Best Writing on Mathematics 2018

I’m honored to be included in Best Writing on Mathematics 2018, Princeton University Press. My essay “How to Play Mathematics” was originally published in Aeon. Here I argue that mathematics can be seen as a performative activity akin to music. Take jazz: while any piece of jazz could be written out as a score, the essence of jazz is an improvisational playing-out. Musicians make music in real-time, and throughout history most music hasn’t been notated. Many of the worlds beloved musicians have been sonically illiterate - not being able to read or write music - Jimi Henricks, Michael Jackson and Muddy Waters were all in this camp. Formal notations are one way to relate to music, yet most of us engage with it sensually, through our ears rather than our eyes. So too, the symbols of equations and textbooks are just one way to relate to mathematics. In my essay, I speak of math as a sensual activity that can often be engaged with through material forms of play. Islamic mosaicists made complex geometric tiling patterns that led them to discover aperiodic tilings long before Western mathematicians came to these forms. African artisans made fractal patterns in textile designs and village architecture centuries before these structures were discovered by Europeans. Moreover, when we look at the natural world, all around us we find that things are doing mathematics. Corals and nudibranchs make hyperbolic surfaces. A peregrine falcon diving towards its prey enacts a logarithmic spiral. And as sound waves bounce around a concert hall they act out the mathematics of the Fourier transform. Here I claim that such performances amount to a form of mathematical “knowing”.

I’m delighted to be in this volume along with some of my favorite math writers, including Art Benjamin, Francis Su, Erik&Martin Demaine. The book is edited by Mircea Pitici.

Description of the book from Princeton Uni Press website:

“This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2018 makes available to a wide audience many pieces not easily found anywhere else—and you don’t need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice—and taking readers behind the scenes of today’s hottest mathematical debates.

James Grime shows how to build subtly mischievous dice for playing slightly unfair games and Michael Barany traces how our appreciation of the societal importance of mathematics has developed since World War II. In other essays, Francis Su extolls the inherent values of learning, doing, and sharing mathematics, and Margaret Wertheim takes us on a mathematical exploration of the mind and the world—with glimpses at science, philosophy, music, art, and even crocheting. And there’s much, much more.

In addition to presenting the year’s most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable pieces on mathematics. This is a must-read for anyone interested in where math has taken us—and where it is headed.”

Screen Shot 2018-12-05 at 8.22.31 PM.png

Best Writing on Mathematics 2018

Gravity Waves, Spacetime, and Crochet

The Nobel Prize in physics for 2017 has been awarded to scientists who detected gravity waves, a long awaited discovery deeply deserving of science's highest honor. Coincidently, just before the Nobel was announced, I was interviewed for the NPR program To the Best of Our Knowledge by host Anne Strainchamps where we discussed the mathematics of spacetime – including gravity waves – and the prospect of crocheting the universe. The interview aired nationally in the US on Oct 7 and can be heard here. Titled Is the Universe A Number?, this episode of the beloved NPR show is devoted to explorations of mathematics and its relationship to nature and culture. Along with me discussing hyperbolic geometry and handicraft, is the Israeli mathematician Shlomo  Maital talking about the Kaballah, and former Nobel prize winner Franck Wilczek discussing his delightful book A Beautiful Equation, about physicists' search for beauty in the laws of nature, which I reviewed here in the Washington Post.

The image below of a beautiful crocheted "hyperbolic plane" reveals an unexpected connection between crochet and gravity waves. Here, the ripples in the model result from the fact that the surface isn't flat, as described by the Euclidean geometry we learn about in school, but is curved in the characteristic manner of hyperbolic space. This gorgeous effect is achieved simply by increasing stitches in a regular fashion to produce ruffles, which at the same time produces a model of what mathematicians call "negative curvature" geometry. Gravity waves are also deviations from flatness in space; in this case, they're ripples in spacetime which undulate out from such cataclysmic events as two black holes colliding. Gravity waves can in theory exist in any of the three basic geometries – Euclidean, spherical and hyperbolic – and one of the great questions on physics today is which one of these structures does the large-scale structure of the universe emulate. Whichever space is rippling, the ripples themselves are curves in the the fabric of that space. So if god is a mathematician, then perhaps He/She/It is also a crocheter.

Crochet model of the hyperbolic plane made by Anitra Menning, based on the method of hyperbolic crochet discovered by Dr. Daina Taimina at Cornell University..

Crochet model of the hyperbolic plane made by Anitra Menning, based on the method of hyperbolic crochet discovered by Dr. Daina Taimina at Cornell University..

This story is also posted on Medium.

