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.

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

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