This essay was originally published on my (now-defunct) Substack newsletter on 14/03/24.

The popular mathematics writer and Guardian puzzle columnist Alex Bellos once described pi as “the most famous number in math.” This is because pi can claim to be the only mathematical constant to:

a)      Get a direct reference in the Bible

b)      Be the subject of a song by Kate Bush

c)      Have its own fragrance by Givenchy

As I’m sure you’ll all recall, I have discussed in depth the ‘Easy as Pi’ scene in The Five Doctors. But that’s not the only reference to the most famous constant in mathematics. Far from it. In fact, the first ever reference to the number pi actually takes place at the start of Episode Four of The Macra Terror.

OFFICIA: What are you doing?

DOCTOR: Oh, just checking.

OFFICIA: Well, I don’t need your help, thank you.

DOCTOR: As you please. But I think you’re going to run into trouble.

OFFICIA: Why? What’s the matter?

DOCTOR: Well, according to my calculations, the pressure gauges are wrong.

OFFICIA: Don’t be ridiculous.

DOCTOR: Ridiculous? Well, that’s as may be. But, after all, three times three is nine, and if you divide by half of its own cubic capacity to the formula pi over four squared, I think you’ll have to agree with me.

We don’t get an exact formula here but the Second Doctor’s casual mention of “pi over four squared” at least indicated that he’s trying to work out the area of a circle, or at least something relating to this. The gas pipes on the nameless human colony are circular in shape so that’s probably why the Doctor is referencing pi. Pi is defined as the ratio between the circumference and the diameter of a circle, which is why it’s commonly known as ‘the circle constant.’

Then there’s 2008’s Midnight, when the Tenth Doctor and a seemingly possessed Sky Silvestry both recite the square root of pi to exactly thirty decimal places – that is, 1.77245385090551602729816748334 – in near synchronicity. This acts as both a test and a boast on the Doctor’s part; not only does he use the square root of pi to test Sky’s ability to copy his speech and thoughts but also as a way to show off his intelligence, not unlike in The Five Doctors. Only here, it eventually backfires as the passengers eventually turn on him.

But the most notable appearance of pi within the televised version of the show has to be 2014’s Flatline. Here the Twelfth Doctor instead decides to use the number pi as a way of communicating with the Boneless. He rationalises this choice by suggesting that “Even in a flat world they would have circles.” Whilst I cannot deny the veracity of this statement, I must hasten to add that there are way more shapes that just circles in the two-dimensional landscape; just ask the residents of Edwin A. Abbott’s Flatland, an 1884 novella that provided writer Jamie Mathieson with some inspiration for this episode.

A slightly better justification would be that Gallifreyan, the language of the Time Lords, can also be written in symbols of interlocking circles. This is also witnessed within the episode when the TARDIS enters Siege Mode. So perhaps the Doctor was actually trying to communicate with the Boneless using his own language via the medium of mathematics?

However.

I think the Doctor’s use of pi to communicate with the Boneless is actually not a wise choice. By critically considering both the history and number theory behind pi, we can actually build an argument against the Doctor. Yes, that’s right, I’m using my free time to write an essay in which I claim to be smarter than Doctor Who! And here’s why…

Firstly, pi is an irrational number.

An irrational number cannot be written as a fraction or ratio of two whole numbers. Whilst this property was suspected as far back as the Ancient Greeks, the irrationality of pi was not formally proved until 1767 by a Swiss mathematician called Johann Heinrich Lambert. In practice, this means that when you write out the decimal expansion of pi (that is, 3.1415926535…) you will see an infinite, non-repeating set of digits.

In order for the Doctor to send the number pi to the Boneless, he would almost certainly have to run a computer algorithm to calculate the never-ending stream of digits before transmitting them. It would seem that the Doctor is counting on the intelligence of the Boneless to recognise the pattern of digits, hoping they too have discovered pi within their two-dimensional realm.

Yet the irrationality of pi has even caused confusion between humans. The Indiana General Assembly gathered in February 1897 to deliberate over House Bill #246, now recognised as the ‘Indiana Pi Bill of 1897’. It had the extremely catchy title of “A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897”.

This frankly absurd bill was proposed by Edwin J. Goodwin, a crank physician who was attempting to profit from a blatant mathematical falsehood by claiming he had in fact ‘squared the circle.’ This is a problem in mathematics that requires you to construct both a circle and a square with exactly the same area, which we know to be impossible (and indeed, was known to be impossible in 1897). Hence the phrase ‘squaring the circle’ now colloquially means to achieve the impossible.

His nonsensical reasoning for believing he had ‘squared the circle’ was contained in the statement “… the fourth important fact, that the ratio of the diameter, and circumference is as five-fourths to four.” This is just a rather obtuse way of stating that pi is equal to four divided by five-quarters, which is 3.2, a number that is clearly not irrational. Given that pi is approximately 3.14, this value doesn’t even round correctly to one decimal place!

Yet the then-politicians of Indiana were so baffled by the bill’s wording that they were reluctant to take any steps to understand what the bill actually meant. It was 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.

