentially is a way to count the magnitudes of infinity. Rather than treat infinity as some flimsy piece of philosophical conjecture or just as an impossibly large number, Cantor decided that infinity should now be regarded as an entirely separate concept, complete with its own set of rules. Some of his contemporaries went as far as to describe this as ‘heretical’, which unfortunately led to him having a series of nervous breakdowns over his lifetime. David Foster Wallace has even identified this historical event as the origin of a stereotype he termed ‘The Mentally Ill Mathematician’ (with the most famous example of course being John Forbes Nash Jr., the pioneer of game theory, as the subject of the Oscar-winning film A Beautiful Mind (2001)) alongside others including the ‘Tortured Artist’ and ‘Mad Scientist’10.

To understand Cantor’s idea of cardinality, consider the set of all whole numbers, also commonly known as the natural numbers, and the set of all even numbers, which are all the numbers wholly divisible by two. If I asked you how big the set of all whole numbers is compared to the set of all even numbers, I would reasonably expect you to say that it is double the size. This seems intuitive because whilst both the sets of whole numbers and even numbers are infinite (as they both go on forever), the even numbers appear half as frequently throughout the whole numbers. Yet I can draw a one-to-one mapping of the whole numbers to the even numbers by pairing each whole number with the double of that number. This one-to-one mapping means they are in fact the exact same size and so have the same cardinality. Infinity then has now been repurposed to more precisely describe the size of any collection of objects and can be used to compare the relative sizes of infinity.

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Pictured: A one-to-one mapping of the natural numbers N to the even numbers E. This illustrates the idea that both sets have the same cardinality and so are ‘equivalent’ in size.  They are both countably infinite. Image take from https://en.wikipedia.org/wiki/Cardinality.

Since mathematicians had run out of Greek letters to borrow, Cantor instead borrowed the Hebrew letter ‘aleph’ (ℵ) and so the cardinalities of infinite sets are written as ‘aleph numbers’ with ‘aleph null’ (ℵ0) being the smallest, and refers to the size of the natural (or counting) numbers. This cardinality of infinity, and any sets of identical magnitude are also known as ‘countable infinities’, with any higher cardinalities known as ‘uncountable infinities’. From this, you can identify the sets of integers and rational numbers as countable infinities, whilst the sets of all irrational numbers and real numbers are uncountable infinities. Whilst this might sound like a lot of work just to get a grasp on what infinity means, these ideas can help us with some more tangible problems ranging from the number of possible ways to slice a pizza to the number of possible chess games that can be played.

As I said earlier, Cantor was criticised by some mathematicians at the time but some came staunchly to his defence. One of those was David Hilbert who described Cantor’s work as “the finest product of mathematical genius” and defiantly exclaimed that “no-one shall expel us from the Paradise that Cantor has created”11. No sensationalism detected whatsoever. Hilbert is expressing Cantor’s work on infinity here in terms of a state of afterlife, a place of eternal happiness, yet it may be worth noting that Hilbert himself was agnostic (he was raised as a Protestant though). In this moment, infinity is not so much what it actually is, mathematically speaking, but rather what you actually believe in.

“There are billions and millions of unlived days for every day we live.” – How Does The Rings of Akhaten Handle Infinity During Its Climax?

We can clearly divide the climax into two key events: the Doctor’s speech that fails to resolve the situation and Clara’s speech that manages to succeed instead. The Doctor offers to the Sun God12 his memories but this fails to satisfy its appetite. His passionate speech conjures up these incredible, awe-inspiring and seemingly impossible imagery such as watching “universes freeze and creations burn” and “universes where the laws of physics were devised by the mind of a mad man”. The Doctor’s strategy here then appears to be to overload the Sun God with these extraordinary tales. But this is a massive oversight on his part. Whether a story is short or long, probable or impossible, factually accurate or entirely fiction, it doesn’t matter: it is still a story. What will end the Sun God’s existence is not the nature of these stories, but the number of them.

But surely the Doctor has an infinity of stories to tell? Indeed, if we are to take all of the licensed expanded universe stories (and, just for good measure, all the unlicensed fan fiction as well) then we can see gaps between adventures that can contain an uncountable infinity of adventures, even in places where logically there shouldn’t be more adventures (otherwise known as The Law of Big Finish). One could suggest a multitude of reasons to get around this, ranging from the television show not considering these canonical to perhaps the Doctor having a finite capacity to his memory, but I think the most reasonable answer is also the simplest one: the Doctor is selecting a finite set of memories to offer. His adventures may take an infinity of forms but his chosen memories are a finite number. In fact, throughout the entire story, he is unwilling to sacrifice anything of his, whether it be his sonic screwdriver, Amy’s glasses, or his entire past, and so he continues to fail at understanding the situation at hand13.

Clara, however, doesn’t just offer her past memories but “a whole future that never happened”, all the uncountable possibilities of the days she could have shared with her mother, “passed on to [her]”. I would argue here that it’s entirely intuitive that she figures this out given that she has already made one sacrifice earlier on in the story, her mother’s ring – in order to gain access to the space moped. Unlike the Doctor, Clara is willing to offer everything, willing to demonstrate the unbounded sentimental value she holds of her most treasured possessions, and this is why she succeeds. It would be impossible to map all the days that could have happened to the days Clara expected her mum to live out with her; an uncountable infinity of days. Just like Hilbert proclaiming that Cantor had created a ‘Paradise’ from his work on infinity, and just like Clara’s mother’s ring, a never-ending circle representing a union that lasts forever, it’s actually the sentiment of infinity that truly counts here. And infinity is too much, even for the Sun God’s appetite.

This emphasis on sentimentalism over mathematical rational is not just present here in The Rings of Akhaten but in Neil Cross’s other work too. In an episode of Luther, Alice Morgan tells John Luther about the size of the observable universe:

MORGAN: Did you know that the observable universe just got bigger? […] Last time I saw you, we assumed there were about 200 billion galaxies. The revised estimate puts it at two trillion, so what we believed to be absolutely everything was basically just a round error. Closer to zero than the true number.14

Yes, Alice. That’s how scale factors work.

The intended effect here is imply that our place in the universe is so much smaller than we had previously thought, but anyone with some sense of mathematical intuition would realise that on such a large scale, even though the absolute difference of nearly two trillion seems a gargantuan number, it is actually relatively tiny. Two trillion is still nowhere near everything, not a scratch on infinity. The emphasis here yet again is not on understanding what it actually means, but on what it makes you feel.

Whilst it may be disappointing to see that the Eleventh Doctor fails to understand what is taking place during The Rings of Akhaten, he does learn his lesson eventually. In The Time of the Doctor (2013), he sacrifices the remainder of what he believes is his thirteenth and final life to defend the town of Christmas on the planet Trenzalore. And just like Clara’s sacrifice of “the most important leaf in human history”, it manages to change their future. Perhaps then that is why we hear a reprise of The Long Song just before the Eleventh Doctor regenerates.

The Borromean Rings of Akhaten – A Conclusion

I have one last piece of mathematics to bring up in this discussion: Borromean rings. The Borromean rings consist of three interlocking circles forming what is known in knot theory as a Brunnian link. What this simply means is that if one of the circles were removed, then all remaining circles would become unlinked. If you inspect the picture closely you may realise that this is a geometrically impossible shape; no-one could physically construct such an object using real rings. It does become possible once you make the rings elliptical but then these aren’t proper rings, are they?

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Pictured: The Borromean Rings. Image taken from https://www.ancient-symbols.com/symbols-directory/borromean-rings.html

The name itself comes from the coat of arms of the Borromeo family, an Italian aristocratic family from around the 17th Century. They certainly weren’t the first to use the symbol as it dates back to the Vikings of Scandinavia, who called it ‘Odin’s triangle’ or a ‘walknot’. Besides Viking runes, it has also been found in early Buddhist art and Roman mosaics. It frequently appears in religious scripture as a symbol of strength in unity and to represent sacred trinities, groups of three-into-one, such as the Holy Trinity of the Father, the Son and the Holy Ghost, for example. These Borromean rings then are not just a symbol of unity, but also of faith and belief. In the final scene of The Rings of Akhaten, the Doctor returns to Clara her mother’s ring:

DOCTOR: They wanted you to have it.

CLARA: Who did?

DOCTOR: Everyone. All the people you saved.

