Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and EverythingElse

by Jordan Ellenberg

Gulliver was fully on board, replying, “No man can talk well unless he is able first of all to define to himself what he is talking about.

The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It’s that the world is full of non-proofs, and grown-ups need to know the difference. It’s hard to settle for a non-proof once you’ve really familiarized yourself with the genuine article.

We encounter non-proofs in proofy clothing all the time, and unless we’ve made ourselves especially attentive, they often get by our defenses. There are tells you can look for. In math, when an author starts a sentence with “Clearly,” what they are really saying is “This seems clear to me and I probably should have checked it, but I got a little confused, so I settled for just asserting that it was clear.” The newspaper pundit’s analogue is the sentence starting “Surely, we can all agree.” Whenever you see this, you should at all costs not be sure that all agree on what follows. You are being asked to treat something as an axiom, and if there’s one thing we can learn from the history of geometry, it’s that you shouldn’t admit a new axiom into your book until it really proves its worth. Always be skeptical when someone tells you they’re “just being logical.” If they are talking about an economic policy or a culture figure whose behavior they deplore or a relationship concession they want you to make, and not a congruence of triangles, they are not “just being logical,” because they’re operating in a context where logical deduction—if it applies at all—can’t be untangled from everything else. They want you to mistake an assertively expressed chain of opinions as the proof of a theorem. But once you’ve experienced the sharp click of an honest-to-goodness proof, you’ll never fall for this again. Tell your “logical” opponent to go square a circle.

What Lincoln had taken from Euclid (or what, already existing in Lincoln, harmonized with what he found in Euclid) was integrity, the principle that one does not say a thing unless one has justified, fair and square, that one has the right to say it. Geometry is a form of honesty. They might have called him Geometrical Abe.

Indeed, in your schoolbook, an “isosceles trapezoid” isn’t one with two equal sides, or with two equal angles; it is one that can be flipped without changing it. The post-Euclidean notion of symmetry has crept in, and it’s there because our minds are built to find it. More and more geometry classes are placing the idea of symmetry at the center, and building structures of proof starting from there. It’s not Euclid, but it’s where geometry is now.

Now we’re ready to correct the lie we told earlier. It is not quite right to say the hole in the top of a straw (a straw-shaped straw, that is) is the same hole as the one at the bottom. But it’s not really a brand-new hole, either. The hole at the top is the negative of the hole in the bottom. What flows into one must flow out the other.

Poincaré: “Mathematics is the art of giving the same name to different things.”

Scronch geometry has laws of “conservation of horizontal and vertical”; if two points are joined by a horizontal or vertical line segment, so are their respective scronches. Lorentzian spacetime is much the same. A point in spacetime is a location and a moment; the special line segments conserved by Lorentz symmetries are those joining two location-moments whose two locations are separated by the exact distance light would cover in the amount of time between the two moments. The speed of light, in other words, is built into the geometry. The question of whether light can reach location-moment A to location-moment B has a definite answer, which is the same whether you get on the bus or not. The scronch plane is like a baby version of Lorentz spacetime. You might think of it as what relativistic physics would look like if, instead of three dimensions of space, there were only one, joining with the one dimension of time to make a two-dimensional spacetime.

To analyze Brownian motion and the stock market and mosquito all at once, with the mathematics of the random walk, is to follow Poincaré’s slogan and give the same name to different things. Poincaré formulated his famous advice in his 1908 address to the International Congress of Mathematicians in Rome. He spoke movingly of the way doing complex computations can feel like “blind groping,” until that moment when you encounter something more: a common mathematical understructure shared by two separate problems, illuminating each in the light of the other. “[In] a word,” Poincaré says, “it has enabled me to perceive the possibility of a generalization. Then it will not be merely a new result that I have acquired, but a new force.”

It was the engineer and mathematician Claude Shannon who first realized that the Markov chain could be used not only to analyze text, but to generate it.

A proof is crystallized thought. It takes that brilliant buoyant moment of “getting it” and fixes it to the page so we can contemplate it at leisure. More importantly, we can share it with other people, in whose mind it springs to life again. A proof is like one of those hardy microbial spores so robust they can survive a trip through outer space on a meteorite and colonize a new planet after impact. Proof makes insight portable.