What do coral reefs know of complex geometry? According to Margaret Wertheim, a science writer, artist, and cofounder of the Institute For Figuring—“an organization dedicated to the poetic and aesthetic dimensions of science, mathematics and engineering”—more than we might think.

As part of their work at IFF, Wertheim and her twin sister, Christine Wertheim, also an artist and a faculty member at the School of Critical Studies at the California Institute of the Arts, developed the Crochet Coral Reef, a project that invites participants to crochet hyperbolic coral forms that are massed into vast reef-like installations. These crochet reefs have been exhibited at art galleries around the world, including the Andy Warhol Museum and the Hayward Gallery in London, and at the Smithsonian’s National Museum of Natural History in Washington, DC. With 8,000 active participants in a dozen countries and over 2 million exhibition visitors, the Crochet Coral Reef is now one of the largest art and science endeavors on the planet.

Through their open invitation to the public to contribute to the project by crocheting corals, the Wertheims lead participants into an exploration of hyperbolic geometry, an alternative to the Euclidian geometry commonly learned in school. Hyperbolic geometric surfaces occur in the forms of many oceanic species, such as coral, kelp, and nudibranchs. Unlike in the Euclidean system, hyperbolic geometric space is not flat but “negatively curved,” creating a completely different set of geometric conditions.

“It turns out that hyperbolic structures are very common in nature, and the place where lots of people encounter them is coral reefs,” Wertheim told me. “Sea slugs, and a lot of other organisms with frilly forms, are biological manifestations of hyperbolic geometry, which is also found in the structure of lettuce leaves and kales, and some species of cactus.”

The crochet reefs are assembled by large groups of mostly female participants, who together construct spectacular, soft, brightly colored organic structures that spill out in undulating seascapes. The Institute For Figuring’s recently published book, Crochet Coral Reef: A Project by the Institute For Figuring (2015), documents this unique body of collaborative female labor, and beautifully articulates hundreds of thousands of hours of collective exploration of artistic and mathematical speculation.

Yet despite the project’s extensive exhibition, thousands of participants, and positive publicity, the Wertheims have struggled to garner support from traditional science funders. During our interview Wertheim recalled the response of one senior program officer, who declared: “I’ll find it hard to convince my board that there is any real science in a bunch of women knitting.”

Clearly, we still have a long way to go in addressing gender imbalances in science and mathematics. Despite the ongoing public discussion on who gets to be invited into these worlds, women accounted for only 40 percent of graduates from bachelors’ programs in the physical sciences, and just under 20 percent of graduates in engineering and computer science, in the United States in 2013. As Wertheim points out, we must continue to ask why women are still underrepresented in these fields, and how alternative ways of presenting science to the public might help to change this.

Wertheim, who was born in Australia and moved to the US in 1991, has authored several books on the cultural history of science, including the influential The Pearly Gates of Cyberspace (1999), which traces the history of Western concepts of space from Dante to the Internet. Her earlier book, Pythagoras’s Trousers: God, Physics, and the Gender War (1995), presented a history of the relationship between physics and religion, and also examined ways in which women were systematically excluded from this domain.

Wertheim’s many and varied pursuits are a rare combination of philosophical and playful, working to tease out our increasingly complex relationship with science and its public representation. I spoke with her by phone while she was at her home in Los Angeles, where she was undoubtedly surrounded by crochet experiments and complex geometric forms rendered in colorful cardboard.

Tega Brain for Guernica

Guernica: You trained as a physicist. What precipitated your love of science?

Margaret Wertheim: As a child I attended a small local school with very few resources on the outskirts of Brisbane, Australia. My school was trialing a new math program, and in grade six I had a very special teacher named Mr. Marshall who allowed me to discover the beauty of math for myself. A seminal experience was a lesson he gave us on circles, and the magical number, pi, that could be found at the heart of all circles. He didn’t tell us what pi was; he led us through a process by which we were enabled to find it for ourselves.