Whilst I think it’s a great shame that we don’t get to see the people of Akhaten do this in person, it nevertheless shows that they greatly value what she has done for them. The return of her mother’s ring then is a symbol of their belief in Clara. But Clara was not alone. She was also supported by Merry, the Queen of Years, and by Ellie Ravenwood, her own mother, symbolised here by “the most important leaf in human history”. In his recent Black Archive, William Shaw argues that it is these three characters who can provide an alternative positive, feminist version of the future in response to the Doctor’s patriarchal assumptions that are present in The Rings of Akhaten14. Here then I propose a new trinity, one that symbolises unity in sisterhood all across the “Seven worlds orbiting the same sun” and beyond. I shall call it the Trinity of Akhaten, and it consists of Clara, Merry, and the Most Important Leaf in Human History. Long may they continue to be with us, now and forever more.

 

References

Whilst all internet references have been highlighted throughout, my primary sources of inspiration and reference points were the following three books:

  • Alex’s Adventures in Numberland by Alex Bellos.
  • Things to Make and Do In The Fourth Dimension by Matt Parker.
  • The Black Archive #42: The Rings of Akhaten by William Shaw.

 

Footnotes

1 Snow, Mark, ‘Doctor Who: “The Rings of Akhaten” Review, IGN, 6 April 2013, https://www.ign.com/articles/2013/04/07/doctor-who-the-rings-of-akhaten-review.

2 Parker, Matt, Things To Make And Do In The Fourth Dimension, p403.

3 Griffths, Peter, ‘The Results in Full!’, DWM #474, cover date July 2014.

4 Maleski, Sam, ‘INTERVIEW – William Shaw, the Leaf and “Rings of Akhaten”’, Downtime, 25 April 2020, https://downtime2017.wordpress.com/2020/04/25/interview-william-shaw-the-leaf-and-rings-of-akhaten/.

5 All transcripts are taken from http://www.chakoteya.net/DoctorWho/33-8.htm and edited for clarity at the author’s discretion.

6 All quotes are taken from The Rings of Akhaten unless otherwise stated.

7 Hutten, Earnest H., The Origins of Science: An Inquiry into the Foundations of Western Thought, George Allen & Unwin Ltd, p. 135. Why not have a listen to this BBC Radio 4 programme to learn more about the Ancient Greeks and the ‘Horror of the Infinite’?

8 Heath, Sir Thomas Little; Heiberg, Johan Ludvig, The Thirteen Books of Euclid’s Elements, The University Press, p. 412 (Book IX, Proposition 20).

9 Bellos, Alex, Alex’s Adventures in Numberland, p400.

10 Bellos, Alex, Alex’s Adventures in Numberland, p400.

11 Parker, Matt, Things To Make And Do In The Fourth Dimension, p404.

12 Or is it a Planet God? See Appendix 1 of William Shaw’s Black Archive on The Rings of Akhaten for more discussion on whether Akhaten is a planet or a sun.

13 For more on how the Doctor misunderstands the events of the episode, see Chapter 1 of William Shaw’s Black Archive on The Rings of Akhaten.

14 Luther, Series 5, Episode 2 (2019). I have never actually seen an episode of Luther but this quote is referred to on pg92-3 in the Black Archive on The Rings of Akhaten.

15 For more on how these three offer a feminised vision of the future, see Chapter 2 of William Shaw’s Black Archive on The Rings of Akhaten.

The Maths of Doctor Who #3: “We would have to consult our top scientists”

Doctor Who has always been recognised as science-fiction show and its earliest serials used time travel as a narrative device in order to tell stories set on either far-distant planets or in the long-distant past that not only aimed to entertain families between Grandstand and Juke Box Jury but also teach children about both science and history. But by 1966, this approach to the show’s production shifted significantly, abandoning history in favour of science. This coincided with the introduction of writer Kit Pedler and script editor Gerry Davis. For this blog entry, I want to have a look at how Pedler and Davis’ serials brought a surge in STEM representation in the show’s format, looking at how mathematics and, more broadly, science, is used in their storytelling. But first, a bit of background.

Christopher Magnus Howard “Kit” Pedler was born on 11 June 1927 and initially worked as a British medical scientist at the University of London, where he was head of the electron microscope department. His first contribution to British television was, perhaps unsurprisingly, Tomorrow’s World and would later go onto co-create and co-write Doomwatch (1970-72). However, he is arguably best known for his work on Doctor Who, for which he has three credited scripts (The Tenth Planet (1966), The Moonbase and The Tomb of the Cybermen (both 1967), provided initial ideas for three further stories (The War Machines (1966), The Wheel in Space (1968) and The Invasion (1969)) and generally acted as the show’s unofficial scientific advisor under Innes Lloyd’s tenure as producer, who wanted to inject more hard science into the show’s format.

Kit’s frequent collaborator was television writer Gerry Davis, who was Doctor Who’s script editor for over a year (running from episode 4 of The Massacre (1966) to episode 3 of The Evil of the Daleks (1967)) and so was part of the production crew that oversaw the transition from William Hartnell to Patrick Troughton. He too was a co-writer for The Tenth Planet and The Tomb of the Cybermen but also contributed The Highlanders (1966-67), which introduced long-serving companion Jamie McCrimmon, as well as Revenge of the Cybermen (1975) during Tom Baker’s first season, though this was heavily re-written by the then-script editor Robert Holmes. The original version, now entitled Return of the Cybermen, will be released as a Big Finish audio drama in November 2021. Together then, it seems we have a duo whose primary skills complement each other: Pedler having the cutting-edge scientific ideas that he wants to fashion into stories but lacking in television experience, whilst Davis has such experience writing TV soaps and drama but can use such scientific ideas to make socially and culturally relevant stories for BBC broadcast. But how did that translate into Doctor Who itself?

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Pictured: Doctor Who writers Kit Pedler (left) and Gerry Davis (right). There seem to few photos of them together, with this being the most common by far.

The War Machines and STEM Representation in Late-1960s Who

Kit Pedler’s first story idea for Doctor Who to get made became The War Machines, written by Ian Stuart Black, and at one stage had the working title of “The Computers”1. Not only is it one of the few complete serials from Season 3, it is also the only entirely complete serial to feature companions Ben and Polly, which also happens to be their debut story. The story is set in contemporary time, which is highly irregular for the show at this point, and the plot mainly revolves around a highly advanced supercomputer called WOTAN (which stands for Will Operating Thought ANalogue) who turns out to be surprisingly malevolent.

Episode 1 sees the Doctor meet WOTAN’s creator, Professor Brett, before later attending a meeting of the Royal Scientific Club, immediately presents scientists as members of the upper echelons of British society, hanging around with the likes of aristocrats such as Sir Charles Summer and security figureheads such as Major Green. It also establishes a link between science and the military that would be become a lot more prominent during the first few seasons of Jon Pertwee’s tenure2. You only have to look as far as Summer’s coining of the term C-Day for Computer Day, which naturally invokes thoughts of the D-Day landings during the Second World War, to further cement the link.

Another interesting and perhaps quite alarming statement from Sir Charles Summer is that WOTAN “is merely a brain which thinks logically without any political or private ends. It is pure thought. It makes calculations, it supplies only the truth.” The complete disassociation between science and logic from politics and ethics here is later demonstrated to be spectacularly misjudged when WOTAN starts hypnotising people to construct the eponymous War Machines and attempt to take control over London, unless one considers total conquest of the world to be an unquestionable truth about how life should be. As Commander Millington remarks about computers in 1989’s The Curse of Fenric, “Whose thoughts will they think?” With plans to link WOTAN to computers around the world, the story presents science not only as a subject that will shape the future of our society, but also one that will be highly influential in the unfolding geopolitical landscape, with Parliament, the Kremlin and the White House all name-checked in the background of Summer’s press conference.

WOTAN’s presence in the story not only brings science into the show but also mathematics. Later on in Episode One, we have the very first maths problem to feature in Doctor Who when the Doctor asks:

DOCTOR: Er, what is the square root of 17422?

(The machine whirrs, then prints a number on a piece of paper.)

BRETT: Correct?

DOCTOR: One moment please. 131 point 993. Yes, that’s near enough.3

I hope I’m not the first person to have actually checked this but if you type that question into your calculator you should get an answer of 131.992424, which to three decimal places would round to 131.992, not 131.993. With this in mind then, the Doctor’s comment of “near enough” lends an alternative reading of the scene. Rather than being in awe of WOTAN’s computational speed and accuracy, the Doctor is actually aware of the machine’s slight calculation error beyond the second decimal place and that he now realises Summer’s complete faith in its calculations are misplaced. It would also imply that the Doctor has superior mental maths skills to the villain.

The plot’s resolution involves the Doctor using his own scientific knowledge to reprogram a captured War Machine and then gives it new orders so that he can use it against WOTAN; the War Machine firing repeatedly at it until WOTAN is destroyed. Far from bringing world peace as was intended, it seems the technology of WOTAN is just yet another new tool that can be used in warfare and is also capable of destroying itself. Only when science and technology are used, not in isolation as intended by Summer, but in conjunction with the Doctor’s ethics can they be used to prevent the invasion of London and so subsequently bring peace.

