The beautiful math of coral - Margaret Wertheim

And here today is June, said to talk about a project that my twin sister and I have been doing for the past three and a half years. We're crocheting a coral reef, and it's a project that we've actually been now joined by hundreds of people around the world who are doing it with us. Indeed, thousands of people have actually been involved in this project in many of its different aspects. It's a project that now reaches across three continents, and its roots go into the fields of mathematics, marine biology, feminine handicraft, and environmental activism.

It's true! It's also a project that, in a very beautiful way, the development of this has actually paralleled the evolution of life on Earth, which is a particularly lovely thing to be saying right here in February 2009. One of our previous speakers told us it is the 200th anniversary of the birth of Charles Darwin. All of this I'm going to get to in the next 18 minutes, I hope. But let me first begin by showing you some pictures of what this thing looks like, just to give you an idea of scale. That installation there is about six feet across, and the tallest models are about two or three feet high.

This is some more images of it; that one on the right is about five feet high. The work involves hundreds of different crochet models, and indeed there are now thousands and thousands of models that people have contributed all over the world as part of this. The totality of this project involves tens of thousands of hours of human labor, 99% of it done by women. On the right-hand side, that bit there is part of an installation that is about 12 feet long.

My sister and I started this project in 2005 because in that year, at least in the science press, there was a lot of talk about global warming and the effect that global warming was having on coral reefs. Corals are very delicate organisms, and they're devastated by any rise in sea temperatures. It causes these vast bleaching events that are the first signs that corals are being sick, and if the bleaching doesn't go away, if the temperatures don't go down, reef stress begins. There's a great deal of this that's been happening in the Great Barrier Reef, particularly in coral reefs all over the world.

This is our invocation in crochet of a bleached reef. We have an organization together called the Institute for Figuring, which is a little organization we started to promote to do projects about the aesthetic and poetic dimensions of science and mathematics. I went and put a little announcement up on our site asking for people to join us in this enterprise, and to our surprise, one of the first people who called was the Andy Warhol Museum. They said they were having an exhibition about artists' response to global warming, and they'd like our coral reef to be part of it.

I laughed and said, "Well, we've only just started it; you can have a little bit of it." So in 2007, we had a small exhibition of this crochet reef, and then some people in Chicago came along. They said in late 2007, the theme of the Chicago Humanities Festival is global warming, and we've got this 3,000 square-foot gallery, and we want you to fill it with your reef.

I naively by this day said, "Oh yes, sure." Now, I say naively because actually my profession is as a science writer. What I do is I write books about the cultural history of physics. I've written books about the history of space, the history of physics and religion, and I write articles for people like the New York Times and the LA Times. So I had no idea what it meant to fill a 3,000 square-foot gallery. So I said yes to this proposition, and I went home and I told my sister Christine, and she nearly had a fit because Christine is a professor at one of LA's major art colleges, CalArts, and she knew exactly what it meant to fill a 3,000 square-foot gallery.

She thought I'd gone off my head, but she went into crochet overdrive, and to cut a long story short, eight months later, we did fill the Chicago Cultural Center, the 3,000 square-foot gallery. By this stage, the project had taken on a viral dimension of its own that got completely beyond us. The people in Chicago decided that as well as exhibiting our reefs, what they wanted to do was have the local people there make a reef.

So we went and taught the techniques; we did workshops and lectures, and the people in Chicago made a reef of their own, which was exhibited alongside ours. There were hundreds of people involved in that, and we got invited to do the whole thing in New York and in London and Los Angeles. In each of these cities, the local citizens—hundreds and hundreds of them—have made a reef, and more and more people get involved with this, most of whom we've never met.

So the whole thing has morphed into this organic ever-evolving creature that's actually gone way beyond Christine and me. Now, some of you are sitting here thinking, "What planet are these people on? Why on earth are you crocheting a reef?" Woollen nests and wetness aren't exactly two concepts that go together. Why not chisel a coral reef out of marble or cast it in bronze? But it turns out there's a very good reason why we are crocheting it.