Through experiences like that, I fell in love with math, and as a child it was clear to me that in some sense math was in the world around us. I became fascinated by what this means. When you look at the shape of the sun and the moon, they’re circles, so every time you see a circular thing, there’s pi embedded in it. How does this come to be, and what does it mean?

Later, I went to university to study physics, and originally thought I’d become a research physicist. But after six years of university science education, which in the late ’70s and early ’80s was very siloed off from other domains of human activity, I felt it was too isolated. So I decided to become a science writer instead.

Guernica: How did you get started with writing?

Margaret Wertheim: I started writing in 1982, when there were almost no science writers in Australia—I only knew of two others. I wanted to improve public communication about science, and, in particular, improve science communication to women. I realized that women read women’s magazines in vast numbers, so I convinced a then-new Australian women’s magazine called Follow Me to let me write a regular column about science and technology. Later, I did the same for Vogue Australia and Elle Australia.

For ten years, I wrote regular columns about science for women’s magazines, and to my knowledge I’m the only person in the world who can say that. This has no kudos in either the science-writing world or the academic world, but it’s one of the most challenging things I’ve ever done. It’s much harder to write about cosmology for a magazine like Vogue than for the New York Times, which I’ve also written for, because you have to imagine that on the page opposite there’ll be an advertisement for eyeliner, or an article about the latest trends in skirt length. You have to keep readers engaged in the context of all this beautiful fashion. At the same time you can’t assume your reader has any of the jargon to draw on. In my experience, it’s harder to write for non-canonical science forums than for canonical ones.

Guernica: What kinds of things did you write about for those magazines?

Margaret Wertheim: One piece was on the discovery that many galaxies are moving away from each other in ways that aren’t just predicted by the Big Bang. These movements suggested that dark matter existed. Another piece was on the problem of how we can communicate to generations to come about the presence of nuclear waste sites. How can we warn people away from these dangerous places 100,000 years from now? This is a big problem because no language lasts that long. It’s a major technological question and also an ethical question. I covered a very wide spectrum of subjects, from cosmology to biotechnology to computation.

There’s something funny about writing an article on dark matter for a context in which dark eye shadow might be a more salient issue.

Guernica: And what would you do in these pieces to keep your audience engaged?

Margaret Wertheim: One has to approach this with a sense of humor. There’s something funny about writing an article on dark matter for a context in which dark eye shadow might be a more salient issue. But you also go about it with a belief that your audience are intelligent people who are deeply interested in the world. I would always look for angles where there was playfulness. I’d certainly try to include women scientists, but that can’t be a necessity. A big challenge was to try and make parallels between scientific ideas and everyday phenomena that the audience might have experienced. Sometimes I’d appeal to notions of beauty. Vogue is a magazine devoted to certain conceptions of beauty, and many scientific ideas are drop-dead beautiful.

Guernica: Is there more science writing aimed at women now than when you started?

Margaret Wertheim: I don’t know of any science writing going on in women’s magazines, unless you count medical stories about things like breast cancer. I still think there’s a huge problem about how we can actively engage a wider range of women. There are now some wonderful women doing things online, such as the young mathematician Vi Hart, and the writer Jennifer Ouellette, who does the Cocktail Party Physics blog, but in terms of targeting female audiences, we’re nowhere near where I’d like us to be. I think it’s critical to add here that, of course, there are women, like me, who love science magazines. I’m not saying women must be a separate audience—I’m just responding to the reality that the majority of people who do read science magazines are male. That’s not a value judgment; it’s a statistical fact.

Guernica: I teach in university-level computer science programs, and this question of gender and diversity remains such an important one. We seem to get such a large number of male applicants no matter how we frame the calls for opportunities and entry to programs. It is almost as if, at the tertiary level, it is too late; we are going against a lifetime of science being talked about in a way which is evidently more appealing to a male demographic.

