So you’re walking down the street carrying a hot slice of pizza on a paper plate. The tip of the slice is so heavy with cheese and sauce that it’s drooping toward the ground. How do you solve this problem?
Naturally, you fold the pizza lengthwise, New York–style. But why does warping the pizza by turning the crust ends toward each other keep the molten cheese at the pizza tip from falling on the street? The answer is geometry.
Pizza has a surface with zero curvature, which means a slice resists bending both vertically and horizontally at the same time (unlike the double curvature of, say, a Pringle chip). Your brain intuits this without any assistance from geometric theory. But the theory exists anyway. In 1827, Carl Friedrich Gauss proved the Theorema Egregium—roughly “Awesome Theorem”—which, extremely simplified, says that you can’t change an object’s curvature and keep its geometry intact. An orange peel has positive curvature, and you can’t flatten it without ripping or stretching the peel. Paper has zero curvature, and you can’t fold it into the shape of an orange. A piece of pizza is like a piece of paper: Fold it horizontally, and it will not droop vertically.
You’ve probably also confronted the Awesome Theorem when looking at a map. The famous Mercator Projection reduces our earthly sphere to a flat plane—or, if you roll it up, a cylinder—in which the parallel longitudinal lines are pulled apart near the poles. This has the effect of making Greenland look bigger than Africa. Why is it so hard to design a perfect map? Because of the Awesome Theorem. Just as a pizza slice, with zero curvature, cannot fold and droop at the same time, a planetary sphere, with positive curvature, cannot fully flatten without breaking its own geometric integrity.
Droopy cheese and the curve of the Earth, the everyday and the cosmic, are beautifully interwoven in the mathematician Jordan Ellenberg’s new book, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. In an interview last week, we talked about all of those topics—including lowercase awesome theorems, gerrymandering, bad COVID-19 projections, and the geometry of our mental frameworks. This conversation has been edited for clarity.
Derek Thompson: Your book is marvelous at pointing out the promiscuity of geometric thinking. It’s not just triangles and squares. Disease spread has a shape. Stock prices have a shape. While reading, I recognized that my frameworks for thinking also have shapes. Whenever somebody asks me a hard question, I default to breaking the answer in two: “on the one hand, on the other hand.” I am enforcing a kind of geometric symmetry in my thinking. Does that resonate at all with you?
Jordan Ellenberg: You’re right that bilateral metaphors are very common when we talk about thinking. We say, “An argument has two sides.” It doesn’t. But we reach for a geometric metaphor to make sense of it.
So here I go: Geometry itself has two faces. On the one hand, it seems like the most formal of subjects. When we teach it in school, it’s the subject where we do proofs, where we rigorously lay out steps of deductions. On the other hand, geometry is the most intuitive of the mathematical subjects. Because it’s about space, which is where we live. It’s about motion, which is what we do. The formal and intuitive sometimes pull against each other, and sometimes they reinforce each other.
Thompson: We’ll get to politics and pandemics in a second, but first, a very important, contentious geometry question that you raise in your book. How many holes does a straw have: zero, one, or two?
Ellenberg: My position is that a straw has two holes, but one is the negative of the other.
Thompson: Wait, it’s one hole! If I say, “Jordan, take this three-hole punch and punch three holes in this piece of paper” and you do it, how many holes does the paper have? Six? No. It has three—three holes that go all the way through. Some holes are through-holes. A straw has one long through-hole.
Ellenberg: Okay, but what if somebody pointed out a big hole in the ground and said “Look at that hole!” Are you one of these people who would say, “No, actually, that’s not a hole, because it doesn’t go through the entire planet”?
Thompson: If a comet smashes into the Earth, it creates a closed hole. Some holes are closed holes. If a comet ever somehow punched a hole through an entire planet—if it smashed through America and blew right through the mantle and the core and came out the other side in China—that would be one long hole, or a through-hole. Like a straw.
Ellenberg: So if all through-holes are just one hole, how many holes does a pair of pants have?
Thompson: (Stupefied silence.)
Ellenberg: So, in math, there are two things you can do when a question is confusing you. You can make the problem simpler, or you can make the problem harder. I like to make the problem harder.
Thompson: Speaking of hard, let’s talk about modeling a global pandemic, which resulted in a lot of terrible predictions recently. For example, last spring Kevin Hassett at the Trump White House Council of Economic Advisers notoriously proclaimed the imminent end of the pandemic with his so-called cubic-fit graph. How would a good geometer explain his and other forecasters’ mistakes?
Ellenberg: That graph was very wrong. But saying things are wrong is what Twitter is for, and you don’t need a book for that. I’m interested in how it was wrong, and why so many of the projections were wrong.
I like the metaphor of tennis here. If you model the flight of a tennis ball with physics, you will never predict that it will go back and forth across the court. Because tennis is physics plus people reacting to physics.
That’s what we see with any large-scale phenomenon, like COVID-19. Anybody who predicted that the disease would behave like physics was proved wrong. Instead you saw an increase in cases, and then scientists modeled the disease, and then people responded to the disease and to news reports of the model.
