Geometry Is the Cilantro of Math

Lex Fridman opens by asking Jordan Ellenberg where mathematical thinking sits in a Chomsky-style cake of mind, with language as the chocolate center and the rest of cognition as icing. Ellenberg, a mathematician at the University of Wisconsin and the author of *How Not to Be Wrong* and the newly released *Shape*, says producing mathematics feels something like producing language. It is hard to imagine doing the subject in a completely non-linguistic way, thinking in propositions rather than talking. Lex pushes on a different channel: visualizing shapes to which language is not obviously attached.

That question reminds Ellenberg of dissection proofs, especially a famous visual argument for the Pythagorean theorem associated with Bhaskara. You take the same pieces, assemble them one way and then another, and the two shapes must have the same area because they were built from the same parts. A disputed story has Bhaskara's manuscript offering only the diagram and the word "Behold." If the proof is a picture, or even a movie of pieces coming together in two formations, is that language? Lex thinks the stitching is what matters: language and math are processes, not piles of symbols sitting still.

Geometry, Ellenberg writes in *Shape*, is the cilantro of math. People are not neutral. Some, like Lex, find that a visual proof is the moment mathematics suddenly makes sense, and they wonder why the rest of school was not like that. Others come to Ellenberg to complain that they could solve for *x* all day, then spent a year lost in geometry and never understood why anyone bothered. What everyone agrees on is that geometry is somehow different from the rest of the subject.

Ellenberg's own conversion happened around age six or seven, staring at a 1970s stereo cabinet: a dark wooden box with a six-by-eight rectangular array of holes to let the sound out. Focusing in and out, he saw six rows of eight and eight columns of six—the same 48 holes. He already knew the times table said 6×8 equals 8×6, but he had not seen why. From an adult vantage that rectangle is a geometric reason for an algebraic fact, and he jokes he may have been fated to become an algebraic geometer. Ninth-grade Euclidean exercises, proving an isosceles triangle has equal base angles, never vibed with him the way that cabinet did.

In ordinary English, Ellenberg says, calling something symmetrical usually means it has an axis: left matches right, top matches bottom, or both. The abstract study of how those operations combine is group theory, one of his first loves in college. Modern mathematics uses the word much more loosely. Any transformation of an image, a space, or an object can count as a symmetry if it is well defined, has an input and an output, and can be repeated or inverted.

In *Shape* he invents an example: stretch a figure to twice its height and squeeze it to half its width. Classically that is not a symmetry, because it changes the look of the thing. He gives the move a silly name, "scrunch," because the language has no word for it, then asks which figures stay the same under scrunch the way some figures stay the same under a left-right flip. The generalization points to what he calls the idea that has dominated contemporary mathematics for about 150 years: when do we consider two things to be the same?

A triangle here and an identical triangle three inches to the left look like two objects. Prove a theorem about the first, though, and it almost certainly still holds for the second. The term of art is congruence; another way to say it is that translation is a symmetry, and we want our theories to be translation-invariant. If moving a figure three inches to the left made the theorems fail, that would be the strange outcome. Poincaré, a slogan factory Ellenberg keeps quoting, put the same thought as "mathematics is the art of calling different things by the same name."

Lex immediately maps the question onto MNIST handwritten digits: what makes every two a two, and which transformations turn a two into a three? Ellenberg says the symmetry groups mathematicians already understand are not enough. If a one were only a Palatino 1 rotated by some angle, a program could detect it. Distinguishing a two from a three is not that. He would be shocked if a classical symmetry formulation captured the difference; he thinks what we do is more complicated. Lex adds that the morph itself carries information: watch a two deform into a three and people can often mark the frame where it stops being a two, while a still of the in-between state might be called a seven.