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Pictured: Sir Charles Summer (left) and Dr. Who (right) arguing whether or not the show was political back in 1966. The apolitical War Machine is required.

STEM Representation After The War Machines

Far from being an outlier in Doctor Who’s cavalcade of serials during the 1960s, The War Machines presents a decisive shift in the characters and themes used in the show’s storytelling. Judging by the progression of serials under Innes Lloyd as producer, he seems to have declared that historical ones are now out and scientific ones are firmly in. Scientists would now feature as key characters in the majority of stories. Starting from Hartnell’s swansong, The Tenth Planet, we would get the introduction of the Cybermen, who would become Doctor Who’s second villain after the Daleks, as well as the trope of teams of scientists being in space stations or other remote locations, extending all the way until the end of the 1960s.

Communities of scientists working together and facing some form of mechanical menace feature in every story where Kit Pedler is credited, clearly showing it is a staple of his storytelling within the show. Pedler would also revisit the ideas of international communications (and magnetic forces) from The War Machines in his final story contribution to the show, The Invasion, which features the remarkable rise of International Electromatics (or International Electromatix if you’re reading the novelisation) and it even features a Head of Security figure called Packer, again linking science with national security.

As for mathematics, Episode One of The Moonbase is the first Doctor Who episode to feature mathematicians as named supporting characters when Hobson says “Nils, our mad Dane, is an astronomer and mathematician as is Charlie here.” The show would also go on to get its first mathematical companion in the form of Zoe Heriot, who introduces herself in The Wheel in Space by saying “I’m an astrophysicist. Pure mathematics major.” However, the juxtaposition of these two sentences is quite striking to a mathematician like myself. The two areas mentioned here could not be further apart. Astrophysics is a subdomain of physics that involves the study of planetary bodies and would involve substantial applied mathematics such as the mechanics of celestial bodies, whereas pure mathematics is generally used to describe the study of mathematics devoid of any context or application, including areas such as geometry, analysis and number theory. Perhaps GCHQ aren’t hiring anyone in 2079?

Whilst the use of scientific ideas in the stories by both Pedler and Davis can be at times wishy-washy and at worst just plain inaccurate, their consistent and topically relevant inclusion in the show’s format was arguably a good thing. It provided the show one of its most iconic villains in the form of the Cybermen as well as several memorable stories that viewers and fans have enjoyed over the years. All but one of Kit Pedler’s six contributions are available to buy on DVD either because they are fully intact or have been completed with animation, and I suspect The Wheel in Space is not far off being animated itself – though that’s just a personal hunch. However, I do have one bone to pick with Pedler and Davis, and it really is a rather petty one, but there’s a certain scene in one of their stories that I find just absolutely atrocious. I doubt most people will find it as annoying as I do, but there’s no harm in trying to explain why. So, let’s talk about Episode One of The Tomb of the Cybermen.

The Tomb of the Cybermen and Appropriating Mathematics as Technobabble

Thought to be yet another long-lost Troughton classic, the serial The Tomb of the Cybermen (1967) was recovered from a TV station in Hong Kong in 1991, and then quickly released by BBC Enterprises on VHS in 1992. With a gap of almost twenty-five years between its initial broadcast and initial commercial release, Tomb has now been available for fans to view longer than it had been lost4. It has received wide acclaim from the fandom, with some proclaiming it as “quite simply, the best [Cyberman] story”5 and “one hundred minutes of sheer magic”6. It was even the first Troughton-era serial to be released on DVD back in 2002, later getting its own Special Edition in 2012. Simply put, it is a highly-regarded serial among fans, coming in at number 23 in a 2014 DWM Poll7, that perhaps, I might dare to suggest, are being a bit too generous towards it. But I digress…

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Pictured: Troughton (centre) may not know what he’s talking about but he certainly knows how to look smart with just a notepad and pen.

For those who need a quick reminder, the opening of Tomb sees the Doctor and co. arriving on the planet Telos at the same time as an archeological expedition. They have discovered an ice tomb which they believe contains the last remains of the Cybermen and, somewhat concerningly, the Doctor keeps drip-feeding them the answers to all the puzzles and traps set for anyone who tries to enter. One such person he assists is Eric Klieg, who delivers this quite remarkable line of dialogue:

KLIEG: But take this mathematical sequence, for example. I’m really no nearer to its solution. I’ve tried every possible combination. You’d hardly call that easy!

At this point in the story, it has already been established that Klieg has helped financed the expedition (so he’s probably very well-off) and we later learn he is a member of the Brotherhood of Logicians, though we never learn what this actually means beyond having sympathies towards the Cybermen. However, the aforementioned line of dialogue suggest quite positively that Klieg is no expert in mathematical logic. In fact, he seems to have a flimsy grasp of the basics of maths itself.

A sequence can be defined as a set of numbers that follow each other in a logical pattern: all we need is a starting point (or first term) and a pattern (or term-to-term rule). Arithmetic sequences involve adding the same number each time and we can use this to generate our times tables (For example, the three times table is 3, 6, 9, 12, 15, 18, 21, 24 …). Geometric sequences involve multiplying by the same number instead, and this can generate the powers of two for example (1, 2, 4, 8, 16, 32, 64, 128…). Other sequences are more playful, such as the Fibonacci numbers, where you get the next number by adding the previous two (1, 1, 2, 3, 5, 8, 13, 21, 34…) or one of my personal favourites, “say-what-you-see” sequences, where the next number is a numerical description of the previous number (1, 11, 21, 1211, 111221, 312211…)8.

We can see that sequences have starting points and rules, but they don’t have solutions, unless perhaps you’re trying to get the next number? But then if you don’t know the next number, how do you know you have a sequence? Furthermore, what are these combinations that Klieg is using to try and solve it? Combinatorics is the area of mathematics that looks at possible outcomes or combinations of events, such as shuffling a pack of cards or rolling a set of dice. It’s certainly not the sort of maths I would use to solve an unknown sequence. I can only begin to understand what he means by his bold claim of trying every possible option by thinking he must be highly incompetent. And to be fair, that’s probably what he’s supposed to be. The arrogant fool who overestimates his intellectual abilities, and requires a much smarter character to help him, who somehow thinks he can become the new leader of the Cybermen. So maybe the writers had intended this all along then… except then the Doctor opens his mouth:

DOCTOR: You see, if you take any progressive series it can be converted into binary notation. If you take the sum of the integrants, and express the result as a power series, then the indices show the basic binary blocks. Only I wouldn’t do it if I were  you. Oh no, I really wouldn’t do it!

If you listen to the DVD commentary of this scene, you will hear Frazer Hines talking about how terrible he was at maths and I’m not surprised as this is what is must sound like to those who don’t understand. Never mind the remarkable logical leap that expects you to convert your numerical sequence into binary numbers (unless we are to believe that Cybermen think entirely like simple computers?), what firmly put this into the realm of nonsense is the word ‘integrants’ – there are no such things in mathematics. You can have integrals, integrands and integration, but not ‘integrants’. However, integrant is a word in the OED relating to something that is integral. This then I would conjecture is a classic case of Patrick Troughton learning an approximation of his lines, rather than what was on the script. Or maybe he did just misremember? Perhaps it was even a typo? We can’t know for sure, but it does seem to fit a wider observation about Troughton’s overall performance.

Even if we substitute it with a near-sounding replacement like integrals or integrands, it doesn’t help elicit any understanding. Why would you consider taking the terms of a sequence and turn them into a sum of integrals or integrands? I should probably clarify these terms first. An integral is an equation that invokes the process of integration, in much the same way that a sum invokes the process of addition, so it’s basically a fancy sum. Integrands are the functions that you are wanting to integrate, like how in a sum you have numbers you want to add. As for integration, well Klieg starts blabbing on about it some more straight after the Doctor’s not-so-subtle hints:

KLIEG: Look! Sum between limits of one and nine one integral into power series. Yes! Yes! Then you differentiate…

At this point, Klieg seems to be your stereotypical mad scientists having some eureka moment, but my final curious observation here is that he has an integral and then… differentiates? This makes little sense since. Differentiation is the process used to find rates of change of mathematical functions, usually the gradients of curves, whilst integration allows you to find the area under the curve. The Fundamental Theorem of Calculus shows us that differentiation and integration are the inverse processes of each other, meaning if you were to integrate a function and then differentiate it you will get back to where you started. If Klieg manages to solve anything here, I haven’t got the faintest idea how.

