Given Apartment613′s focus on arts and culture, some fascinating parts of city life can get overlooked. I was thinking about this recently while reading a math-related novel. While I am no number whiz, I appreciate math’s inherent beauty, and strongly believe that mathematical ideas can be as captivating as a beautiful painting or catchy song. In fact, I came to realize that mathematical concepts manifest themselves in the city’s art and cultural life in some unexpected ways.
To uncover Ottawa’s hidden math scene, I chatted with Prof. Brett Stevens from Carleton University, while from the University of Ottawa I spoke to Prof. Robert Smith? (the question mark is not a typo as he legally changed his name).
The ensuing conversations introduced me to some fascinating ideas that I will share below, as well as revealing new things about Ottawa that I would have otherwise overlooked.
Infinity and the National Gallery
Let’s begin at the National Gallery on Sussex, where one can see the work of the late Vancouver-born artist Kazuo Nakamura. In his painting Square Infinity that hangs in the Gallery, as well as Number Structure and Fractals that is stored in the museum’s warehouse, Nakamura draws patterns that if extended outwards would go on forever.
“We would use (mathematical) induction to prove that those patterns go to infinity, even if the painting doesn’t go to infinity,” says Prof. Stevens, who has written on Nakamura’s work.
This connection with mathematical induction is particularly cool.
“(This is) a technique we use in mathematics,” explains Prof. Stevens. “It’s a handle on infinity …. It’s a way we can prove that an infinite number of statements is true.”
In other words, it’s impossible to write out an infinite sequence of numbers. If you want to prove that a statement is true for all natural numbers, i.e. the infinite set of counting numbers 1, 2, 3 and so on, you must present a finite argument that fits on paper. Cue mathematical induction that allows humans to prove an infinite series in finite terms (see here and here for details).
Similarly, Nakamura used a finite canvas to represent infinity, which is pretty mind-blowing when you think about it. For us finite creatures, it is amazing that we can even discuss the infinite, let alone see a representation of it at the National Gallery of Canada.
Chaos Theory and Civil Servants
Are you a policy analyst for the government? If yes, then you might have engaged in reasoning that mimics chaos theory.
Let me explain.
In everyday usage, the word chaos refers to a random event with no discernible pattern. The mathematical definition, in contrast, refers to a particular type of system, explains Prof. Smith? (see here, here, and here for details).
When mathematicians say a system is chaotic, they do not mean it’s random, but rather that it’s highly sensitive to initial conditions. A good example is the weather, where even a small event can have massive consequences. This is the so-called butterfly effect: a butterfly flaps its wings in one part of the world, starting a sequence of events that cause a hurricane in another.
To perfectly predict the weather, one must measure every molecule in the atmosphere, which is impossible. That is why weather forecasts cannot be 100 per cent accurate. Climate, however, does behave in predictable patterns. For instance, while we cannot know if it will rain in 30 days, we do know that February is colder than August. As such, while it is impossible to predict with certainty the state of a chaotic system, one can model its behaviour.
“History does not repeat but it rhymes,” notes Prof. Smith?, who models the spread of infectious diseases and has written a book on zombies. “Things that go away must come again.”
So what does this have to do with civil servants? Well, government policy is developed for society instead of specific individuals. When evaluating a policy, civil servants analyse its impact on social behaviour rather than on individual people – similar to chaos theory.
“That’s right,” says Smith?, when asked to comment on the chaos theory-civil service analogy. “We can’t predict what you and I will do, but we can predict what the group will do.”
Pigeons, Your Hair and a Manual for Life
A beautiful thing about math, notes Prof. Stevens, is its ability to transform simple, almost childlike statements into powerful tools. Consider the Pigeonhole principle.
In plain English, this principle states that if there are n holes and n + 1 pigeons, and every hole has a pigeon, then because there are more pigeons than holes you will have at least two pigeons in one hole. This statement is so simple it sounds trivial.
This principle, however, can prove interesting results. Like the fact that at least two people in Ottawa-Gatineau have the exact same number of hair strands on their head.
Here’s the argument: A typical human head has between 90,000 to 150,000 hairs (see here and here), so it is reasonable to assume that nobody has more than 1 million hair strands. The latest population for Ottawa-Gatineau is estimated to be 1.2 million. Given that there are more people in Ottawa-Gatineau than the maximum number of hairs on a head, we can conclude that at least two people in the National Capital Region have the exact same number of hair strands.
Puzzle lovers will recognize this form of reasoning.
“Sudoku players are doing this all the time,” says Prof. Stevens. “When Sudoku was first introduced newspapers would say, ‘It has numbers but don’t worry, it has no math.’ But of course it has math. It may not have addition and subtraction but it has mathematical logic.”
Prof. Stevens adds that mathematical logic is also found in such literary groups as Oulipo, a group of French writers and mathematicians who experimented with constrained writing techniques. For instance, each line in a poem could consist of a single word, with each successive line being one letter longer. Another hypothetical technique could exclude one or more letters, e.g. a short story without the letter B.
A brilliant use of constrained writing is in the novel Life A User’s Manual by George Perec. This masterpiece is based on a single moment around 8:00 p.m. on June 23, 1975, and tells the individual stories of the inhabitants of a 10-storied building in Paris. The book contains a number of constraints, including an ingenious knight’s tour that is used to navigate the building.
So if you are curious about what happens when math and literature combine, go to the Ottawa public library and pick up Perec’s book, and explore how mathematical logic can produce great art.