That attention to transforming shapes, Ellenberg says, is also an insight of modern geometry, and it leads straight to Henri Poincaré and the three-body problem. Two masses under gravity are a clean story Newton already knew: they either orbit in an ellipse, the stable case, or one swings past the other on a hyperbola and flies away. Add a third body and the zoo opens. Poincaré won a prize offered by the king of Sweden in the 1880s for the first real progress, and 150 years later the problem is still a huge mystery. What he recognized is now called chaotic dynamics: stable and unstable solutions wound among each other, so a tiny change in initial conditions can make the long-term behavior completely different.

Ellenberg did not set out to make Poincaré a central character of *Shape*, but the man keeps showing up. He came of age in late-nineteenth-century France after the Franco-Prussian War, when the country decided it had to get strong in math the way Germany was—a Sputnik moment. Lex connects that mood to the Soviet Union, where excellence in science felt like a way to save the world, not only to throw a ball through a hoop. Ellenberg points him to Amir Alexander's *Duel at Dawn* and to Évariste Galois, the romantic inventor of group theory who died in a duel in his early twenties and, in modern terms, wrote extremely emo letters about the fire within him. Math is made of people, Ellenberg says, and they work inside the morals of their era. Poincaré, unusually among the salty characters in the book, seems to have been a modest, regular guy who did math four hours in the morning and four in the evening.

Until Poincaré, geometry had mostly been the study of two- and three-dimensional objects, the ones we see. To track three bodies he needed more. Each body has an *x*, *y*, and *z*, but also a velocity, so its state is better thought of as a point in six-dimensional phase space. He needed a geometry flexible enough for any number of dimensions, and he pioneered the subject he called *analysis situs*—roughly, analysis of location—a name that did not catch on. Today it is topology. The famous first line of the paper, as Ellenberg recalls it, is that no one doubts nowadays that the geometry of *n*-dimensional space is an actually existing thing.

Poincaré never strayed far from physics, but physics drove him into higher dimensions, and the formalism that let him work there also included non-physical objects. You can call those parts trash, Ellenberg says, or you can take the mathematician's attitude: if the framework is elegant, works well, and already contains the physics, maybe the whole thing is worth studying. Relativity and spacetime later made four-dimensional geometry look suddenly real, and the toolkit was already sitting there. The Poincaré conjecture itself is a classification question about curved three-dimensional spaces. Using a mug as a prop, Ellenberg explains simple connectedness: a loop drawn on the bowl of the mug can be pulled shut; a loop around the handle cannot, not without breaking the surface. The conjecture says that a three-dimensional space with no interesting loops of that kind is the standard one we think we live in.

Topology, Ellenberg stresses, is an intrinsic theory: you can answer the mug question without ever leaving the mug. Live on a circle and walking long enough in one direction brings you home; live on a line segment and you hit the end of the world. Those are facts about your world from inside it. Whether you live on an unknotted circle or on a knot floating in three-dimensional space is different. To the inhabitant both are circles; only a creature with three-dimensional eyes sees the knotting, which is about how the circle sits in an ambient space. Lex admits he loses no sleep over not knowing which one we occupy. Ellenberg is the same.

A Möbius strip is trickier. Lex's first test—walk around and see if your friend comes back mirrored—collapses because you may not recognize the mirrored friend, or even know whether you yourself flipped. A better intrinsic test, Ellenberg says, is a loop that cannot be pulled shut after one circuit but can after two, a feature of the Möbius strip and of the real projective plane. Lex then wonders whether a creature living inside a sphere would see the back of its own head in every direction. Ellenberg is unsure, because "see" smuggles in the physics of light. The metaphor of little creatures trying to apprehend a larger world, he notes, keeps returning, and he knows it best from Edwin Abbott's *Flatland*.

Abbott's comic novel, narrated by a square, has a small circle appear in the square's house and grow, a horror movie for two-dimensional people. The visitor is a sphere passing through the plane. After a sterile argument the sphere yanks the square out, and suddenly the whole world is visible from above. Abbott was a minister, so the Christian subtext is a higher being lifting you from the earthly world to see the truth. The subversive ending is that the square, now fluent in cubes, asks to reason about a four-dimensional version of himself, and the sphere is outraged: there is no fourth dimension, there are three, I can see. We cannot visualize a six-dimensional cube, Lex says, but we can count its corners. That split between seeing and reasoning is the point.