Whilst one might consider commending the use of mathematics within a popular children’s TV show, for those who understand the language being used may be frustrated by the lack of any coherent logic to it. And as for those who don’t, like Frazer Hines’ comment stated earlier for example, it brings back school day memories of mathematical anxiety, where people remember have frustrated and confused feelings about not understanding what is happening in the lessons. I would therefore be inclined to draw the conclusion that such representation does more to hinder than to help the subject’s image. I have already highlighted some instances where Pedler and Davis’ representation is much more, shall we say, integrated into the stories they tell, but this scene falls below the mark in my opinion.

But this is just the start for the show’s relationship with mathematics, and more broadly science. Doctor Who will go on to have far more sophisticated representations of mathematics in stories like Castrovalva (1982) by Christopher H. Bidmead, which employs recursion and Escher’s art, Flatline (2014) by Jamie Mathieson, which sees creatures transcending between the second and third dimensions, and Extremis (2017) by Steven Moffat, which employs the not-so random nature of random number generators to help deliver a key plot revelation in that particular story. But I think it’s somewhat fair to say that it all got started back in 1966 when Pedler and Davis wanted to bring their interests and ideas into the stories of Doctor Who. Just so long as you don’t start peer-reviewing their work.

Footnotes

  1. Source: http://www.shannonsullivan.com/drwho/serials/bb.html
  2. For more on this, I would recommend Robert Smith?’s Black Archive on The Silurians (1970) which investigates further the link between science and the military.
  3. All quotes are taken from the transcripts provided on http://www.chakoteya.net/ with a few minor spelling and grammar edits by myself.
  4. Source: https://www.herocollector.com/Article/doctor-who-tomb-recovered
  5. Martin Day in Cloister Bell 10/11, dated March 1985
  6. Jeff Stone in TSV 29, dated July 1992.
  7. http://doctoroo.blogspot.com/2014/06/from-doctor-who-magazines-first-50.html
  8. If you haven’t quite understood this then here’s a longer explanation. The first number is 1, which can be described as one one, so the next number is 11. This can be described as two ones, so the next number is 21. This can then be described as one two and one one so the next number is 1211… and so on. Wikipedia calls them Look-and-say sequences but they are the exact same thing!

The Maths of Doctor Who #2 – “Don’t they teach recreational mathematics any more?”

The Ambassadors… Of Maths (twang!)

Over its long and varied history, Doctor Who has had a few mathematically minded writers producing scripts for the show. Perhaps you would say that the most prominent of these is Christopher H. Bidmead, who served as script editor during the show’s eighteenth season, the last to feature Tom Baker as the incumbent Doctor, and also produced three scripts for the show: Logopolis (1981), Castrovalva (1982) and Frontios (1984). The first of these concerns itself with a society of mathematicians holding the universe together (indeed, Toby Hadoke has jokingly referred to this serial as ‘The Maths of Death’ on his Who’s Round podcast), whilst the second one draws upon the mathematically themed artwork of M.C. Escher. The third one has some funky gravity shenanigans. A near hat-trick then.

Looking more recently at the revived era of the show, Stephen Thompson (sometimes credited as Steve Thompson) also has a background in mathematics, as he himself was a former maths teacher before entering television writing. He has previously talked to the media about how the plotting of Time Heist (2014) was somewhat based around the River Crossing Problem, a classic logic problem involving the transportation of a fox, a hen and a bag of grain, and the movie-style poster specially made for Journey to the Centre of the TARDIS (2013) also has strong M. C. Escher vibes (or Castrovalva vibes, if you prefer).

However, I would actually argue that the most mathematical writer is none other than the current showrunner himself, Chris Chibnall. Three of the episodes penned by him so far have made reference to three distinct groups of numbers: happy primes in 42 (2007), cube numbers in The Power of Three (2012), and pentagonal numbers in The Tsuranga Conundrum (2018). Even more curiously, if you look at the order of solo-penned Chibnall episodes (meaning we ignore Rosa (2018) here) then each of these episodes is separated by three episodes without a numerical reference.

Should this pattern continue into the next series of the show, then that would mean Spyfall: Part Two (2019) should be the next Christ Chibnall episode to have a numerical reference. Will this trend continue to hold? Watch this space [UPDATE: The episode featured an early counting machine, Charles Babbage and the Enchantress of Numbers herself Ada Lovelace – I see this as an absolute win!]. I am also willing to propose another conjecture on the back of this here:

Chris Chibnall is actually the most mathematical writer in the history of (televised) Doctor Who. So far.

In order to examine this suggestion, let’s travel back in time and have a look at each of these mathematical references from the aforementioned Chibnall-penned episodes in turn and see if we can learn anything along the way. After all, Doctor Who has its very roots in educating the kids about science and history during Saturday teatime viewing, but why stop at those subjects? Why not keep the learning streak going? Why break the habit of a lifetime? (Or is it several lifetimes?) I’d like to think one more lesson wouldn’t hurt anyone. I’ll start with the most recent of the three…

Count on a Bomb, It’s Fifty-One!

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The Tsuranga Conundrum: Or How I Stopped Worrying And Love the Pting.

In the climatic moments of The Tsuranga Conundrum, the Thirteenth Doctor and Yasmin plant a bomb in an escape pod as part of a trap for the Pting, who has been menacing them throughout the episode by eating parts of the spaceship. It’s at this particular point that the Doctor decides to utilise Yaz for a bit of Random Number Generation (RNG)1:

 

DOCTOR: Pick a number between 1 and 100.

YASMIN: 51.

DOCTOR: Pentagonal number. Interesting.

 

DOCTOR: Get in that corner.

YASMIN: What was the number for?

DOCTOR: Number of seconds before the bomb goes off. I moved it forward a bit.

YASMIN: What? I would’ve gone higher!

 

Humans aren’t particularly random when it comes to picking a ‘random’ number, with some numbers being far more preferable than others. A reddit user asked people to pick a random number between 1 and 100 and after thousands of responses the top three responses (in third, second and first respectively) were 7, 77 and 69. People appear to have a fascination with the number seven as ‘the most random number’ (and indeed, some appear to have a juvenile sense of humour).

It also seems that more often than not people go for odd numbers rather than even numbers, and prime numbers rather than composite (non-prime) numbers. Here we can see that Yaz has also picked an odd number but it isn’t prime since 51 = 3 x 17, though it can be mistaken for being prime given that similar-looking numbers like 11, 31, 41, 61 and 71 are all prime. I hope to write more on random number generators and their use in the episode Extremis in a later blog post, but let’s get back on track here.

Perhaps frustratingly for a mathematician like myself, the Doctor never actually defines what a pentagonal number is within the episode. Whilst Chibnall in Series 11 could be seen to be harking back to the William Hartnell days with a triage of companions and alternating the adventures between sci-fi and historical (if you ignore the episodes set in the present day), he could go one step further by including such definitions to strengthen the ‘educating the kids’ part that here. But no matter, this is where I come in!

Before we actually get onto pentagonal numbers, let’s start off with a simpler but related group of numbers: the triangular numbers. Triangular numbers are the number of dots needed to make an equilateral triangle of increasing side length. This is more easily seen using a diagram so below here is one I’ve borrowed from Wikipedia. The first triangle (T1) has just the one dot for each side, the second triangle (T2) has two dots for each side so it needs three dots altogether, the third triangle (T3) has three dots for each side so it has six dots altogether, and so on. You may have also noticed that you can predict these by counting the first n whole numbers (1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, 1 + 2 + 3 + 4 + 5 = 15, etc).

First_six_triangular_numbers.svg
The first six triangular numbers. Source: Wikipedia.

Naturally, we can extend this to other shapes. The square numbers are similarly the number of dots needed to make a square of increasing side length, though you may also recognise them as the numbers you get when you multiply every whole number by itself (1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, etc). And then we get to pentagonal numbers which are, as you’d expect, the number of dots needed to make a pentagon with increasing side length. Unfortunately, the general formula is not quite as straightforward as the previous groups; the nth pentagonal number is actually equal to (3n2 – n)/2. The first six of these are 1, 5, 12, 22, 35 and 51, so we now know that 51 is not just any pentagonal number; it’s the sixth pentagonal number.

Pentagonal_number
And the first six pentagonal numbers – Source: Wikipedia

This has left me somewhat annoyed because The Tsuranga Conundrum was the fifth episode broadcast but it was the seventh episode to enter production, though I suppose it’s the sixth appearance of Jodie’s Doctor if you include Twice Upon A Time. If you can find some other possible hidden meaning as to why it’s the sixth pentagonal number, then leave me a comment down below.

That’s The Power of Three

doctor-who-the-power-of-three.jpg
The Power of Three: Did you know… the BBC tried to see these cubes at £39.99 a piece!?!