Because many organisms in coral reefs have a very particular kind of structure—those freely crenelated forms that you see in corals and kelps and sponges and nudibranchs—are a form of geometry known as hyperbolic geometry. The only way that mathematicians know how to model this structure is with crochet. It happens to be a fact; it's almost impossible to model this structure any other way, and it's almost impossible to do it on computers.

So what is this hyperbolic geometry that corals and sea slugs embody? Over the next few minutes, we're all going to get raised up to the level of a sea slug. This sort of geometry revolutionized mathematics when it was first discovered in the nineteenth century, but not until 1997 did mathematicians actually understand how they could model it.

In 1997, a mathematician at Cornell, Dana Tamina, made the discovery that this structure could actually be done in knitting and crochet. The first one she did was in knitting, but you get too many stitches on the needle, so she quickly realized that crochet was the better thing. What she was doing was actually making a model of a mathematical structure that many mathematicians had thought was actually impossible to model. Indeed, they thought that anything like this structure was impossible per se. Some of the best mathematicians spent hundreds of years trying to prove that this structure was impossible.

So what is this impossible hyperbolic structure? Before hyperbolic geometry, mathematicians knew about two kinds of space: Euclidean space and spherical space, and they have different properties. Mathematicians like to characterize things by being formulas, so you all have a sense of what a flat space is—Euclidean spaces—but mathematicians formalize it in a particular way, and they do it through the concept of parallel lines.

So here we have a line and a point outside the line, and Euclid said, "How can I define parallel lines?" I asked the question, "How many lines can I draw through the point but never meet the original line?" You all know the answer to that; someone want to shout it out? One! Great! Okay, that's our definition of a parallel line; it's a definition really of Euclidean space.

But there's another possibility that you all know of—spherical space. Think of the surface of a sphere, just like a beach ball, the surface of the Earth. I have a straight line on my spherical surface, and I have a point outside the line. How many straight lines can I draw through the point that never meet the original line? What do we mean to talk about a straight line on a curved surface?

Now mathematicians have answered that question, and they've understood there's a generalized concept of straightness; it's called a geodesic. On the surface of a sphere, a straight line is the biggest possible circle you can draw, so it's like the equator or the lines of longitude. So we asked the question again: How many straight lines can I draw through the point that never meet the original line?

Does anyone want to guess? Zero! Very good! Now mathematicians thought that was the only alternative. It's a bit suspicious, isn't it? There are two answers to the questions so far—zero and one. Two answers; there may possibly be a third alternative. To a mathematician, if there are two answers and the first two are zero and one, there’s another number that immediately suggests itself as the third alternative. Does anyone want to guess what it is? Infinity! You all got it right!

Exactly! There is a third alternative. This is what it looks like: it is a straight line, and there's an infinite number of lines that go through the point and never meet the original line. This drawing nearly drove mathematicians bonkers because, like you, they're sitting there feeling bamboozled, thinking, "How can that be? You're cheating; the lines are curved." But that's only because I'm projecting it onto a flat surface.

For several hundred years, mathematicians had to really struggle with this; how could they see what it meant to actually have a physical model that looked like this? It's a bit like this: imagine that we'd only ever encountered Euclidean space, and then our mathematicians come along and say, "This is a thing called a sphere, and the lines come together at the North and South Pole." But you don't know what a sphere looks like.

Then someone comes along and says, "Look, here's a ball!" And you reply, "I can see it, I can feel it, I can touch it, I can play with it!" That's exactly what happened when Dana came in 1997 and showed that you could crochet models in hyperbolic space. Here is this diagram in crochet, and I've stitched Euclid's parallel postulate onto the surface, and the lines looked curved.