Margaret Wertheim: Absolutely. After my physics degree I wasn’t ready to leave science, so I did another degree majoring in mathematics and computer science. This was the 1980s and computer science was a new academic field that we all assumed would soon be equally practiced by women and men. That was thirty years ago and the percentage of women doing computer science degrees has gone down, not up. It’s worth asking why. I’ve been in the field of science communication for three decades, and although there are definitely more women science writers now, I would still characterize the general spectrum of science communication as overwhelmingly masculine.

Guernica: When did art come into your work?

Margaret Wertheim: I never imagined myself operating in the art world, and yet today that’s where I spend most of my time. It happened because I became very disenchanted with the public representation of science. There are vast numbers of people out there who want to engage with science. They’d come up to me at parties and say things like, “I bought A Brief History of Time, and I couldn’t get past chapter one. Could you recommend a book about physics that I can actually understand?” And I couldn’t. It became like a ghastly joke. I realized that the normative strands of science communication were not reaching broad sections of our population. In 2002, I did the formal research and gathered statistics on who reads science magazines, and they were pretty stark. In a nutshell, it was white men over forty years old, from high-earning economic brackets. I thought we needed to reach a wider spectrum than that, and in order to do so, I believed I’d have to invent a new kind of forum for doing innovative science engagement. That’s when I started the Institute For Figuring. Its acronym is IFF, which is the logical symbol for “if and only if.” I never imagined I’d be exhibiting at the Hayward Gallery or the Andy Warhol Museum. It taught me a fantastic lesson: you never know how life will play out.

Guernica: It’s a wonderful name for an organization. Could you describe the work of the Institute to date?

Margaret Wertheim: The mission of the IFF is to engage people with science and mathematics by looking at the poetic and aesthetic dimensions of these fields. In the classical sense, the IFF wasn’t conceived as an art and science project where one decides on a scientific subject, say nanotechnology or cosmology, and then recruits artists to engage with it. The IFF’s mission is rather different. I ask: How can we use techniques from both science and art to produce objects that have scientific or mathematical content but are also aesthetically wondrous? It has become an “art and science” project by virtue of the fact that most of the places we work in are art galleries. But the whole thing was conceived as a science outreach project.

One thing that’s critical here is that I do this with my twin sister Christine, who comes from the art world. When I was studying physics she was at art school, and now teaches at CalArts. She used to be a painter and is now an experimental writer. Through me she’s learned about science, and through her I’ve learned about art. Her understanding of the art world, and of art practices, has been pivotal to the IFF’s success.

What we want to do is to physicalize and visualize scientific and mathematical ideas—to literally bring abstract ideas into material reality so that they are concretized for people. Chrissy and I believe that seeing and playing with physical objects can enable access to symbolic ideas. When I studied physics and math at university it was all done through equations and textbooks—that’s the classical way science education operates, and for good reason—whereas artists like her go to art school and start making stuff; they fling paint at the walls, they dance and bash things together with giant bits of metal. Our society has come to think of science as being a very abstractified thing, and art as being a materialized thing. Part of Chrissy’s and my interest is to break down this divide and show how abstract ideas—things like hyperbolic surfaces and fractals—have physical manifestations in the world. In that sense, the IFF’s work is about trying to bridge the divide of what is perceived to be “science” and what is perceived to be “art.” We want to bring together symbolic and material methodologies of production.

Guernica: The IFF’s best-known project is the Crochet Coral Reef, a large-scale, participatory work that explores hyperbolic geometry through handicraft. Can you tell me a bit about the science that underpins the project?

Margaret Wertheim: Mathematicians spent a long time trying to prove that hyperbolic geometry was impossible. Then, in the early nineteenth century, they were finally forced to realize that there is this alternative geometry. Logically speaking, it was possible—yet they didn’t have a way of physically modeling hyperbolic surfaces. At that point they hadn’t understood that these structures were all around them in nature. In 1997, a mathematician at Cornell University, Dr. Daina Taimina, showed how she could model hyperbolic surfaces using crochet. You can crochet these forms and stitch theorems onto the woolen models to demonstrate fundamental points about this geometrical structure; for instance, basic things like triangles and parallel lines have different ways of behaving in hyperbolic space. Taimina and her geometer husband, Dr. David Henderson, used these crochet models to teach their students at Cornell, and I learned of their work around the same time we started the Institute For Figuring. I thought this combination of mathematics and handicraft would be a wonderful way to make an abstract branch of mathematics real and tangible. Historically speaking, hyperbolic geometry was a tremendously important development in mathematics, because in the mid-ninteenth century, it opened up the doors to the whole field of what’s called non-Euclidian geometry, which turns out to be the mathematics underlying general relativity.

