Mathematician Sarah Hart explores the use of mathematics in literature and sees mathematics as a new perspective for understanding literature.
For thousands of years, people have explored the connection between mathematics, **, and the arts. Sarah Hart is now looking at literature through the lens of mathematics.
From an early age, Sarah Hart had a keen eye for the ways in which mathematics had infiltrated other fields. As a child, she was struck by the ubiquitous number 3 from fairy tales. Hart's mother, a math teacher, encouraged Hart to find patterns and gave her math problems to kill time.
Hart received his PhD in group theory in 2000 and later became a professor at Birkbeck College, University of London. Hart's research explores the structure of the Coxter group, a more general structure that can describe the symmetry of polygons and prisms. In 2023, she published the book Once upon a time, there was a prime number, which talks about the application of mathematics in ** and poetry.
Since we are part of the universe, it is only natural that our forms of creative expression, including literature, also exhibit a tendency towards patterns and structures," Hart wrote, "and mathematics is therefore the key to understanding a whole new perspective on literature." ”
Since 2020, Hart has been Professor of Geometry at Gresham College, London. There is no traditional curriculum at Gresham College, but rather a few public lectures each year by professors. Hart was the first woman in history to hold the position (428 years old), which had been held in the 17th century by Isaac Barrow, who was best known for teaching another Isaac (Newton). Most recently, the position was held by mathematician Roger Penrose, winner of the 2020 Nobel Prize in Physics. Hart talks to Quanta about how math and art influence each other. Interviews are condensed and edited to ensure clarity and ease of understanding.
Why did you write a book about the connection between mathematics and literature?
These connections are less explored and understood than the connections between fields such as mathematics and **. The connection between mathematics and ** can be traced back at least to the Pythagoreans. However, while academia has written a few articles and studies on specific books, authors, or genres, I have yet to come across a book that speaks to a general audience about the broader connections between mathematics and literature.
How should people in the art world view mathematics?
There is a lot in common between mathematics and (let's call it other arts). In literature, **, and art, you never start with a blank slate. If you're a poet, you're going to make a choice: do I write a haiku that strictly follows the syllabic limits, or do I write a sonnet that has a specific number of lines, rhyme, and metrics? Even if there is no rhyme, there will be lines and rhythms. There will always be some constraints to stimulate creativity and help you focus.
In mathematics, we have the same situation. We have some ground rules. Within this range, we can explore, play, and prove theorems. What mathematics can do for art is to help find new structures and show what's possible. What would a ** look like without tonality? We can think about the different permutations of the 12 tones, and all the ways in which it can be done. There are different color schemes to design here, and there are different forms of poetic rhythms.
How was mathematics influenced by literature?
Thousands of years ago, Indian poets tried to think about possible rhymes. In Sanskrit poetry, you will have long and short syllables. Long syllables are twice as long as short syllables. If you want to figure out the syllable combinations that take three units of time, you can have short and short, or long or short and long. There are three ways to get 3. There are five ways to make a phrase of four lengths. There are eight ways to make a phrase of five lengths. You get a sequence like this, where each term is the sum of the first two terms. This is exactly what we now call the Fibonacci sequence. But this predates Fibonacci by centuries.
So how does mathematics affect literature?
A very simple sequence, but with very powerful effect, is Luminous Body, published in 2013 by Eleanor Catton. She used the sequence of 1, 1 2, 1 4, 1 8, 1 16. Each chapter in the book is half as short as the last. This creates a very captivating effect, as the pace is picking up and the choice of characters is more limited. Everything is speeding up towards the end. By the end, the chapters become very short.
Another example of a slightly more complex mathematical structure is the orthogonal Latin phalanx. The Latin phalanx is a bit like a Sudoku lattice. In this case, it will be a 10x10 mesh. Each number appears only once in each row and column. Orthogonal Latin squares are formed by superimposing two Latin squares, so there is a pair of numbers in each space. The grid formed by the first number in each pair is the Latin square, and the grid formed by the second number in each pair is also the Latin square. Also, in the pairing grid, no pairing appears twice.
These are very useful in a variety of ways. You can use them to generate error-correcting codes that are useful for delivering messages over noisy channels. But the greatest thing about these squares of particular size 10 is that Leonhard Euler, one of the greatest mathematicians of all time, thought they could not have existed. It was one of the few mistakes he made; That's why it's so exciting. This conjecture was disproved in 1959 long after he suggested that these things could not exist for a particular size, and the discovery was published in Scientific American magazine that same year.
Years later, French writer Georges Perec was looking for a structure for his book, Life: A User's Handbook. He chose an orthogonal Latin phalanx. He set his book in an apartment building in Paris with 100 rooms forming a 10x10 square. Each chapter is in a different room, and each chapter has a unique style. He lists 10 things – various fabrics, colors, and many more. Each chapter uses a unique combination. It's a very fascinating way to build a book.
