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Art and science

Death by prime numbers

05 Mar 2020
Taken from the March 2020 issue of Physics World. Members of the Institute of Physics can enjoy the full issue via the Physics World app.
Prime Suspects: the Anatomy of Integers and Permutations

Two people are dead, and the police are baffled. Arnie Int, the 60-year-old lieutenant to the godfather of the Integer crime family, is found brutally murdered in a drainage tunnel. Later, the petite body of young ballet dancer Daisy Permutation is also found and brought to the morgue. Gruff and grizzled lead detective Jack von Neumann suspects a link between the two, and has brought in a consultant on the case – legendary mathematician and professor of forensic science C F Gauss.

So begins the unique graphic novel Prime Suspects: the Anatomy of Integers and Permutations, an imagined world “where detectives work closely with mathematicians”. A forensic crime drama, mixed in with number theory, as well as an exploration of student–mentor relationships, all in the graphic novel format, Prime Suspects was written by the Canada Research Chair in Number Theory at the University of Montreal mathematician Andrew Granville and writer, educator and director Jennifer Granville; with illustrations by Toronto-based graphic designer Robert J Lewis. Bringing in elements from film noir, TV police shows and famous movies, coupled with some amazing art work, subtle mathematical humour and corny science jokes, and what you have is a one-of-a-kind creation – indeed, Prime Suspects has it all from minus to plus infinity.

Early in the story, Gauss involves two of his most promising students – the snooty Sergei Langer and the hip, red-haired heroine Emmy Germain – while a pair of documentarians have arranged to record the team’s work. Eager to impress (and become the professor’s new research assistant), Langer and Germain both attempt to uncover the details of the murders. At the morgue, the team soon begins to find some surprising links between the two bodies, including peculiar cuts on the chest, and both hearts having been surgically removed.

While Langer barfs at the gruesome sight, Germain reaches into Int’s chest and pulls out a bloody clump of tissue with the distinct shape of “7309”; and so the book is off to the races. “Primes are the fundamental constituent parts of integers,” she tells the confused film producers, “their genetic code, if you like.” We also learn that the ballet dancer’s family business is the Alternating Group – a tongue-in-cheek description, though not as visceral as the primes. A bit corny, perhaps, but corny in the pursuit of larger truths.

A bit corny, perhaps, but corny in the pursuit of larger truths

As the story moves to Gauss’s luxury penthouse (don’t all legendary mathematicians have one?), the trio play billiards and discuss the similarities between primes, the fundamental constituents of integers, and cyclic permutations (or cycles), the fundamental constituent parts of [Daisy’s] permutation. Langer, ever anxious to show off his smarts, says “they’re about as similar as apples and iPhones”. In fact, as the rest of the book goes on to show, primes and cycles play similar roles in the study of integers and permutations, respectively. Even many of the equations describing their traits are similar; there’s a cyclic equivalent to the 15-year-old real mathematician Carl Friedrich Gauss, who found that the number of prime numbers among the first N integers is near log(N). (Physicists use the notation ln(N)).

As a student of physics who received their career-worst grade in undergraduate topology class (in my defence, I did join the class two weeks late), the properties of numbers and sequences come more easily to me than the more abstract properties of permutations. We all understand prime numbers. A permutation of, for example, the four integers (1 4 5 8) is (4 5 8 1); another is (5 8 1 2), and so on. M items can be permuted in M! (M factorial) ways, and every permutation can be broken up into “cycles”.

These cycles (a bit too involved to include in this review; see Wikipedia’s entry on “permutations”) all containing M or fewer items, uniquely represent a permutation much as primes uniquely represent an integer. Exactly one in every M permutations on M letters is a cycle. Phew.

Just when you start to fear the book is going over your head, and that you’ve missed something and can’t keep up with the maths, you thumb through the rest of the book to discover a chapter near the end called “The mathematics of prime suspects”. There, the mathematics of both primes and cycles is laid out in traditional mathematical garb, lingo and rigour. It felt like arriving home after a foreign trip.

It was then I realized the graphical storyline wasn’t meant to teach the details of the maths so much as to wet my whistle and motivate me to learn more. Which it did, not just via the later, textual chapters but by going online to Wikipedia and YouTube and reading a few papers (some by one of the authors, Andrew Granville). As it emerges, the central question here is: why are the anatomies of integers and permutations so similar?

Later in the book a third murder victim, Polly Nomial, is found whose body has been spread over Finite Fields…. I won’t give away the murderer or the story’s conclusion, but I will give away its success: it entertains while provoking one’s curiosity. Every page has something witty on a background sign or screen, and the book is chock full of homages to famous mathematicians, right down to the character names. With its beautiful and expressive artwork Prime Suspects is truly “mathematics as you’ve never seen it before”.

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