Is the universe many more dimensions than we experience, the way unification attempts in physics keep climbing to ten or eleven, or is high dimension only a bookkeeping device? Ellenberg thinks extra dimensions are possible. The only way to find out is to ask what effect they would have on the things we can perceive. If the answer is none, the question starts to go sterile, because you can posit as many unobservable entities as you like. You cannot mess with the other dimensions from inside three-space the way Abbott, a three-dimensional writer, could mess with Flatland.

Ellenberg's favorite classroom fight is how many holes a straw has. Ask a room and something comic happens: everyone thinks the answer is obvious, then discovers that other people have a different obvious answer, and people get heated. Zero-holers are rare. They say you make a straw by rolling a rectangular sheet of plastic, the sheet had no hole, and you never poked one. Ellenberg grants the engineering consistency and then asks them to say a bagel has no hole either, because you can form one by joining the ends of a cylinder of dough. If you are really committed to that, he says, who are you?

One-holers say there is a single region of space going all the way through. Two-holers say there is a hole at the top and a hole at the bottom. A common demo is a water bottle: it has one hole at the mouth; poke another so the water spills and it has become a straw, so how can you still count one? The one-holer replies by asking where the first hole ends and the second begins. Faced with a confusing problem, Ellenberg says, mathematics has two moves: make it simpler, as when they dropped a dimension to talk about circles, or make it harder. He stops talking about straws and asks about pants.

Most two-holers for the straw would say pants have three holes: waist and two legs. His daughter said pants have two, because the waist is just the two leg holes stuck together. That child's answer, he argues, captures something central to homology: holes have an arithmetic; they can be added. "Waist equals leg plus leg" is an equation, but not an equation about numbers. A modern mathematician, speaking like a rabbi who dodges the original question, would say the straw has two holes, and one is the negative of the other. Sip a milkshake and the amount flowing in one end equals the amount flowing out the other; they are not independent. Through a pair of pants, left-leg inflow plus right-leg inflow equals waist outflow.

Poincaré and Betti were already moving toward relations among holes. The truly modern view that holes have an arithmetic, Ellenberg says, was Emmy Noether's: she put the subject on the footing it has now. He is frank that a book like *Shape* cannot deliver a semester of algebraic topology on the page. Some topics he can only flavor; others he tries to do up close, so the book is not only about the math but has some math in it for the reader to touch.

Lex names Grant Sanderson of 3Blue1Brown as someone who puts the idea on the page, or rather on the screen, instead of talking around it. Ellenberg calls the flowering of math YouTube a wonderful thing because teaching lives on a curve of engagement versus reach. A PhD advisor spends three to five years with one student; a lecture hall holds thirty or five hundred for a semester; a book reaches more people for the hours it takes to read; a YouTube video can outsell any hardback not written by a Kardashian or an Obama. Eugenia Cheng gets thirty seconds on television. Every point on that curve matters, because brains have different hooks. Sanderson draws in Python, and you have to understand a topic, and think about it newly, to visualize it that way. Academic mathematics, Ellenberg says, still has not quite figured out how to value visualization, or the building of computational infrastructure, as mathematics rather than as a service to theorem-proving. Brown's Institute for Computational and Experimental Mathematics, an NSF-funded institute, did a whole theme semester on visualizing math, which he takes as a sign of buy-in.

Lex then asks whether an AI system will one day win a Fields Medal, the Nobel of mathematics. Ellenberg says no: that refusal is his CAPTCHA, the proof that he is a human. The more serious answer is historical. Computers have already eaten work that used to count as research. In 1890 a PhD thesis might be a year spent computing the invariants of a polynomial ring under a finite group. Today a few lines of Macaulay, Sage, or Magma—specialized computer-algebra systems kin to Maple and Mathematica—spit the answer out, and we no longer call that math. Those tools do not prove theorems, but they are now part of most mathematicians' process. Attempts to get AIs to prove interesting theorems have not done well. He has seen a neural net find counterexamples to some conjectures, which is cool, but the conjectures were not famous, and five graduate students might have found the same examples.