Nature rarely does anything mathematically perfect.

Guernica: And how did the Crochet Coral Reef project actually get started?

Margaret Wertheim: Crocheting with Dr. Taimina’s formula, you get surfaces that are mathematically pure, but after a while they all look pretty similar. Christine started saying, “I’m not going to stick to the formula, I’m going to branch out and do wonky ones that aren’t precise.” As soon as she started deviating from mathematical perfection, the models began to look like living organisms, particularly like corals, and that’s because they are doing what real corals are doing. Nature rarely does anything mathematically perfect, and naturalism—what Donna Haraway describes as “liveliness”—results from imperfections and deviations.

At the time we started the project, in 2005, Chrissy joked that if the Great Barrier Reef ever died out, our Crochet Coral Reef would be something to remember it by. That was meant in jest, but a decade later, coral scientists are saying that reefs might actually die out in this century if we don’t curb greenhouse gas emissions.

Everyone is conversant with global warming now, but ten years ago, it was still a highly contentious issue and only just coming into public consciousness. Reefs were one of the first indications that global warming is not some problem for the far future, it’s happening here and now. We intended the Crochet Coral Reef project to engage people with mathematics, and also with one of the biggest environmental issues of our time. Of course we wanted to make something beautiful, so it was an artistic project too, but the primary aim was to communicate science in new ways. We thought the science world would love it and the art world would ignore it, and the opposite happened. The art world has embraced it, yet the science world has virtually ignored it.

Guernica: Why do you think that is?

Margaret Wertheim: I can only speculate about this. My supposition is that gender has a lot to do with it. Ninety-nine percent of the people who participate in the project are women. We’ve done it all over the world: all across the USA, across Europe and Australia, in Abu Dhabi, Latvia, Germany, and many other countries. Our exhibitions have been seen by millions of people, and I imagine most of them have been women, too. Yet no science foundation we have approached has supported the project. I think that’s a reflection of the masculine face of science and the fact that when women do things, it’s too often invisible-ized. Each time I have approached science foundations, I’ve been told not to bother putting in an application.

Guernica: I am so surprised to hear this.

Margaret Wertheim: I haven’t been on public record saying this, but I think now it’s important to discuss. We had an exhibition at the Smithsonian and another at the Science Gallery, in Dublin, but although they are science venues, all of the IFF’s funding has come from the art world.

Guernica: You often talk about the practice of play, which is perhaps more commonly associated with art than with science. Do you think the lack of complex technology used in the production of these handmade coral structures relates to this idea of play as being crucial?

Margaret Wertheim: I do. The technology involved in crocheting a coral reef is a piece of bent metal, a crochet hook. There’s nothing that plugs in. There are no circuits. And this is another issue: there is too much of a conception these days that science communication has to be computationally mediated. And given that our field here is mathematics, that’s ironic. Mathematicians are proud of the fact that, generally, they do their work with a piece of chalk and a blackboard. They value hand-done proofs above all else. A big question in mathematics today is whether or not computational proofs are legitimate. Some mathematicians won’t accept computational proofs and insist that a real proof must be done by the human hand and mind, using equations. The reef project is in line with this attitude; it is valorizing what humans can do. That aspect of the project is very close to my heart, and it can be seen, I believe, as a worldwide experiment in collective mathematics.

Guernica: What do you mean by that?