Next up we have The Power of Three, an episode from 2012 that features the “invasion of the very small cubes”, as the Doctor quite succinctly puts it. Whilst the story is largely concerned with this unusually slow invasion and the Doctor having to spend time living on Earth with Amy and Rory, the maths reference here comes in right at the end of the episode’s final scene:

 

AMY: So that was the year of the slow invasion, when the Earth got cubed, and the Doctor came to stay. It was also when we realised something the Shakri never understood. What cubed actually means. The power… of three.

 

If you listen very carefully, you can still hear the audible groans of people who witnessed that last line to this day…

We’ve already talked about square numbers, the sequence of numbers where each integer is raised to the power of two, and similarly you can get the sequence of cube numbers, sometimes referred to as just ‘cubes’, by just raising to the power of three instead (13 = 1 x 1 x 1 = 1, 23 = 2 x 2 x 2 = 8, 33 = 3 x 3 x 3 = 27, etc). Given that the world we live in consists of three dimensions (at least, it did the last time I checked), an understanding of cube numbers helps us to understand space and volume as well, so it’s certainly handy stuff.

Mathematicians have been fascinated for centuries by what we can and cannot do with cubes. For example, it is a proven fact that there is no cube number that can be written as the sum of two other cube numbers. This is in accordance to Fermat’s Last Theorem, a maths problem that was only solved relatively recently in the history of maths by Sir Andrew Wiles in 1997, around 358 years after the problem was first proposed by Pierre de Fermat himself! This precise theorem was also referenced by the Eleventh Doctor during his Skype conference in The Eleventh Hour in order to demonstrate how intelligent he is by claiming to have ‘the real proof’ for it.

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The Eleventh Hour: A prime example of how an impromptu maths conference can save the world.

This in itself is a reference to the proposed existence of a more straightforward proof that Fermat could have potentially conceived, as he once famously claimed to “have discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain” – a hilariously outrageous statement for someone who hardly ever bothered to rigourously prove his mathematical discoveries (perhaps I’ll cover his biographical life in more detail within a future blog post – stay tuned!). Anyway, the proof devised by Wiles at the end of the twentieth century used mathematics far beyond what was known hundreds of years ago and so it seems that his rather simple proof either never existed (which is my personal belief) or it is indeed forever lost to the mists of time…

Unless you’re an alien time traveller from the planet Gallifrey, of course.

Another quite interesting fact is that it is impossible to ‘double the cube’, that is to geometrically construct a cube that has precisely double the volume of another cube, since the cube root of two is not a ‘constructible’ number. If you’d like to learn more about constructible numbers as well as the tragically ignored mathematician who discovered this truth, check out my previous maths blog post on Flatline and the number pi here.

Platonic Relationships

Cubes also belong to a rather select group of objects known as the Platonic solids. A Platonic solid is defined as ‘a regular, convex polyhedron’. To put this more plainly, that means any 3-D solid where every side is the same regular polygon (so the square sides on a cube for example) and that all the sides around the solid and all the angles within the solid are ‘congruent’, meaning simply that they are just the same size.

The precise origin of this concept is unknown as it can be traced as far back as to several ancient civilisations, and even authorship of the idea within Ancient Greek society is disputed, with some sources suggesting they should be referred to as Pythagorean solids rather than Platonic. There are only five Platonic solids: the cube (or hexahedron), the tetrahedron, the octahedron, the icosahedron, and the dodecahedron.

4634757632_8174c6db41_b.jpg
The Gang’s All Here: The Five Platonic solids as dice.

Any fans of board games or role-playing games might recognise these shapes as the shapes of dice used when playing these games: a tetrahedron is a four-sided die, a cube is a six-sided die, an octahedron is an eight-sided die, a dodecahedron is a twelve-sided die, and an icosahedron is a twenty-sided die. Their natural geometric properties make them ideal for games of chance where the probability of rolling each number should be equally likely, and also you might find them quite aesthetically pleasing to look at!

The mention of a dodecahedron also brings up an unexpected connection to the 1980 Tom Baker serial Meglos, a story that sees a shape-shifting cactus from Zolfa-Thura attempting to steal the power source of the planet Tigella, which happens to take the shape of a dodecahedron. The power source was originally meant to be in the shape of a pentagram according to the writers Flanagan and McCulloch, but was changed at the insistence of the then-script editor Christopher H. Bidmead.

It is perhaps worth observing that a pentagram is a star with five points which itself has symbolic links to the societies of Ancient Greece, and this may have then prompted the mathematically-literate Bidmead to suggest a Platonic solid instead. He also chose the dodecahedron; the only Platonic solid made up of five-sided shapes, those being regular pentagons. Arguably, this is his way of keeping to some of the writers’ original idea whilst still changing it to something he much more preferred.

The story of Meglos also concerns the ongoing dispute between two opposing groups: the scientific Satants, who utilise the Dodecahedron as a source of energy for the society, and the religious Theons, who believe the Dodecahedron is a crystalline gift from the great god Ti. This may also have further stimulated Bidmead to choose a Platonic solid for the shape that acts as a power source but also an object of worship since archaeologists and anthropologists have referred to the Platonic solids as “sacred geometry”.

The group of Platonic solids are recognised by cultures throughout history as having ‘divine properties’, and were well known amongst famous scientific societies, like the Ancient Greeks and Babylonians, as well as famous scientific thinkers such as Leonardo da Vinci. The five solids were believed to represent the four major elements and the universe: the tetrahedron represents fire, the cube represents earth, the octahedron represents air, the icosahedron represents water, and the dodecahedron represents the universe2. It seems unsurprising yet again that Bidmead chose this particular solid over the other four candidates.

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The Sacred Texts: Da Vinci’s drawings of the five Platonic solids, and a sphere.

This observation actually rather neatly brings us back to The Power of Three once again, which started the discussion of this entire subsection. Here, the plot in this story involves the Shakri using cubes to gather knowledge about planet Earth and the behaviour of the human race before using this information to invade more tactically. Could it be that the Shakri themselves knew about the history of Earth civilisations that led them to choose the cube as the symbol of their invasion of Earth, or does the cube represent something entirely different to them? And could it be that the writer Chris Chibnall consciously chose the shape of the cube here, given that it symbolises the earth, further cementing my theory that he is in fact Doctor Who’s most mathematical writer?

The Answer To Life, The Universe, And Also The Pub Quiz

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School of Hard Suns: The pub quiz trivia knowledge here is absolutely on point.

Lastly, we come to what is probably the most overt maths reference in the history of modern Who so far (at the time of writing this Series 12 has yet to broadcast so this could well change in the near future), and it comes from Chris Chibnall’s Doctor Who writing debut. The episode 42 (2007), which incidentally is the only Doctor Who episode title to consist of solely just a number3, takes place in faux real time (oxymoron much?) as the Tenth Doctor and Martha have just 42 minutes to help a cargo spaceship called the SS Pentallion avoid crashing into a nearby sun.

To do this, Martha teams up with crew member Riley to work their way through thirty deadlocked doors in order to reach the ship’s controls, and each door will only open if they correctly answer a pub quiz question set by members of the crew. Almost nine minutes into the episode’s 45-minute runtime and we get the following dialogue exchange that has been immortalised, if not quite correctly transcribed, by this selection of GIFs on the Official Doctor Who Tumblr page4 .

 

RILEY: Find the next number in the sequence: 313, 331, 367, …? What?

MARTHA: You said the crew knew all the answers.

RILEY: The crew’s changed since we set the questions.

MARTHA: You’re joking.

DOCTOR: 379.

MARTHA: What?

DOCTOR: It’s a sequence of happy primes. 379.

MARTHA: Happy what?

DOCTOR: Just enter it.

RILEY: Are you sure? We only get one chance.

DOCTOR: Any number that reduces to one when you take the sum of the square of its digits and continue iterating until it yields one, is a happy number. Any number that doesn’t, isn’t. A happy prime is a number that is both happy and prime. Now type it in! I don’t know, talk about dumbing down! Don’t they teach recreational mathematics anymore?

 

At last, we have an episode that provides the definition of a mathematical concept mentioned within the actual dialogue – hurrah! Education is back on the agenda. But perhaps you still find this a little too technobabble for your tastes, so let’s expand a bit on the explanation given by the Doctor. And since I’ve already covered what prime numbers are back when I discussed The Tsuranga Conundrum earlier so I’m going to focus exclusively on happy numbers here.

The best way to see how happy numbers work is to go through an example so I’m going to pick the number 28. To check if 28 is happy, I need to add the square of each digit in 28 in an iterative sequence and see if we reach the number one. You should already know what square numbers are because I’ve covered those earlier in the blog as well (Isn’t planning a wonderful thing?).