Symbiotic Earth - A glorious film about Lynn Margulis

Enrico Fermi, one the pioneers of particle physics, wrote that in science “there are two possible outcomes: if the result confirms the hypothesis, then you’ve made a measurement. If the result is contrary to the hypothesis then you’ve made a discovery.” Biologist Lynn Margulis spent her life making discoveries by finding things contrary to the hypotheses of Neo-Darwinism, the set of views that evolution happens at the genetic rather than organismal level, a notion made popular by Richard Dawkins’ book The Selfish Gene. In challenging Neo-Darwinism, Margulis created a paradigm shift that's reverberating through the foundations of biology, and beyond.

A few nights ago I had the privilege of seeing a pre-release screening of a new documentary about her ideas on symbiosis and how they are revolutionizing our understanding of the origin and evolution of life. This splendid film gives us insights into Margulis’ prolific career; and into the woman, whose vitality explodes across the screen with the force of a slow-motion supernova. I was by no means the only person who came away feeling I’d just witnessed a fifth force of nature.

The film, Symbiotic Earth: How Lynn Margulis Rocked the Boat and Started a Scientific Revolution, due for release in 2018, was shown at the NOVA cinema in Melbourne in a special event organized by local sustainable building maven Daryl Taylor. Like Taylor, I’ve long been an admirer of Margulis, and her symbiosis framework helped to inspire my work on my Crochet Coral Reef project. Corals, of course, are prime examplars of interspecies entanglings and a beautiful case of Margulis’ view that critters from across different kingdoms of life can assist and sustain one another through interdependent relations.

It is a mark of Margulis’ brilliance that symbiosis no longer seems radical, or even a surprising concept – as Mark Twain noted, great ideas are always obvious once they’ve been accepted. But when I was starting my career as a science writer, biological thinking was so dominated by the Neo-Darwinist paradigm of gene-level competition that the slightest suggestion of anything approaching “co-operation” between organisms was viewed as romantic sentimentalism, a far worse sin to be accused of in science than “heresy,” which at least has the advantage of potent symbolic associations (Copernicus, Galileo and so on.) Dawkins’ “selfish gene” concept niftily encapsulated a philosophy, made orthodox in the 1940's, that evolution “really” happens at a chemical level, and that the “true” units of natural selection are not species or organisms but the “molecular machines” known as genes.

To all this Margulis voiced a quiet insistent No! From the late 1960’s on she churned out a stream of scientific papers and books arguing for a view of life in which organisms bond together symbiotically, in the process creating new kinds of organisms. According to Margulis, the very eukaryotic cells making up the basis of plant and animal life were formed from a symbiotic melding of simpler, more primitive bacteria.

In the late 1970’s gene sequencing began to prove her right. But resistance remained entrenched in many areas of the academy; so much so that during her life Margulis never received a grant from the National Science Foundation and many other foundations to whom she applied for funding. At one point feedback on a research proposal bluntly declared: “Your work is crap. Don’t bother applying again.” Despite the growing importance of her ideas and clear evidence for them she wasn’t awarded a MacArthur or a Nobel. At some point in the 2000’s rumors began to circulate she was in line for a Nobel, but tragically in 2011 (just as this film was getting under production), she suddenly died of a stroke, and science’s greatest honor can’t be granted posthumously.

I will restrain myself here from writing a long essay about the film and its goddess of a subject – Margulis was also the co-architect with James Lovelock of the Gaia theory of our biosphere – as I’d like to do fuller justice to both when the film launches next year. For the moment I want to make a few brief remarks about gender and science. Filmmaker John Feldman has astutely chosen not to include an overt discussion of this topic, opting instead to show us a powerful female mind at work, and it’s impossible not to be struck by the force of Margulis’ intellect and will, a quality we are so used to seeing in men and are so rarely exposed to in public representations of women. Yet at this moment in 2017 when the subject of women and science is so much on the international radar it would be a lost opportunity not to say something.

One of the few allusions to Margulis’ gender in the film is a brief reference regarding her early marriage to astronomer Carl Sagan, and their son Dorion Sagan’s gently strained on-camera comment about his father being a “typical 1950’s husband” who expected his wife to be at home taking care of the housework and the kids, but that she too “had huge ambitions.” It’s a delicate moment for lovers of science as Sagan is revered in contemporary science communication, yet his fame makes her achievements all the more amazing. How did she break free – and break through – to such a degree?

The following comment isn’t included in the film, however Margulis (who had two husbands) once said: “I left my job as a wife twice. It’s not humanly possible to be a good wife, a good mother and a first-class scientist. Something has to give.” That she exhibits not a shred of anger or regret, during this two-and-a-half hour epic, about the challenges she faced as a woman – or as a wildly original thinker – and radiates to all around a luminous sense of generosity, is a capstone of appealingness on an already wonderful life.