You obviously place a lot of emphasis on good writing. How would you rate the quality of writing for a math study**?
The difference is huge! I know we value brevity, but I think sometimes that gets over-interpreted. There are too many ** examples that don't have any usefulness.
What we really value is a clever argument because it cleverly covers all the situations, so it's also short and elegant. This is different from compressing a lengthy argument into a smaller space than it needs to be, by shortening the symbol by overwriting the mystery symbol you created on the page, but not only the reader, but even yourself may have to struggle to explain it again to understand what's going on.
We don't give enough thought to help the reader remember the symbols of meaning. The right notation can revolutionize math and also make room for promotion. Think about the history when it was easier and even possible to start thinking about 2 and 3 when you went from writing the unknown, its square, and its cube in three different letters, to when you started writing x, x 2, and x 3.
Do you see the connection between math and art evolving?
There's something new coming out all the time. Fractals were ubiquitous in the 1990s. On the wall of every student residence hall is a Mandelbrot set or similar**. Everybody is like, "Oh, this is awesome, fractals." You'll find, for example, that homemakers and composers are using fractal sequences in their works.
When I was about 16 years old, these new things called graphing calculators appeared. Very exciting. A friend of my mother gave me a program to draw a Mandelbrot set on one of the small graphing calculators. It has about 200 pixels. You program this thing in, and then I had to leave it for 12 hours. It plots those 200 points at the end. So even elementary school students in the late 80s and early 90s were able to get involved and make these ** for themselves.
It sounds like you're very interested in hardcore math even when you're in school.
I think I've been interested even before I knew it meant I was good at math. Like, I've loved making patterns since I was a kid.
When I was a kid, my favorite toys were some very simple wooden painted tiles. They come in a variety of different colors. I'll make them patterned, and then I'll look at it proudly for a day or two before making another one.
When I was a little older, I would play with numbers and observe patterns. I would go to my mom and say, "I'm bored." Then she'll say, "Can you figure out the pattern of the number of points that make up a triangle?" Or something like that. She'll make me rediscover triangle numbers or something, and I'll be very excited.
My poor mother, how many amazing inventions I brought to her. "I've developed a whole new way of doing things! And then she'd say, "Okay, that's good. But, you know, Descartes thought of this hundreds of years ago. Then I left; A few days later I came up with another amazing idea. "Fantastic, dear. But the ancient Greeks already had this. ”
Do you remember any particularly satisfying moments in your mathematical research career?
It's always satisfying when you finally understand the patterns you're seeing, and when you find how to accomplish the proof you've been working on. The happy moments I remember the most, probably because they were felt for the first time, at the beginning of my research career. But when you finally understand what's going on, that "aha" feeling is still a wonderful feeling.
Very early on, I tried to prove something about the infinite Coxter group. I solved some cases, and while working on others, I came up with a technique that could be applied if certain conditions were met. You can write these relationships as a graph, so I started collecting a collection of graphs that could apply my technique. It's a year during the Christmas season.
After a while, my **set started to look like a specific set of ** listed in a book about the Coxter group in my office, and I began to wish it was this exact set of **. If so, then it will fill the gap in my proof and my theorem is complete. But I can't be sure until you go back to college after Christmas – when you haven't been able to Google everything. I guess having to wait to confirm my hunch and make it even more exciting when compared to the charts in the book at the end, they do match.
How do you see mathematics as a matter of creation or discovery? Hardly anyone would argue that any of the ** families you write about in your book "discovered" their work. Is this the fundamental difference between mathematics and literature?
Maybe, but there are some resonances.
Doing math feels like discovery. If we're inventing math, it's certainly not going to be that hard to prove something! Sometimes we very much hope that something is true, but it is not. I don't think we can avoid logical consequences.
When you do, everything feels like discovery. Some of the choices reflect our experience in the real world, such as the axioms of geometry that we use, which are chosen because they seem to roughly conform to reality – although even there, there are no "points" or "lines" (because we can't draw something that doesn't take up space, whereas lines in geometry have no width and extend infinitely).
To some extent, a similar continuum exists in literature. Once you've defined the rules of a sonnet, you'll have a hard time writing a sonnet that ends with "orange" or "chimney" in the first line.
But I couldn't resist sharing jr.r.Tolkien said of writing The Hobbit: "It all started with me reading exam papers to earn some extra money. ......One day, I came to a blank page of an exam book and scribbled on it: "In a hole in the ground lives a hobbit." I don't know anything about these creatures except this one, and his story won't grow in a few years. I didn't know where the word came from.
The Hobbits – Did he create them or did he discover them?