22 + 82 = 2 x 2 + 8 x 8 = 4 + 64 = 68

It is clear that 68 is not the number one, so we have to repeat the process again:

62 + 82 = 6 x 6 + 8 x 8 = 36 + 64 = 100

Again, this isn’t the number one so we try one more time:

12 + 02 + 02 = 1 x 1 + 0 x 0 + 0 x 0 = 1 + 0 + 0 = 1

This shows not only that 28 is a happy number, but also every number along this sequence is also happy, meaning 68 and 100 are also happy numbers. And since the order of the digits does not matter when calculating these, it is also clear that 82 and 86 are also happy numbers. Below I’ve borrowed a handy tree diagram from the STEM Learning website that shows all of the happy numbers between 1 and 100.

happy_numbers_graphic
Happiness Will Prevail: All of the happy numbers between 1 and 100.

But what about the unhappy numbers, such as 2, 16, and 89? Well if you try this iterative process of adding the squares of digits with them, you will end up in a never-ending sequence of numbers, as you will never reach the number one. I’ve also included this other handy tree diagram from the STEM Learning website that shows all the unhappy numbers between 1 and 100. Notice that all the numbers pictured below all eventually end up in a cyclical loop shown by the dark blue numbers at the centre: 145, 89, 58, 37, 16, 4, 20 and 42. Rather tragically, it seems that 42 is in fact an unhappy number5. We can cap off this explanation by recalling that “a happy number is a number that is both happy and prime”.

unhappy_numbers
Side by side with sadness: All the unhappy numbers between 1 and 100, and a few greater than 100 too.

And as if that wasn’t enough, Chibnall’s scripting of episodes entitled 42 and The Power of Three perhaps pre-empted a previously undiscovered link between the two ideas, that is until 2019. Since 1954, mathematicians have been able to write every number between 1 and 100 as the sum of three cubes, except for one particular number – 42. Whilst some have conjectured (starting with Roger Heath-Brown in 1992) that every whole number can be written as the sum of three cubes this has yet to be proven, but a recent discovery has brought some hope. A computer algorithm in September 2019 found the first ever set of three cube numbers that sum to 42, as if you look at the thumbnail below you can see they are pretty large ones as well – no wonder it took us so long to find! There’s still much work to be getting on with in the world of number theory but for a small group of mathematicians on that day of discovery, 42 was indeed the answer to Life, the Universe and Everything.

Teaching Recreational Mathematics In A Fun But Irreverent Way

By examining three episodes penned by Chris Chibnall, we have ended up discussing triangular, square and pentagonal numbers, cube numbers, as well as happy numbers and prime numbers. We have also talked about the importance of the cube as one of the five Platonic solids, linking it to the 1980 serial Meglos, which also features a Platonic solid and was script-edited by a writer with a penchant for science and mathematics, and then discovered that the cube itself is an ancient symbol for the earth element. All of this has led us to the lament of the Tenth Doctor, and the title of this very blog: “Don’t they teach recreational mathematics anymore?”

It’s hard to be sure what he exactly means by the term ‘recreational mathematics’ here; some would even argue this is an oxymoron. I am somewhat amused by the definition for this term given by Wikipedia: “Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity”. Evidently whoever wrote this had not really considered the likes of Fermat who, despite becoming one of the big names within mathematics, largely played around with numbers in order to pass the time. The term ‘recreational’ seems to allude to an idyllic notion that maths can be an enjoyable pastime and/or something done for non-academic purposes6, like trying to solve BBC Radio 4’s Puzzle For Today, or just simply finding unusual number patterns and sequences, such as happy primes for instance. I personally rather enjoy thinking and learning mathematics, but it sadly seems rare that other people feel the same way.

Nevertheless, while the definition of ‘recreational mathematics’ seems to leaves us in ambiguity, I feel the definition of Chris Chibnall as a ‘mathematical writer’ for Doctor Who is actually far from being ambiguous, most certainly when stood in comparison to the pantheon of writers throughout the show’s 56-year history (and counting!). If you still doubt my claim that Chibnall is indeed the most mathematical writer, rather than the likes of Bidmead or even Thompson, then consider my one final point on the matter.

Perhaps when, as the SS Pentallion hurtles towards an all-consuming sentient fireball, the Tenth Doctor cries out about the lack of recreational mathematics in the education of those from either the 21st or 42nd centuries, it isn’t actually the vain, pompous and arrogant voice of the Tenth Doctor speaking here. Perhaps instead, it is the vain, pompous and arrogant voice of the then future Doctor Who showrunner Chris Chibnall bleeding into the script right here, as he tries to bring the show back to its early educational roots7. I hope he hasn’t changed his mind.

 

All references are linked or specified throughout the article.

Footnotes

1 All episode quotes throughout this blog are my own transcriptions based on viewing them.

2 For more background on platonic solids as ‘sacred geometry’, have a read of this Mathematics Magazine blog post.

3 The next closest in my mind is 100,000 BC as an alternative serial title for An Unearthly Child, but this itself is a year rather than a number. Also, Chibnall is the only writer to have two episodes with a number in the title (42 and The Power of Three).

4 Don’t forget to click below to reblog the Official Doctor Who Tumblr page.

5 For a bit more background to happy and unhappy numbers, have a read of this STEM Learning blog post here.

6 For more content on recreational mathematics, have a read of this New York Times article.

7 If Chris wants to combine the show’s educational roots with the demands of a modern audience – is there Space For Chib ‘n’ All?

 

The Maths of Doctor Who #1: “I don’t mean edible pie, I mean circular pi.”

Dissecting The Narrative

As I recall, Flatline (2014) was the first episode of the Capaldi era that I didn’t watch on the live broadcast. Rather than tune into the Doctor and Clara’s latest adventure, I was too busy having an adventure of my own in a restaurant, somewhere in Exeter, with three Daleks (who were terrifically friendly) and Polly Wright herself (who was terrifically bonkers). A few hours earlier we’d just wrapped on a very successful mini-convention event that I helped organise at the university and we’d decided to go out for a meal to celebrate. It remains possibly the best excuse I’ve ever had for missing an episode of the show.

Naturally, the episode that I missed had to be a really good one. Jamie Mathieson quickly established himself as one of my favourite writers of the revived era with his first two episodes, Flatline and Mummy on the Orient Express (2014), both of which are full of inventive ideas, memorable characters and some pretty good jokes along the way. No wonder he topped the Doctor Who Magazine poll for the best writer of Series 8 then. There are numerous ideas and themes to dissect (much like the monsters themselves did) in Flatline, but there’s one scene in particular that I’d like to dissect myself.

Around the halfway point in the episode, whilst Clara and her newfound gang are trapped inside the train shed, the Doctor, who himself is trapped inside the shrunken-down TARDIS, suggests that the creatures are “reaching out, attempting to talk… Trying to understand.” Perhaps the situation unfolding is not an alien invasion but all a big misunderstanding because there is no way for them to communicate with the humans, besides flattening every unfortunate person that crosses their path. Even when Clara remarks that usually the TARDIS translates alien languages, the Doctor reasons that “their idea of language is just as bizarre as their idea of space. Even the TARDIS is confused.”

It’s at this point the Doctor attempts to communicate with the creatures, which he will later refer to as the Boneless, and he starts by using the number pi (π). For those who need a refresher of their GCSE maths, pi is the constant value you get when you divide the circumference (C) of a circle with its diameter (d). We can also write this as an equation, C/d = π. This statement is true no matter how great or small the circle is; the ratio between the circumference and the diameter is always the same. It’s also used to calculate the area of any circle – just multiply pi by the square of the radius (Area of a circle = π x r2).

The Doctor makes this sound like a reasonable choice – after all circles exist in two dimensions (“Even in a flat world they would have circles”) and so pi could in theory be a recognisable constant to them. But pi is also arguably one of the strangest and most baffling numbers we have in mathematics. It belongs to three rather unusual groups of numbers: the irrational numbers, the transcendental numbers, and the non-constructible numbers. Put simply, it is not a very nice and easy number to understand.

In fact, in the Pearson novelisation of Flatline, the dialogue referencing pi is entirely absent from the book. Instead, the author has decided to replace this with the rather fabulous line “Let’s start with some numbers. Even in a flat world, they’ll have numbers.” Such is the complexity of the number pi that they felt it was necessary to censor it from young teenage readers, though it’s probably more likely that they wanted to make the text itself more widely accessible to an audience learning to read and analyse literature.

Therefore, whilst borrowing from the rhetoric style of the Twelfth Doctor himself, I suggest the following statement…

Proposition: The Doctor actually antagonised, and perhaps even declared war on the Boneless, using the number π.

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Attack Eyebrows: Presumably this is how the Twelfth Doctor would react to the argument being proposed here.

An Irrational Choice?