The story that fascinates him is the Conway knot and whether it is slice, a technical question about three-dimensional surfaces in four dimensions. It stumped topologists for decades until Lisa Piccirillo, then a graduate student at UT Austin and now at MIT, solved it in a nine-page paper of which two pages were pictures. Was the problem therefore easy, or was it hard because the world's top people could not do it for twenty years? Maybe we need a category for things that are difficult to figure out are not difficult. Lex hopes the universe will turn out to have simple laws. Ellenberg will spend the next stretch of the conversation pushing back on that hope.

Lex wants Fermat's last theorem, the simple-looking claim that $x^n + y^n = z^n$ has no positive integer solutions for $n$ greater than two, and the romance of a proof too wide for a margin. Ellenberg pours cold water on the romance. Historically, he says, Fermat did not have the proof. He later proved special cases and wrote to people about the problem in ways that make clear he did not know how to do the whole thing. He may once have had a deep intuition that is lost; what we can know is that later he did not think he was concealing a complete argument. Special cases via infinite descent are lovely undergraduate material. They are not big. Work on the Fermat problem—Ellenberg prefers that name, because it was not really Fermat's theorem—is what built the richness of number theory we live in now.

Andrew Wiles, with Richard Taylor, finally proved it, but the failures matter as much as the success. A famous nineteenth-century false proof by Lamé led Kummer, in understanding the mistake, to something extraordinary. Unique factorization into primes feels like God's law: 12 is two 2s and a 3, never a 5, never four 3s. Euclid essentially knew you can factor at all; the depth is that there is only one way. In number systems slightly more general than the integers, and relevant to Fermat, unique factorization collapses. If you loved the simplicity, you might call that bad. Ellenberg calls it good. "I'm highly in favor of complication," he says, because every complication is an opportunity to study new phenomena you cannot see through the old lens.

At the heart of the Wiles–Taylor argument is deformation theory, a subject Ellenberg's advisor Barry Mazur helped create. The objects in the Fermat problem—Galois representations and modular forms, words he knows will not help the listener—can be moved a tiny bit. If you understand the infinitesimal directions a thing can move, you may piece that together into the whole global space of possible moves. Their argument shows two such global spaces are the same, the fabled "R equals T." That "tiny bit," though, is not the distance from calculus. Distance, a theme of *Shape*, is not given by God. We already use many distances in English: a close relative need not live next door. Natural language processing uses a semantic distance that has nothing to do with dictionary order.

Number theory has a wilder one, the 2-adic distance: two numbers are close when their difference is a multiple of a large power of two. Then 1 and 49 are close, because the difference is 48, a multiple of 16; 1 and 2 are far, because the difference is 1, not even a multiple of 2. Worse, 1 and 1/64 are astronomically far, a negative power of two. Lex, briefly broken, finds the key: it is almost binary written backwards. In ordinary binary a small number starts with many zeros after the point; in the 2-adic metric a small number *ends* with many zeros. Ellenberg says he should have used that metaphor. Wiles's deformations are tiny in this 2-adic sense. Change the distance and you can talk about nearby changes that ordinary closeness would never see, then pass from local to global. Lex generalizes: define two people as close when they love each other, analyze seven billion of them, and you might see things geography hides. Neural nets, he adds, are in the business of learning a distance; dictionary order would make a terrible translator.