Margaret Wertheim: To crochet a mathematically pure hyperbolic surface, you use Dr. Taimina’s very simple algorithm: crochet n stitches, then increase one. So, you might crochet three stitches increase one, crochet three stitches increase one, and keep repeating. Or you could do it once every four stitches, or every five stitches. As long as you increase at a regular rate, you get a mathematically pure hyperbolic structure—you’re creating what’s called a constant curvature surface.

One way to think about a pure hyperbolic surface is that it’s the geometric opposite of a sphere. If you look at a sphere, the curvature is the same everywhere, as opposed to, say, an egg, which clearly does not curve the same everywhere. This is what makes spheres geometrically important. Mathematically speaking, a sphere has positive curvature and a hyperbolic surface has negative curvature, but both have constant curvature everywhere. Now, just as there are lots of things like eggs and sea urchins that are wonky spheroids, so are there lots of wonky hyperbolic surfaces that are not geometrically perfect. That’s what things like corals are doing. In the Crochet Coral Reef project, everybody starts off making the mathematically pure surfaces and then they begin to branch out and ask, What happens if I zoom out a bit here, or if I get more frilly over there? So it’s become a kind of experiment, a form of applied mathematics, and no one knows what’s going to happen in advance.

Nature is an open-ended experiment based on morphing a DNA code, and ours is an open-ended experiment based on morphing a crochet code.

What Christine and I didn’t realize initially is that we’d created a framework in which people can take a basic mathematical algorithm and start playing and making it more complex. And that’s what also happened with the history of life on earth. Simple DNA gradually morphed and evolved, so that you had the coming into being of ever more complex and diverse creatures, until one day you wake up and find there are peacocks and giraffes. Nature is an open-ended experiment based on morphing a DNA code, and ours is an open-ended experiment based on morphing a crochet code. Of course, the crochet code is a lot simpler than the DNA code, but it’s a similar idea in principle. It’s also interesting that what we’re doing has parallels with the computational field of “artificial life.”

Many times, early on in the project, we reached a point where we thought we’d seen everything, and then some contributor like Helen Bernasconi, who lives on a sheep farm in Bonnie Doon, Australia, would send us a box of astounding things she’d made by playing with the code. It would be as if someone had invented an entire new branch on the crochet tree of life. We now realize that just as evolution of life on Earth is a never-ending tree of possibility, so is there no end to the novelties on the crochet tree of life. The people doing our project are not just passively imbibing knowledge, they are actively engaging in a process of experimentation and discovery. And that’s the very essence of science. I believe that’s one reason the project has been so successful. It is an active, dynamic, intellectual, as well as material, experience. I think this has enormous, rich potential for the whole field of science communication.

Guernica: Aside from your work with the Institute For Figuring, we’ve discussed some of your career as a science writer. In your 2011 book Physics on the Fringe: Smoke Rings, Circlons, and Alternative Theories of Everything, you explore what you call “outsider physics”; you examine people who are mostly untrained in science who develop alternative theories of the universe. You observe dominant, codified science, then profile these strange, peripheral practices. How does this book reflect your interest in these eddies that happen at the edges?

Margaret Wertheim: I have an abiding interest in how ordinary people produce knowledge, and what it means for individuals to know the world. I thought I’d be a theoretical physicist because I love physicists’ views of the world—I find general relativity and quantum theory thrilling—but I have always felt uneasy with the idea of an Ultimate Truth. One of the functions of science is to help us instrumentally; it helps us to build things like microchips and GPS satellites. But another function of science in the modern world is to help us feel “at home in the universe”—that phrase was famously put forward by the complexity theorist Stuart Kauffman in a book he wrote in 1995.

The question at the heart of Physics on the Fringe is: Who gets to feel at home in the universe articulated by contemporary physics? General relativity, quantum theory, and string theory are inaccessible to 99 percent of our population, and that’s because they are difficult to understand. The people I write about in my book make an interesting claim: We’re intelligent, thoughtful people, they say; we love science and the way it works. We read Stephen Hawking and Brian Greene and we can’t understand what they say. We believe nature speaks a language that a thoughtful, intelligent person who doesn’t necessarily have a PhD in math ought to be able to understand.