Pi is probably the most famous example of an irrational number. This is when a number cannot be expressed as the ratio of two integers (that means any whole number – positive, negative or zero), and so cannot be written in decimal form using a finite or repeating set of digits after the decimal point.

For example, one third can be written as 0.333333… where the 3 digit just repeats forever, and so is a rational number. Whilst one seventh can be written as 0.142857142857… where the set of digits ‘142857’ also repeat forever, and so is also rational. In contrast, pi in decimal form starts with 3.1415926535… and then just keeps going – it will never loop back round and start regularly repeating any set of digits. There just simply is not enough time in the universe (or indeed any other universe) to write it out in full.

Due to its never-ending nature, there is no precise way to express it numerically but there are also several robust methods for calculating these digits of pi with increasingly greater accuracy. The computation of pi is something of a mathematical badge of honour for applied mathematicians and computer scientists, requiring them to combine the most advanced computing power they have with the most efficient algorithms they can write. In fact, at the time of writing, the world record for approximating pi is currently held by Emma Haruko Iwao, a software engineer at Google, who on Pi Day 2019 (that is the US date of 3.14.19) calculated the value of pi to 31.4 trillion digits. How neat!

I do, however, wonder what method the Doctor used to calculate pi in the episode. Perhaps the TARDIS has access to more powerful mathematical methods developed far in the future or from a far more advanced civilisation. I would also ask whether or not he could even communicate to the Boneless what he was precisely trying to calculate; bearing in mind communication was already the key barrier to begin with. Without any sense of forewarning, sending a never-ending number could be interpreted as a way of antagonising the Boneless by trying to overload their senses as they try to grapple with our dimensional space and interpretation of number.

Transcending All Understanding

Next up, we have the fact that pi is also a transcendental number. The concept of a transcendental number is quite a modern one given that it was only first defined in the 18th Century by a well-known mathematician called Leonard Euler (who incidentally also has an irrational number named after him – the number e).

Transcendental numbers are briefly mentioned in a Big Finish Main Range audio drama starring the Sixth Doctor called  …ish (2002), written by Phil Pascoe, who incidentally has also recently published The Black Archive entry on the Sixth Doctor serial Timelash. The script itself is quite a verbose one and is sure to appeal to any budding lexicographers as the plot itself involves a character trying to discover the ‘Omniverbum’, a word that is infinitely long and transcends all meaning. Anyone who uncovers its existence would then cease to exist as they fall victim to, and I mean this quite precisely, a literal black hole. However, Phil’s script rather unfortunately gets the definition… not quite correct. Here’s what one of the characters states near the beginning of Part Three:

“A transcendental number, such as pi, can only be approximated since it is impossible to write down as a finite or repeating sequence of digits.“

Those of you following along so far will have noticed that this matches the aforementioned definition of an irrational number. It’s a common misconception, since not only is pi both irrational and transcendental, but also because transcendental numbers are a subset of irrational numbers. This just means that all transcendental numbers are irrational but not all irrational numbers are transcendental. I strongly suspect Phil included this idea in …ish because of its use of the phrase transcendental, rather than the actual concept behind it. It’s actually not too uncommon for mathematical terms to mean something entirely different to a similar sounding concept from another discipline. For example, mathematical induction is not the same as inductive reasoning in philosophy; it’s actually a form of deductive reasoning! But nevertheless, I still have huge respect for Phil including the idea within the script and would recommend you give his audio a listen.

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Omninumerum: Does this Big Finish audio drama have an incorrect definition of transcendental numbers? Well, …ish.

A Crash Course in Algebra

So ‘what is a transcendental number?’ I hear you all crying out. Well before I define it, I think it would be best to introduce another concept to you first, which is algebraic numbers. Algebraic numbers are simply all the numbers that are solutions to an algebraic equation. So for example, if I had the equation x – 6 = 0, then the solution here is x = 6, and so we can say that 6 is definitely an algebraic number. It is easy to see that any whole number is algebraic. If you pick any whole number and call it a, then that number solves the equation x – a = 0.

This also works for any fractional number, though mathematicians more typically call these rational numbers. If I have the number two-fifths (or 2/5) then that number solves the equation 5x – 2 = 0, hence two-fifths is an algebraic number. In this example, the five is an integer coefficient, which just means a whole number in front of any x terms. Similar to the whole numbers, if you pick any fraction and call it a/b then that number solves the equation bx – a = 0, hence all the rational numbers are algebraic.

Now here’s where it gets interesting. Consider the equation x2 – 2 = 0. Then the solution to this equation is the square root of two (or √2). If you type √2 into a calculator you will get a never-ending set of digits after the decimal point. Just like pi, the square root of two is another well-known example of an irrational number. Hence, we can see that there are at least some irrational numbers that are also algebraic numbers. Hopefully you are feeling fluent with the concept of an algebraic number now!

So let’s bring this back to the original idea of transcendental numbers and explain how they are related to this. Transcendental numbers are the opposite of algebraic numbers; they are all the numbers that are not a solution to any algebraic equation. This also means that every number that you can think of is either an algebraic number or a transcendental number – there are no exceptions. In general, mathematicians just love to partition things in two distinct groups such as positive and negative numbers, even and odd numbers, rational and irrational numbers. It’s just what they do!

A few of you might be thinking, well what about the equation x – π = 0? Surely that has pi as a solution? Well it does… but it’s not an algebraic equation. The unspoken rule of algebraic equations is that we can only use whole numbers, also known as integers, in them, and so the example given here is not an acceptable equation. This here is the crucial counterexample, which clearly demonstrates that the definition given in …ish doesn’t quite work. Irrational and transcendental numbers are not the same thing; the terms are not interchangeable.

If you managed to follow all this then very well done because this is something usually taught to those in the final year of their undergraduate maths degrees. Give yourself a  gold star!

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Badge of Mathematical Excellence: Found in the wreckage of a space freighter around 65 million years ago. Seeking new owner.

Simplicity In Complexity

As we have seen, showing that a number is algebraic is quite straightforward in that you just need to find an equation that it solves. But transcendental numbers are very hard to prove because you have to show that no such equation exists. When I studied algebraic number theory at university, we were simply allowed to state that pi was a transcendental number without further reasoning, due to the advanced nature of the concept. This certainly lacks rigour in approach to learning but it was still great when revising for the final exam.

But I was still curious. So I took out a book from a university library that contained the first proof that showed pi is a transcendental number by Ferdinand Von Lindemann in 1882. I have no qualms in saying that I hadn’t got the faintest idea what was happening on the page. In fact, it was much like that bit in the novelisation of Shada (2012) where Ship has to read out The Worshipful and Ancient Law of Gallifrey to his Lord Skagra: “Squiggle, squiggle, squiggle, squiggle… squiggle, line, squiggle, squiggle…” I have since found a more accessible text online that explains the proof pretty clearly, which you can read here if you so wish. It’s not for the faint-hearted.

People, and presumably other intelligent life, gain an understanding of the world around us, all of its systems and functions and processes, by using equations to model and predict their outcomes. However, the transcendental numbers are the group of numbers that do not solve ordinary kinds of equations. Why then would the Doctor try to solve a communication problem by sending a number that is known for its inability of being a solution to the most common kinds of mathematical equations? It seems that the Doctor’s strategy does not appear to add up. The plot thickens…

A More Constructive Approach

Lastly, we come to constructible and non-constructible numbers. Whilst transcendental numbers have only been around for the last two hundred years, constructible numbers have been around for about two thousand years, dating back to the ancient Greek mathematicians of old. Moreso, the ancient Greeks were clever enough to realise that the theory of constructible numbers was the key to solving several problems they had stumbled upon within their study of mathematics, even though they could not prove the answers for themselves.

So how do we know if a number is constructible? Constructible numbers are defined as all the possible lengths of line segments that can be created using a straightedge ruler and pair of compasses in a finite number of steps. In other words, imagine you have a piece of paper and a pen. Draw two dots on the paper and then join them with a straight line using the ruler – this is now a line of length one (and so one is a constructible number, obviously). But now you can only add new points on to the page, using your straightedge ruler and pair of compasses, as long as they follow these three rules:

  • Any new point you construct must be the intersection of two lines, two circles or a line and a circle.
  • All lines are drawn with the ruler (no measurements!) and must pass through two points you have already constructed.
  • All circles must be centred on points you have already constructed and their radius must be the distance between two points you have already constructed.

If you would like to see some proof that all the integers, rational numbers and square roots of numbers are in fact constructible numbers, have a look at this site here.

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A handy tree diagram that shows the relationship between the different number groups we’ve discussed here. Sadly, trees have no moving parts and don’t communicate.