The goal, Ellenberg says, is not a theorem factory with a production quota. He cites Bill Thurston's essay "On Proof and Progress in Mathematics": the point is to help humans understand, and new theorems are the benchmark, not the aim. Lex quotes the line often attributed to Feynman, that if you cannot explain something simply you do not understand it. Is that true? "Flatly no," Ellenberg says, then grants a profound fragment: learning to explain something helps you understand it. Real things are not simple; a few are, most are not. He doubts Feynman literally believed the slogan, even if he liked how saying it felt. Physicists, Lex observes, often take joy in believing the world can be explained simply even when the explanation is a poem about a complicated thing rather than a distillation. Ellenberg agrees that no poet would call a poem an explanation; they might call it a description.

Paul Erdős talked about The Book, where God keeps the right proof of every theorem, and praised an elegant argument as being from the Book—while calling God the Supreme Fascist. Hilda Hudson, who helped develop the epidemic models behind today's SIR framework and was devout, said mathematical truths are not the most important of God's thoughts, but they are the only ones we can know precisely. Poincaré was colder: if a divine being came down and told us the universe in the language of the divine, we would not understand it, so it does not matter. Our job is the parts that are not inhumanly complex. Human history's seasons of hate, Lex notes, often ran on extremely simple stories. Simplicity is as pleasant as happiness, and as unreliable a test of truth.

Primes, Ellenberg says, are the atoms of the integers: factor until you cannot, and what is left is prime. One is not currently deemed prime, a choice of definition, not a discovery of truth. Grothendieck once needed an example of a prime and said 57, which is 3×19, a joke that still lives in some circles; 91 feels prime and is 7×13. Fermat's little theorem gives a one-sided test: raise 2 to the *n*, divide by *n*, and if the remainder is not 2 then *n* is definitely not prime. If the remainder is 2, *n* is likely prime but not certain. The smallest pseudoprime that fools the test is 341. Euclid already knew there are infinitely many primes. They get rarer as numbers grow: four of the first ten numbers are prime, 40 percent, which will not happen among ten-digit numbers.

Twin primes are pairs two apart, 3 and 5, 11 and 13. Whether there are infinitely many is open. People hear the conjecture and think mathematicians believe in a deep structure that keeps pulling primes together. Ellenberg says the opposite. To believe the twin-prime conjecture is to believe primes are strewn around pretty randomly; if they were, chance would give infinitely many twins, and "we expect them to behave just like they would if they were random dirt." Primality is deterministic. Pretending they come from a random process is still phenomenally useful, first as a way to guess what is true so you do not spend years proving something false. In an imaginary argument with a Sam Harris-style determinist, Ellenberg is happy to let the other person insist on the metaphysics while he extracts insight from the pretense. Treat people as beings with wants, even if that is not the deep story, and you understand them better. Lex finds the same anthropomorphism useful in robotics, against the field's allergy to naming and gendering machines.

John Conway, for Ellenberg, is one of the twentieth-century mathematicians whose personality is wound into the work. Outsiders sometimes think mathematics is a cold activity separate from being a person. No: your personality is in there the way it would be in a novel, a painting, or the way you walk down the street. Conway treated everything as a game, invented games with crazy names, then analyzed them mathematically, eventually collaborating with Donald Knuth on the surreal numbers, a system in which each number is a two-player game. Bishop Berkeley's "ghosts of departed quantities," the infinitesimals that made people queasy about Newton, live there now as games rather than ghosts. Ellenberg, as a postdoc at Princeton, would find Conway in the common room, ask a question, and get a rich tour that usually had nothing to do with the question. Turning him on was like winding a knob.

The Game of Life, developed in the 1970s, is not a game against an opponent. You mark squares on graph paper—Conway really did it with a pen—and a one-line rule decides birth and death from the eight neighbors. From a small, innocent configuration you get astonishing richness: gliders that travel, glider guns that emit them, species that interact in a language borrowed from biology. Ellenberg recommends the app Golly, which runs Conway's rule and hundreds of variants faster than the eye. The rebuke to anyone who thinks rich structure cannot come from simple laws is obvious. The less advertised fact is the one he stresses: pick a random rule and you get nothing interesting. Typical rule sets generate no interesting behavior. Some do, and we have no clear way to tell which in advance.