Thompson: And many times, in many parts of the world, the disease seemed to go away, but then people relaxed, and the virus surged again.
Ellenberg: Disease models are almost certainly going to be wrong. But does that mean you shouldn’t model diseases? No, you should try to model diseases! It’s like, if you’re a tennis player, you’re better off understanding or intuiting the physics of a ball hit with a certain slice and velocity. That tells you where the ball is going to go, which is highly relevant to a tennis player. But physics doesn’t give us all the information we need to explain what happens in a tennis match.
Thompson: You have a very long chapter on the most famous geometric catastrophe in American politics, which is partisan gerrymandering. A lot of gerrymandering critics spend a lot of time talking about all the most bizarre-looking gerrymanders. But you point out that the emphasis on weirdly shaped districts conceals the fact that, in many states, liberals are so concentrated in densely populated downtown areas that even normal-looking districts might strategically undercount them. We have to focus not only on what the geometry looks like but also on who lives in those political polygons.
Ellenberg: Look at Wisconsin. It has roughly a 50–50 distribution of Democrats and Republicans. It also has several urban areas like Madison and Milwaukee that are very Democratic. So if you just cut the state into rectangles, it’s very likely that Republicans would win a majority of districts every election, because urban Democrats would be so concentrated inside a few of those rectangles. All of our mathematical analyses say so.
I think in the end, districting is a political process, not a purely mathematical one. But it needs a political solution that is informed by mathematics. What mathematicians can do is use their analysis to eliminate the outliers, the worst and most grievous abuses of districting, by identifying which maps are most likely to yield an enduring, disproportionate, partisan advantage.
Thompson: Do you think geometers could at least design a map that is ideally representational?
Ellenberg: It’s very difficult for geometers to design a perfect map. I don’t even know how to define a perfect map. Plus, a lot of representation will never be perfectly proportional, and I’m not sure that’s a problem.
For example, about one-third of Massachusetts residents are Republicans. So are 30 percent of the state’s congressional representatives Republican? No. All nine of them are Democrats. That’s not necessarily an unfair outcome. It’s hard to find a large region of Massachusetts that’s majority Republican. Just as it’s hard to find a large region of, say, Wyoming that’s majority Democratic.
The solution to gerrymandering won’t look like a perfectly proportional government at every level. But since we’re going to carve the country into polygons, those shapes shouldn’t be drawn for the explicit purpose of creating a lasting majority for one party over another.
Thompson: You write about the golden ratio, which some people claim is a sort of skeleton key to physical beauty. Some say the golden ratio describes the proportions of the Great Pyramid of Giza, Stonehenge, the Mona Lisa, Beethoven’s Fifth Symphony, the shape of leaves, and the Pepsi logo. What makes the golden ratio so golden?
Ellenberg: I do think there is something profound about the number. I don’t think it necessarily explains the beauty of the Mona Lisa, or the Pepsi logo, or the perfect flower, or—as the Journal of Prosthetic Dentistry has claimed—the perfect false teeth.
But the golden ratio is a number that possesses a great deal of charm, at least in part because it appears again and again in geometry. If I draw a regular pentagon and then inscribe—perhaps because I am doing some weird satanic rite—a pentagram in it by drawing on the diagonals, the diagonals of the pentagram are longer than the sides of the pentagon, and that ratio is the golden ratio.
Or if you draw a rectangle whose length-to-width is the golden ratio, and then you lop off a square, then what’s left is a rectangle of the exact same dimensions. And the golden ratio is the only ratio for which this is true.
Here’s another favorite fact of mine. So an irrational number is a real number that you can’t write as a simple fraction. The golden ratio turns out to be the most irrational of all the irrational numbers. That is, it’s the hardest to approximate with a fraction. Pi is irrational, but a mathematician discovered that you can get very close to pi with a simple fraction: 355 divided by 113. That’s much harder to do for the golden ratio.
So I think there is something profound about the number. But it’s a little above my pay grade to state why math is often bolted onto mysticism.
Thompson: And it is! People seeking a theory of everything have often looked to geometry to provide proof that the world is the sum of mathematical equations. There’s the golden ratio in design. There’s the algebraist James Joseph Sylvester, who wrote an entire book, The Laws of Verse, to quantify the beauty of poetry. There are “Fibonacci retracement lines” in investing, whereby professional investors claim that an approximation of the Fibonacci sequence can predict turning points in the market. Geometry explains a lot about the world, but there is also a fascinating instinct to garnish mystical bullshit with geometry to make it look profound.
Ellenberg: My favorite example is the Torah codes. The entire idea with the Torah codes is that you can use mathematical analysis to find proof that God exists and that he wrote the Torah. I think it’s funny because, in the Torah, God makes it very clear that he wants people to know whenever he’s there. Oh, the bush is on fire? And it’s talking? Okay, God’s here.
But I approach the project of bolting math onto mysticism with a bit of sympathy. Thinking about things geometrically is useful. It’s been tremendously useful in studying natural language for machine translation, and for understanding the nature of social networks. So my feeling is: Yeah, try it! Write your book on geometric poetry! Try to make it work.