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Cinema Science: PI AND THE RAPTURE OF COMPLEX MATHEMATICS: While its title already indicates one of its central mathematical topics of concern, Darren Aronofsky's cult paranoid thriller also offers ample prompts for senior secondary students to explore sequences, phenomena such as the golden ratio and Pythagoras' theorem, and more intricate concepts like irrational numbers and recursion, as DAVE CREWE outlines.

To commemorate the eighth iteration of Cinema Science, this issue we're doing something a little different. The typical mode of this column is to identify the scientific and mathematical content within a contemporary film - the kind of film that you can expect to find fresh on physical media, streaming services and in your students' memories at the time of publication. But, two years down, I wanted to delve into the archives to examine Darren Aronofsky's debut feature, the mathematically minded Pi (1998).

This necessitates a slightly different approach for a few reasons, with the main consequence being that the topics addressed here are primarily - if not totally - aimed at senior secondary Maths students. (Think: Mathematical Methods and Specialist Mathematics classrooms, or whatever your local equivalent might be.) That's not entirely because of the content; many of the ideas covered herein - irrational numbers, recursive sequences, Pythagoras and his many contributions to mathematics - could comfortably be incorporated into earlier Mathematics curricula. Rather, this is mostly a reflection of the film itself.

Cinema Science typically centres on modern movies for purely logistical reasons; few Science or Maths classrooms can accommodate a viewing of an entire feature Elm, so choosing well-known, recently released films allows teachers to play clips or simply refer to the movie rather than playing it in its entirety. Not so here; with two decades having passed since Pi's release, you'd be hard-pressed to find any high school student who would've heard of, let alone seen, the film. While Aronofsky has made some major films since - Requiem for a Dream (2000), Black Swan (2010) and, more recently, mother! (2017) - his debut is most fondly remembered by arthouse aficionados.

That's for good reason - Pi doesn't make for easy viewing. Aronofsky worked mathematical concepts into the scaffolding of this paranoid thriller, incorporating the fleshy, cyber-body-horror aesthetic of Tetsuo, the Iron Man (Shinya Tsukamoto, 1989) to produce an overexposed, monochrome monstrosity. As Aronofsky's first film is rated M by Australia's classification board, you'd in any case struggle to justify screening the film to students outside of the oldest years of secondary schooling.

Thus, this Cinema Science instalment is built around the assumption that you'll be showing the complete film - at a slender 84 minutes, not an impossible ask - to a class with a solid background in Mathematics. Perhaps not warped geniuses like protagonist Max (Sean Gullette), though you never know; maybe you'll spark a few mathematical obsessions along the way.

The life of pi

Where better to start with Pi than pi itself? Pi may be an important figure in a range of mathematical disciplines, and even in mathematics communities - fanatics annually celebrate Pi Day on 14 March (for those who write their months before their days) and/or Pi Approximation Day on 22 July (for those of us who write dates the correct way and happen to prefer a more accurate approximation than 3.14') - but it also figures prominently in pop culture. In Yann Martel's novel Life of Pi - as well as Ang Lee's 2012 film adaptation - the protagonist, Piscine, earns the moniker 'Pi' after memorising the number to an intimidating number of decimal places. In Twilight (Catherine Hardwicke, 2008), vampire heart-throb Edward (Robert Pattinson) tries a similar trick with the square root of pi. Dig into the internet and you'll find plenty of YouTube videos of people performing the same trick to thousands of decimal places.

That pi's digits continue to infinity without pattern or termination is at once pedestrian - the same is true of any irrational number in our decimal system, after all - and undeniably fascinating. In part that's because, I suspect, pi is most people's first exposure to the intersection of irrational numbers and infinity. Whatever the reason, the centrality of pi to Pi presents a prime opportunity to explore the notion of irrational numbers in more complexity.

To sidestep Aronofsky's debut for a moment, rewind a year to 1997 and the release of Contact, an adaptation of Carl Sagan's eponymous novel directed by Robert Zemeckis. In Sagan's novel, pi holds the secret to the universe. The protagonist finds a binary sequence that, when represented in base 11, (2) produces a circular pattern, which suggests that the universe is the product of intelligent design and propels the ensuing storyline. I'm personally a big fan of this as a way to stimulate thinking around of infinity. Ask your students: Is this a plausible conclusion? Or is it more reasonable to expect that an infinite string of numbers would eventually produce any pattern you could think of?