However, for the purposes of this blog post, I am far more concerned with what the property of constructibility tells us about numbers. I’ll show you what I mean with a famous example. It’s an ancient maths problem first proposed by some ancient Greek geometers. You might have already heard of this as it gives us a well-known saying…

Squaring The Circle

People use the phrase ‘squaring the circle’ as a metaphor for doing the impossible, or perhaps something extremely difficult at least. In mathematics, it refers to the following problem: can you construct a square with the exact same area as a given circle in a finite number of steps using a straightedge ruler and pair of compasses? You can probably infer from the language used in this question that this has literally everything to do with constructible numbers.

We can actually simplify this problem even further using a handful of key facts we have discussed during the course of this blog post. First off, we know that the area of a circle is equal to π x r2. Suppose we want to ‘square’ a unit circle, which is a circle with a radius of one (r = 1). Then the area of this circle is equal to:

π x 12 = π x 1 = π

This means that in order to prove that we can square the circle, we need to construct a square that has the same area as this circle, which is an area of π. Squares are defined by as a quadrilateral shape that has the same length and width, and since the area of any quadrilateral is the length (l) multiplied by the width (w) then the area of the square is just l2, since l = w here. We want the area of the square to be π so this means that l2 = π and so l =π. Therefore, if we can construct a line that has length √π then we have solved the problem. But, we also know from earlier that we can construct square roots of numbers, and so this means we just need to construct a line with length π in order to prove you can square the circle.

What this line of mathematical reasoning has hopefully just demonstrated to you is that being able to square the circle and being able to geometrically construct a line of length π are actually equivalent statements. They mean the same thing. And so if we can prove or disprove either the statement about constructing a line of length pi, then we have equally proved or disproved the statement that you can or cannot square the circle.

The only question that remains unanswered here is whether or not π is indeed constructible. It isn’t. This is because all constructible lengths must be algebraic numbers, as was first proven by the much neglected French mathematician Pierre Wantzel in 1837. Wantzel was also the first to prove that it was impossible to ‘double the cube’ and ‘trisect any angle’ using his knowledge of constructible numbers; astounding results that were largely ignored by the mathematical community for nearly a century.

As an aside, I found out in my research that Wantzel died tragically young at only 33 years of age, apparently due to overworking himself and his abuse of coffee and opiates. There is very little biographical detail to be found about him, with perhaps several key documents still only available in French. His Wikipedia page fills just one screen and he doesn’t have an entry in any of the 27 volumes of the Complete Dictionary of Scientific Biography. He is regrettably yet another example of a mathematical prodigy who died quite young – see also Niels Henrik Abel, Évariste Galois, Srinivasa Ramanujan, and Maryam Mirzakhani. And whilst probably by coincidence, Doctor Who itself has not been immune to mathematical prodigies who die tragically young either.

Combining Wantzel’s result with the proof that pi is transcendental (von Lindemann, 1882) we can conclude that pi cannot be algebraic and, hence, cannot be constructible. Normally, this would be the point to wrap up the discussion on the constructibility of pi; we’ve solved the problem after all. It would be absolutely ridiculous to imagine that somebody could possibly be so stupid and ignorant as try and overturn this concrete mathematical fact. But naturally that didn’t stop the United States of America trying to pass the counterstatement into law.

A Potty Bill

In February 1897, the Indiana General Assembly were deliberating over House Bill No. 246 of that year, more easily identifiable as the Indiana Pi Bill. This frankly absurd bill was the brainchild of Dr. Edwin J. Goodwin, a physician by trade yet a crank by reputation. Within the text of his bill, Goodwin claimed to have ‘squared the circle’, a maths problem that we already know was rigorously disproved fifteen years prior (von Lindemann, 1882). His nonsensical reasoning for this incredible statement was the following line about the diameters of circles:

“… the fourth important fact, that the ratio of the diameter, and circumference is as five-fourths to four.”

This is a somewhat obtuse way of saying that pi is equal to four divided by five quarters, so that’s 4 ÷ 5/4, which is equal to 3.2, a number which is clearly not irrational, transcendental, or non-constructible. Given that pi is approximately equal to 3.14, this value of 3.2 doesn’t even round correctly to one decimal place! The only way I can begin to rationalise this behaviour is to imagine this is some sort of 19th century precursor to ‘shitposting’.

Understandably, the politicians were utterly baffled by the wording of this bill, leading it to be bounced from the House of Representatives, to the Finance Committee, to the Committee on Swamplands, and then finally to the Committee on Education, who then proceeded to pass the bill without any objection because none of them possessed the wisdom to fathom what it actually meant.

However, rather fortunately, the then head of the Mathematics Department at Purdue University in Indiana, Professor C. A. Waldo, happened to be visiting the statehouse to discuss matters of academic funding when, by pure chance, someone mentioned this bill to him. The committee then offered to introduce Professor Waldo to Goodwin, but he simply replied he already knew enough crazy people. A quick lecture to the senators showed them the error of their ways and so after a second debate the Indiana Pi Bill was postponed indefinitely. It remains in a filing cabinet, somewhere in the basement of the Indiana statehouse to this very day, just waiting for the next idiot willing to revive it.

Teaching Maths in a Fun But Relevant Way

Whether the Doctor actually remembered, or considered, or even knew of any of this to begin with, is purely speculative. But I would like to think that in their many years travelling the universe the Doctor would have come across at least some, perhaps most, of this knowledge regarding pi. As a self-proclaimed maths tutor to the medieval people of Essex in The Magician’s Apprentice (2015), before going onto lecturing on just about anything he wishes in The Pilot (2017), I’d like to think the Doctor is an informed and well-educated individual about many things in the universe, and that most definitely includes mathematics.

Returning to that critical moment in Flatline, when the Doctor sends the number pi to the Boneless, which itself has led to this entire discussion about the nature of pi, perhaps the Doctor was being more optimistic here than I had initially expected? By trying to communicate with the Boneless using the language of numbers, trying to impart knowledge and understanding about the world they happen to find themselves in, he was really trying to save the world. He was trying to be a good man all along.

There are other aspects of the number pi that I haven’t had the time to touch upon, such as its surprising and unexpected occurrences in areas such as quantum physics and the natural world. Perhaps the most mind boggling of them all is this sliding block physics puzzle that can generate digits of pi without any (obvious) reference to circles.  These are arguably all things that would suggest that pi was a reasonable choice to establish communication, by thinking of it as a universally recognised constant that consistently appears throughout the universe, whatever level of it you may be operating on.

But as we have also seen, the irrational, transcendental and non-constructible nature of pi makes it a lot less friendly than you initially thought. Indeed, we see the Boneless trying to reinterpret and construct themselves in three dimensions, having previously existed in only two, and so sending them a number that isn’t geometrically constructible could be seen as a form of attack or challenge in their own terms. If the Doctor chose the number pi knowing about these properties then he would be party to antagonising the monsters of this story, and so facilitating himself to play the role of “the man that stops the monsters”.

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A good man: Did the Doctor try to antagonise the Boneless in Flatline?

This would then suggest that the Doctor is declaring war on the Boneless. To use the words of Danny Pink in the episode The Caretaker (2014), the Doctor is “an officer”, that “he’s the one who lights” the fire, whilst Clara and her gang of survivors are the soldiers fighting the battle on his behalf. Meanwhile, the Doctor is rather conveniently trapped inside the TARDIS console room, away from the main battlefield here.

However, perhaps the conclusion is even more straightforward than this. Perhaps the Doctor, whilst having the best of intentions, just doesn’t have a clue what he is doing. This itself is the resolution we get from the man himself at the very end of Series 8 in Death in Heaven (2014). He proclaims that is “not a good man”, “not a bad man”, “not a hero (nor) a president (nor) an officer.” After all his travels through time and space, this is who the Doctor thinks he is:

“I am… an idiot, with a box and a screwdriver. Just passing through, helping out, learning.”

I expect that, with reference to this case, he means passing through Bristol, helping out the locals, and learning more about the number pi. Well, we can only hope.

 

Appendix: Jokes About Pi

  • My friend decided to get a tattoo of the symbol pi. It was an irrational decision.
  • Did you know that 3.14% of sailors are pirates?
  • Teacher: What is the area of a circle?
    Student: Pi r squared.
    Teacher: No. Pie are round. Cakes are square.
  • Me: Doctor doctor, I keep having nightmares about the digits of π
    Doctor: Is it a recurring dream?
  • Don’t ever have an argument about pi. You’ll just go round in circles.

 

References

  1. The subsection ‘A Potty Bill’ which discusses the Indiana Pi Bill is largely based on material within Chapter 3: Are you π-curious? in The Simpsons and Their Mathematical Secrets by Simon Singh.
  2. All online references are linked to at the relevant points throughout this blog post.
  3. The links to Phil Pascoe’s work are not references but unpaid advertising.