Is that claim valid? It may or may not be, but it is one worth exploring sociologically. What is important here, I think, is the insistence that science is not just for an elite few. I don’t have a resolution to this issue, but if we care about science we need to think about this because science’s vision of the universe is becoming increasingly inaccessible. That doesn’t mean it’s wrong. On some levels, general relativity is right, because we have to take its calculations into account to produce accurate GPS systems. Instrumentally, it’s correct to something like twenty decimal places. But epistemologically and psychologically, it is not available to most of our population. This is what I call the “cosmological problem” of science. Science has the instrumental function that has given us computers and so on, but its cosmological function is to give us a picture of the world we inhabit as human beings, and on that level it’s failing a vast number of people.

Guernica: It is really interesting how almost all of the outsider physicists you profile are male, and they’re standing on that podium claiming they have “the answer,” some universal theory that explains everything. Is there a relationship between that sort of impulse and gender?

Margaret Wertheim: I thought a lot about this issue while I was writing the book, which I was working on at the same time that Christine and I were originally developing the Crochet Coral Reef project. I sometimes felt as if I was living two parallel lives. One was a life giving voice to outsider women in art, because most of our Reef women are not professional artists, yet we have gotten them exhibited in places like the Hayward, one of the world’s foremost contemporary art galleries. At the same time, I was trying to give voice to outsider men in science. I didn’t originally see it in a gendered way, but eventually I realized that in the outsider science world, most of the participants were men, while in the craft-art realm of the Reef, it was almost all women. I’m still not quite sure what to do with these facts.

Guernica: But they’re vastly different ways of participating. In your Coral Reef project, there’s not a lot of individual authorship going on—yet for the outsider physicists, their work is strongly tied to their individual identity or public profile.

Margaret Wertheim: Actually, in the Reef project there’s a lot of individual authorship going on, and we make a huge point of giving credit to every one of them. In our Crochet Coral Reef book, there’s a whole chapter that lists the names of all 7,000-plus contributors worldwide, and some of them have done works that are absolutely unique. What’s different about Reef contributors and outsider physicists is that the Crochet Coral Reef, overall, is a collective project. Its very strength lies in the fact that it’s being done by thousands of people. Outsider physicists don’t want to share the limelight with anyone else because each of them is sui generis. There is no collectivity to their overall effort, and thus little ultimate effect.

When you’re in a room with twenty people who’ve all got the True Theory of the universe, it’s difficult to know what to do.

Guernica: I often wonder what we are losing, or what goes unrealized, because of so much focus on individual achievement. It doesn’t allow for these other ways of working, such as working collectively.

Margaret Wertheim: In the Reef project, thousands of women all over the world participate in making installations together, and in that sense they parallel what living reefs are doing, because real reefs are built by millions, or billions, of coral polyps working together. The Crochet Coral Reef project offers a kind of feminist metaphor for how we might approach the problems of global warming through collective action. Rather than relying on a few individual geniuses inventing some technological solution, let’s try and think about this together. That sense of the inherent value of thinking and acting collectively is one of the lessons Christine and I learned from our mother, Barbara Wertheim, who was a pioneering Australian feminist in the 1970s and ’80s.

Outsider physicists are in the opposite boat: they are all reinventing the world for themselves. When you’re in a room with twenty people who’ve all got the True Theory of the universe, it’s difficult to know what to do. And the paradox is that so much important science is now being done by huge teams. With Higgs boson, for example, it’s estimated there were around ten thousand scientists and engineers involved in building the machines that made this discovery possible. Two of them were granted the Nobel Prize, but it took thousands of people to produce this knowledge. Similarly, if you look at the search for gravity waves, and work in genetics, these are vast collective endeavors. Almost all major scientific projects today are huge collaborations, yet we still have this public obsession with the idea of the individual scientific genius. One of my goals as a science communicator is to celebrate the collaborative dimensions of science, which I think will be critical for facing the ecological and resource challenges ahead. In a sense, we are all corals now.


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