But look! I can prove to you that they're straight because I can take any one of these lines, and I can fold along it, and it's a straight line. So here, in wool, through a domestic feminine art, is the proof that the most famous postulate in mathematics is wrong. You can stitch all sorts of mathematical theorems onto these surfaces.

The discovery of hyperbolic space assured in the field of mathematics that’s called non-Euclidean geometry, and this is actually the field of mathematics that underlies general relativity and is actually ultimately going to show us about the shape of the universe. So there is this direct line between feminine handicraft, Euclid, and general relativity.

Now, I said that mathematicians thought this was impossible. Here are two creatures who never heard of Euclid's parallel postulate, didn't know it was impossible to violate, and they're simply getting on with it. They've been doing it for hundreds of millions of years. I once asked mathematicians why it was that they thought this structure was impossible when sea slugs have been doing it since the Silurian age, and their answer was interesting. They said, "Well, I guess there aren't that many mathematicians sitting around looking at sea slugs."

That's true, but it also goes deeper than that. It also says a whole lot of things about what mathematicians thought mathematics was, what they thought it couldn't and couldn't do, and what they thought it couldn't represent. Even mathematicians who, in some sense, are the freest of all thinkers literally couldn't see not only the sea slugs around them but the lettuce on their plate because lattices and all those clearly vegetables—they also are embodiments of hyperbolic geometry.

So in some sense, they literally had such a symbolic view of mathematics, they couldn't actually see what was going on on the lettuce in front of them. It turns out that the natural world is full of hyperbolic wonders, and so too we've discovered that there is an infinite taxonomy of crochet hyperbolic creatures.

We started out, Christine and I and our contributors, doing the simple mathematically perfect models, but we found that when we deviated from the specific set, the mathematical code that underlies it—the simple algorithm: crochet three, increase one—when we deviated from that and made embellishments to the code, the models immediately started to look more natural.

All of our contributors, who are an amazing collection of people around the world, do their own embellishments and, as it were, we have this ever-evolving crochet taxonomic tree of life. Just as the morphology and the complexity of life on Earth is never-ending, little embellishments and complexifications in the DNA code lead to new things, like giraffes or orchids.

So too, little embellishments in the crochet code lead to new and wondrous creatures in the evolutionary tree of crochet life. This project really has taken on this inner organic life of its own—that's the totality of all the people who've come to it and their individual visions and their engagement with this mathematical mode.

We have these technologies; we use them. But why? What's at stake here? What does it matter for Christine and me? One of the things that's important here is that these things suggest the importance and value of embodied knowledge. We live in a society that completely tends to valorize symbolic forms of representation—algebraic representations, equations, codes.

We live in a society that's obsessed with presenting information in this way, teaching information in this way. But through this sort of modality—crocheting and other plastic forms of play—people can be engaged with the most abstract, high-powered theoretical ideas, the kind of ideas that normally you have to go to university departments to study in higher mathematics, which is where I first learned about hyperbolic space.

But you can do it through playing with material objects. One of the ways that we've come to think about this is that, with the Institute for Figuring and projects like this, we're trying to have kindergarten for grown-ups. Kindergarten was actually a very formalized system of education established by a man named Friedrich Froebel, who was a crystallographer in the 19th century.

He believed that the crystal was the model for all kinds of representation, and he developed a radical alternative system of engaging the smallest children with the most abstract ideas through physical forms of play. He is worthy of an entire talk on his own right. The value of education is something that Froebel championed through plastic modes of play.

We live in a society now where we have lots of think tanks where great minds go to think about the world, and they write these great symbolic treatises called books, papers, and op-ed articles. We want to propose, Christine and I, through the Institute of Figuring, another alternative way of doing things, which is the play tank.

The play tank, like the think tank, is a place where people can go and engage with great ideas, but what we want to propose is that the highest levels of abstraction—things like mathematics, computing, logic, etc.—all of this can be engaged with not just through purely cerebral algebraic symbolic methods but by literally physically playing with ideas. Thank you very much!