Case in point: the 'Feynman point'. Named for famous physicist Richard Feynman - though more reliably credited to Douglas Hofstadter's book Metamagical Themes: Questing for the Essence of Mind and Pattern - it refers to a string of six nines found from the 762nd decimal place of pi on. In his book, Hofstadter jokes about memorising the preceding 761 digits so that he can recite them up to this point and then imply that pi is a terminating decimal - and thus rational. (3) This is an excellent example of how non-recurring decimals like pi don't necessarily follow the kind of consistent 'non-pattern' pattern we've been conditioned to expect.

From here, you can ramp up to an exploration of irrational numbers in general. I would hope that students at this level are familiar with the idea that irrational numbers can be broadly separated into surds - roots of natural numbers that can't be simplified to a natural number - and transcendental numbers like pi and Euler's number, e. But, at the senior secondary level, you can explore how we know these numbers are irrational, going through the classic proofs that the square roots of two and three (and beyond, if you like) are irrational.

Ah, but let's not forget pi. While the proof that pi is irrational - first proffered by Johann Heinrich Lambert, and soon expounded upon by a suite of mathematicians - is likely too complex for secondary students, the question of how we can actually generate this transcendental number to a dizzying number of decimal places is one worth exploring. While pi's initial definition is, of course, linked to the ratio between a circle's diameter and circumference, mathematicians have since found an intimidating array of formulas and series (4) to define the number. It's these definitions - such as the Gregory series - that can be used by computers, much like Max's 'Euclid' in Pi, to algorithmically approximate pi to whatever degree of accuracy is required.

* How do we know that pi is a transcendental number and not a surd?

* Is it necessary for a good mathematician to know pi to a given number of decimal places?

* Is it reasonable to assert that any sequence of numbers, no matter how long, could be found in the decimal places of pi?

Good as golden ratio

Like any good mathematician, Pi's Max outlays his assumption early. These are:

1. Mathematics is the language of nature.

2. Everything around us can be represented and understood through numbers.

3. If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.

While Max is plugging away at identifying the patterns buried within the stock market (or, per his example, 'the wax and wane of caribou populations'), the film itself is densely populated with numerical patterns - the most prominent being the Fibonacci sequence and its associated golden ratio. The Fibonacci sequence, taking the nickname of creator Leonardo Bonacci, has a simple, recursive definition: take one as the first term, one as the second term, and then find each subsequent term by adding the two preceding terms. This produces: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and so forth - with the latter two terms getting shout-outs in the film as the numerical translations for 'Kadem', the Garden of Eden, and 'Aat ha haim', the Tree of Knowledge, respectively.

While I can't speak to the Fibonacci sequence's relationship to the Hebrew language or Jewish theology, there are enough opportunities to explore its appearances in nature and beyond - enough to fill an entire Cinema Science column, to be honest. I'll single out two: firstly, in mathematical patterns, Pascal's triangle. (5) Pascal's triangle has a plethora of famous patterns nestled within it, but line up your diagonal lines just right and sum the totals and, yes, you can find the Fibonacci sequence hiding there, too. Then there are natural spirals, as flagged in the film: count the spirals found in plenty of plants - sunflowers, pine cones, etc. - and you'll find that the number of spirals is (almost) invariably of a Fibonacci nature. More broadly, Fibonacci is a great jumping-off point for a broader explanation of sequences, series, convergence and mathematical patterns in general.

The real fun of Fibonacci comes when you explore its common ratio; as Max suggests in the film, 'If you divide 144 into 233, the result approaches 9 [theta] - the Greek symbol for the golden ratio.' Granted, there're two mistakes in that sentence: the conventional symbol for the golden ratio is (p [phi], not 9, and really Max should say that if you divide two consecutive terms of the Fibonacci sequence, they'll approach phi... but we take his point. As the Fibonacci sequence progresses, the ratio between terms gets closer and closer to said golden ratio, which appears so often in nature that it has attained a kind of mystical quality in maths.

The golden ratio, or phi, is equal to half of the sum of one and the square root of five; or, if you prefer, it's the positive solution to the quadratic equation ([[phi].sup.2] - [phi] - 1 = 0. A more interesting definition is found in exploring where the equation comes from; typically, it's derived from the spiral of rectangles defined in a similar way to the Fibonacci sequence (as seen, if erroneously, in Pi). Even more interesting is how often the golden ratio pops up, whether in art and architecture, or mathematical applications such as continued fractions and the Euclidean algorithm for finding the greatest common divisor of two integers. All of these provide a wealth of opportunities in the classroom (indeed, I've assigned a term assignment in the past based entirely around the golden ratio).