However, rather serendipitously, a local mathematics professor happened to be visiting the statehouse to discuss matters of academic funding and, by chance, someone mentioned this bill to him. The committee then offered to introduce him to Goodwin, but the professor replied that he already knew enough crazy people. He gave a short lecture to the senators in order to show them the error of their ways, and so after a second debate the Indiana Pi Bill was postponed indefinitely.

Given that a bunch of humans, albeit local American politicians, were once so baffled by a piece of duff legislation on the number pi, it’s really saying something that the Doctor thinks the Boneless will have no qualms in understanding his message to them.

Secondly, pi is a transcendental number.

A transcendental number is a number that CANNOT be the solution to an algebraic equation. This makes them the counterparts to algebraic numbers, the numbers which CAN solutions to algebraic equation. All numbers within mathematics must be either algebraic or transcendental.

The idea that the number pi is mathematically transcendental draws a thematic parallel with the Doctor’s own TARDIS, which is itself dimensionally transcendental. One might go as far to speculate that a “Boneless Doctor”, a hypothetical version of the Doctor belonging to the Boneless’s two-dimensional world, would perhaps travel about in a time machine that takes the form of the number pi.

That’s actually not that absurd within the context of Doctor Who. The villain Nobody No-One, a Word Lord from a parallel universe that is governed by laws of language instead of physics, travels about in a CORDIS which takes the form of a repeated meme, the number forty-five. Nobody features in two Big Finish audio dramas, The Word Lord (2008) and A Death In The Family (2010), both written by Steven Hall. Personally, I highly recommend both of these audios.

But not all Big Finish scribes can gain such high praise. A ‘definition’ of a transcendental number features in the Big Finish audio drama ish… (2002) written by Phil Pascoe. My sense of pedantry regrets to inform you that the definition given is actually for an irrational number, but pi is still correctly identified as an example of a transcendental number.

The transcendental nature of pi has also been a source of conflict in the past as it managed to start one of most bizarre and petty disputes ever in academic history. During his later years, the 17^th Century political philosopher Thomas Hobbes developed an amateur interest in geometry and, at the age of 67, claimed he had managed to square the circle. Ah yes, that old chestnut. We’ve already established that this is impossible and so Hobbes’s solution was entirely fallacious.

John Wallis, an Oxford maths professor described as “the finest British mathematician before Issac Newton”, soon challenged the validity of Hobbes’s ‘proof’ in a short pamphlet entitled Elenchus geometriae Hobbianae (The refutation of Hobbes’s geometry). Wallis was perhaps spurred on by Hobbes’ scathing attack on the Oxbridge academic system in his political totem Leviathan.

Hobbes later responded to Wallis in an addendum to his book entitled Six Lessons to the Professors of Mathematics. Wallis would counter this with Due Corrections for Mr Hobbes in School Discipline for not saying his Lessons right. Hobbes then retaliated with Marks of the Absurd Geometry, Rural Language, Scottish Church Politics and Barbarisms of John Wallis. Wallis then retorted with the publication of Hobbiani Puncti Dispunctio! or the Undoing of Mr Hobbes’s Points. Hobbes then subsequently died in 1679, and so Wallis claimed victory by the virtue of still being alive. It’s incredible to think that British academic life was consumed by a squabble over the transcendental nature of pi for the best part of a quarter of a century.

Thirdly, and finally, pi is a non-constructible number.

A number is constructible if and only if a geometric line of that length can be constructed using a ruler (with no markings!) and a pair of compasses in a finite number of steps. Any number that cannot is non-constructible. This is a very old category of number, dating all the way back to the Ancient Greeks. Using these rules of geometry, you can construct any whole number, any fraction and even the square root of any whole number, but not pi.

The inability to construct a geometric line of length pi is the reason why it is actually makes it impossible to ‘square the circle’. This mathematical titbit is actually the consequence of a result in an area of mathematics called Galois Theory; the Eleventh Doctor and Valerie meet Evariste Galois in the Big Finish Short Trip The Galois Group (2023) and face the very real consequences of Galois Theory for themselves, much to their (mathematical) horror.

The non-constructability of pi is thematically relevant to the events of Flatline. The Boneless are reconstructing themselves in three dimensions, appropriating the forms of their victims. If we assume that this process of translating themselves from two dimensions into three dimensions is a mathematical one, and that they operate on the same rules of mathematics as ourselves, then it is conceivable that the transmission of a non-constructible number isn’t going to help them achieve their goal here. If anything, it’s going to be a hinderance to this; a mathematical form of blockade!

Whilst this might act as a valid form of defence, it would not establish the peaceful solution that the Doctor is hoping to achieve here. Mathieson has even acknowledged this himself in an interview (Doctor Who Magazine – Issue 493) after reading a comment on Reddit, stating “For all we know, Pi was a declaration of war in their language!”

So there you have it, that’s why I think the Doctor was wrong to use pi to broker peace in Flatline. In summary, like alcohol and matches, the number pi should be used responsibly. And if you don’t like numbers… well, there’s always the arts.

Pass the spray can.

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“And incidentally, a Happy Pi Day to all of you at home!”