The same day the film screened in Melbourne, a long interview appeared in Quanta Magazine with physicist Nigel Goldenfeld, director of the NASA Astrobiology Institute for Universal Biology, who is using principles from physics to study the emergence and evolution of life. Throughout the interview Goldenfeld talks about concepts Margulis championed. He stresses the need for biologists to get over the “modern synthesis of biology,” the formal term for Neo-Darwinism. Goldenfeld notes the limitations of gene-level determinism and argues that life evolved from primitive, pre-nucleic cells through gene transfer, a process central to Margulis’ later thinking. Though he doesn’t mention Margulis by name, the spirit of her work shines through his discussion; and again it's a mark of her brilliance that ideas she once championed against the mainstream are now being taken up leaders of an emerging mainstream alliance between physics and biology.

Welcome to the "Science Goddess" blog with a tribute to mathematical goddesses, including Maryam Mirzakhani

Hi everyone and welcome to my new blog Science Goddess. Here I'll be writing about whatever I feel like on scientific subjects, be it a pointer to something I love or a rant about something I hate. The blog will reflect my two-fold belief that science is a set of conceptual enchantments that stir our minds and senses, and it's also a socially embedded activity that necessarily reflects wider currents in politics, society, and culture. This plays out in ways that are at times fantastic and beautiful and at others troubling, or even outrageous. As someone who's committed my life to communicating better about STEM subjects, I believe its our duty – and a sacred one - to highlight what's great about science while not being afraid to shy away from its darker affiliations.

I'll begin on a high note as this website has been a long time coming and there's much to celebrate! In honor of my blog name, I'm devoting my first post to mathematical goddesses. No-one epitomizes this more than the brilliant mathematician Maryam Mirzakhani, the first and only woman (so far) to win the Fields Medal, the math Nobel. Mirzakhani died a few weeks at age 40, after an long battle with breast cancer, a tragedy that's been widely mourned in the science press. But lets take a moment to marvel at her achievements. Much of her work was in the area of geometry and topology, and she was an expert on the properties of hyperbolic surfaces - which makes her resonate particularly in my heart, given my work with hyperbolic crochet. Mirzakhani studied the ways curves behave on hyperbolic surfaces that have holes, a quality known as their genus. Holey surfaces are hugely challenging from a mathematical perspective even when they're Euclidean, and are much harder to understand if they're hyperbolic. In her final years, Mirzakhani had realized that the properties of holey hyperbolic surfaces could be related to the mathematics of billiard balls on polygonal tables, a famously alluring and difficult problem that her research was illuminating. Isn't it great what mathematicians spend time on?

In addition to being the first woman to win a Fields Medal, Mirzakhani was also the first Iranian, and in this sense she reminds us of the great, too-often glossed over, history of science and mathematics in the Islamic world. Her death has lent fuel to two important debates in Iran: one is whether children born to Iranian mothers should automatically qualify for Iranian citizenship. Currently, only children born to Iranian fathers qualify, which means Mirzakhani's own daughter doesn't. The other social debate she feeds into is that in Iran the press is barred from showing images of Iranian woman abroad unless they're wearing head scarves. Mirzakhani didn't wear scarves but the Iranian press have widely used her image as national PR. Now the question arises as to why shouldn't other Iranian women also be shown with bare heads, including Persian actresses at Hollywood red carpet affairs. It's interesting that it's a mathematician rather than an entertainer who's pioneered this path and is hopefully helping to open up channels for all Iranian women. And while we're on the subject of gender, lets pause to ponder the implications of the fact that its Persia that's produced the first female Field's Medalist, maths' highest honor.

Many tributes to Mirzakhani have been written and the American Mathematical Society has set up a page to honor her and collect them. Here's an insightful piece about her work in Quanta by Erica Klarreich, and another in Forbes by Paul Helpern, plus a lovely essay in Scientific American  by Evelyn Lamb, herself something of a math goddess. The New York Times, Psychology Today and the Tehran Times all ran pieces, testifying to Mirzakhani's near-mythical status. One of my favorites was the New Yorker essay by Siobhan Roberts, author of two biographies about math legends: geometer Donald Coxeter (who famously taught M.C. Escher about hyperbolic space, immortalized in his Circle Limit series of prints), and a terrific book about John Horton Conway, who invented the "game of life" based on cellular automata.

Which brings us to the point that there are an amazing number of women today writing superbly about mathematics and math-related sciences. In addition to Lamb and Roberts, we can also admire the work of Natalie Walchova, a writer for Quanta and Nautilus, among others. See her gorgeous recent pieces on solving the mystery of pentagonal tiling and on a physics theory of life. All around, women are proving that the beauties of mathematics can be enjoyed, shared and celebrated by everyone.

Speaking of brilliant women, I'd like to thank Caitlyn Parry for her awesome design of this website.