But let's return to spirals, given how they're specifically referenced in Pi's dialogue. Many use the presence of Fibonacci numbers in natural spirals as evidence of intelligent design. Much like Sagan in Contact, they see these patterns as evidence of some grander pattern. And that's a compelling-enough hook if you'd like to entertain your class with such a narrative, but I personally prefer to examine why the Fibonacci numbers pop up in these spirals. As entertainingly explained by Vi Hart in a short YouTube series, (6) plants have evolved to grow in this fashion to maximise their cumulative exposure to sunlight - in part, by using the 'most irrational angle' defined in terms of the golden ratio. An investigation of this would be an excellent way to underline just how emphatically mathematics is the language of nature.

* Explain the relationship between the Fibonacci sequence and the golden ratio.

* Which artworks or architecture can you find the golden ratio in? Consider critically the claim that the golden ratio is 'aesthetically pleasing'.

* Max claims that 'there are patterns, everywhere in nature'. Is it reasonable for him to extrapolate that he could therefore find the patterns to predict the performance of the stock market?

Cult of Pythagoras

Pythagoras' name is linked to the golden ratio explicitly within Pi. Max offers the following reflection in voiceover halfway through the film:

Remember Pythagoras. Mathematician. Cult leader. Athens. Circa 500 BCE. Major belief: the universe is made of numbers. Major contribution: the golden ratio. Best represented geometrically as the golden rectangle. Visually, there exists a graceful equilibrium between the shape's length and width. When it's squared, it leaves a smaller golden rectangle behind with the same unique ratio. The squaring can continue, smaller and smaller and smaller, to infinity.

There are, again, minor mathematical errors here; in fact, I'd suggest replaying this clip for your students to see if they spot them. (For reference, the use of 'squaring' is misleading at best, while the diagram that appears on screen is erroneously labelled if you're trying to represent (p.) So is Max correct in his assertion that Pythagoras' major contribution is the golden ratio?

Most would be familiar with Pythagoras from his eponymous theorem relating the length of the hypotenuse of a right-angled triangle to its adjacent sides; among recalcitrant Mathematics students (past and present), it's a favourite example of a formula that they'll never 'use in the real world' - the unfortunate side effect of an introductory curriculum that implies that you're likely to need to use long division to calculate your grocery bill. But, yes, Pythagoras can be credited with both this theorem and the golden ratio, as well as many other discoveries such as the link between musical harmonies and the lengths of instruments' strings ... provided you're happy to accept a degree of historical uncertainty.

That uncertainty is almost certainly a consequence of Pythagoras' 'cult leader' status, as flagged by Max. Pythagoras was the head of a notorious, numbers-obsessed community that worshipped the number ten (as the 'perfect' triangular number) and generally shared Max's belief that numbers were the key to the universe. Their secrecy means that much of the historical record regarding Pythagoras is contradictory and confusing - a mix of rumours, legends and myth.

But that's no reason not to explore! Far too often in the Mathematics classroom, we offer up dry definitions, theorems and proofs because that's how it's done, but diving into history can be a compelling way of providing flavour and narrative to concepts lacking both. There are enough interesting stories lurking in libraries and across the internet regarding Pythagoras that you could easily spend a lesson asking your students to conduct research about in small groups, before sharing their most interesting findings - with conscious effort to link these stories back to the curriculum, of course.

My favourite (and almost certainly false) story regarding Pythagoras is that of the student who attracted his ire by proving the existence of irrational numbers. Enraged - as, allegedly, one of his core beliefs was that all numbers could be represented as a perfect ratio - Pythagoras drowned his follower to keep the story from spreading. Given that the story has spread, one can understand my scepticism about this tale's validity, but it nonetheless makes for a great yarn and an easy introduction to a unit centring on irrational numbers.

Max's increasing obsession with numbers as Pi progresses mirrors, in many ways, the beliefs of Pythagoreans - and their contemporary equivalents. As Max's mentor, Sol (Mark Margolis), scolds, 'As soon as you discard scientific rigour, you are no longer a mathematician. You're a numerologist.' Numerologists share with mathematicians a love for numbers and patterns, but are better described as cultists rather than academics, finding patterns everywhere with the kind of desperation that defines conspiracy theorists.

While Mathematics teachers are known to indulge these inclinations from time to time - in my first year of teaching, I recall having seen a senior assignment built around biorhythms, and there'd be more than a few teachers who make the golden ratio seem magical to engage their classrooms - in the context of senior secondary students, I'd use numerologists as a cautionary tale, with Max's increasing paranoia a case in point. If you want more evidence in movie form, you could even dust off the forgotten Jim Carrey flick The Number 23 (Joel Schumacher, 2007), which similarly wraps numerology (in this case, the '23 enigma', also briefly alluded to in Pi during Sol's rant) around the paranoid-thriller genre.

* Among his many accomplishments, Pythagoras is sometimes credited with linking mathematics to music. Research this claim, investigating the links between Pythagoras, ratios and musical instruments.

* What is the '23 enigma'?

* The geometric relationship described by Pythagoras' theorem was known before Pythagoras 'discovered' it, but Pythagoras is said to be the first to have formally proved it. Find three different proofs of this theorem.

Odds and ends

In one of Pi's earliest scenes, Max demonstrates a prodigious ability to mentally multiply numbers. Hollywood movies love to use such scenes to demonstrate the genius of their mathematically minded protagonists, even if fluid mental arithmetic is, in the scheme of things, not especially necessary at the highest levels of the discipline. Max, after all, has his trusty personal computer to crunch the numbers for him; provided he's fluent with the concepts, mental calculation is largely unnecessary.

Still, why not take this opportunity to test out your students' mental-multiplication skills with a game? Refer your students to the ancient past of mathematics - when mathematicians would demonstrate their prowess with public duels, and effective algorithms were prized secrets to be kept - and hold a few one-on-one challenges to multiply complicated numbers swiftly. Discuss the shortcuts that students might take to find these values without access to a calculator or a piece of paper. This is a good way to have a little fun while developing your students' confidence in the subject.

Pi hinges on Max's ability to accurately predict the behaviour of the stock market. It's safe to say that, if anyone in the real world has accomplished a similar achievement, they've kept it very close to their chest. Of course, the whole notion is ludicrous, but it's worth examining why. After all, Max is correct to assert that mathematics is the language of nature. Why can't we accurately predict the stock market or, say, predator-prey populations? That's a hard question to answer, but I'd start with an introduction to chaos mathematics: the mathematics of systems that are deterministic yet, due to the precision of their initial conditions, algorithmically impossible to predict.

Finally, I'll direct you towards my favourite mathematical theorem and associated proof of all time: Godel's incompleteness theorem. In the early twentieth century, Kurt Godel proved that we couldn't prove everything - that certain mathematical theorems were necessarily undecidable, meaning that we'd never know for sure whether they were true or false. This is a shock to the system for any student who's spent their schooling learning about the precision and purity of mathematics, but where's the link to Pi? That can be found in Godel's proof for this theorem, wherein he uses numbers to represent the possible operations within mathematics and thereby demonstrates its fundamental incompleteness - much as Max's kabbalist kidnappers obsess over expressing Hebrew as numbers, or binary language encodes any possible expression as numbers.

Dave Crewe is a secondary school teacher and film critic based in Brisbane. Queensland. His writing can be found at SBS Movies, The Brag and Metro magazine, or his own website. <https://ccpopculture.com>.

Endnotes

(1) The fraction 22/7, or three and one-seventh, is a common approximation for pi that's slightly more precise than 3.14. Hence: 22 July, written as 22/7.

(2) There's an extra lesson or two here on exploring our decimal system and the definitions of recurring and terminating decimals, if you'd like - but, given that this is a piece on Pi, not Contact, I digress.

(3) Douglas R Hofstadter, Metamagical Themes: Questing for the Essence of Mind and Pattern, Basic Books, New York, 1985, p. 125.

(4) A good resource for this can be found at MathWorld: <http://mathworld.wolfram.com/PiFormulas.html>, accessed 11 December 2018.

(5) If you're unfamiliar with how to generate Pascal's triangle, here's a simple explanation: <https://www.mathsisfun.com/pascals-triangle.html>, accessed 11 December 2018.

(6) See, for example, 'Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]', YouTube, 21 December 2011, <https://www.youtube.com/watch?v=ahXIMUkSXXo>, accessed 11 December 2018.
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Date:Mar 1, 2019
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