Stephen Wolfram has gone furthest down that Darwinian path of playing with simple rules for a thousand pages. Ellenberg thinks that if you put ego aside, the work is quite brilliant, even though Wolfram sits at the edge of the math and physics communities. The Rule 30 challenge—predict the middle column of a one-dimensional automaton—has no solution yet. Wolfram talks about irreducibility; Ellenberg calls that poetry, not a theorem. Conway himself had an ambivalent relationship with Life. It was what the outside world knew him for, and it grieved him a bit to be the Game of Life guy when he had proved deeper theorems and invented the surreal numbers. He was exactly the person you would never want to reduce to one achievement.

Group theory, returning as a theme, starts for Ellenberg as the general theory of symmetry. A person has two: flip and do nothing. A rectangle has four: do nothing, flip left-right, flip top-bottom, or both. A square has eight, because a 90-degree rotation still looks like the same square, which a rectangle does not allow. The real observation is that the operations combine: do one, then another, and the result is a third. A group is a collection of transformations closed under that combination. Shuffling a deck is the same world: every ordering is a symmetry of the original order, there are vastly more than eight, and a shuffle followed by a shuffle is another shuffle. The symmetries of the line are already infinite: reflect about any point, or slide any distance.

Lex asks for gauge symmetry, the thing physicists keep invoking. Ellenberg is not a mathematical physicist, but he offers a contour. Physics is always asking what the symmetry group of the world is. If gravity works here, it should work over there, so translation should be a symmetry of physical law. One way to tell the Einstein story is that we changed our mind about that group. Lorentz contraction, in the old frame, is an object changing shape near the speed of light. In the new frame it never changed shape; we were wrong about what counted as a symmetry. Once you have the Lorentz group, the deformation was an illusion. Lex still wants a minimal programmable example, and Ellenberg says that hunt is exactly how they train students.

The Poincaré conjecture's proof, completed by Grigori Perelman in a tradition that includes Richard Hamilton and many others, is Ellenberg's favorite case of going one level up. You think you are studying a three-dimensional space. The geometry you actually need is the geometry of the space of all three-dimensional geometries. Ricci flow lets a mystery space move along a path in that higher space; the argument is that it always flows to standard three-space, without sharp transitions, so it was the standard space all along. The hard analytic step is proving there are no singularities, no sudden kinks. As with Wiles, the keystone sits on an arch built by others: Kummer, Gerhard Frey, Mazur, Ken Ribet. You only see the one object clearly, Ellenberg says, once you have been jerked out of Flatland and can see the whole space of geometries at once.

Perelman turned down the Fields Medal. In some vast spiritual sense, Ellenberg says, awards are not important the way understanding the universe is important; on the other hand most people who are offered that prize accept it, so the choice is unusual. Peter Scholze has turned down a large monetary prize, perhaps because it sent the wrong message about what mathematics is for. Lex takes the episode as a reminder that most lives contain a moment when society wants one thing and a principle wants another, and standing there is integrity, whether or not it makes sense to anyone else. He is not sure Perelman's reasons, and Ellenberg does not pretend to know them. The International Mathematical Olympiad, Ellenberg adds, is false to research in another way: canned problems with known answers, scored on speed. Gold medals go to the top twelfth of contestants, not to a single winner. His first IMO, in 1987 in Havana, was a Cold War production; the American team was taken to the Museum of American Imperialism.

How should a young person, or an adult who wants math back, learn it? Ellenberg will not give one answer, because the work is a journey of self-knowledge and what works differs. Some people at any age just start reading textbooks. For others the gateway is Martin Gardner's decades of "Mathematical Recreations" columns in *Scientific American*, collected into books that stay fresh for each new generation and embody Conway's spirit of play. Contest problems, especially Russian and Bulgarian collections, motivate some people; well-produced videos motivate others. The honest reply is that you have to find what resonates.

Then he tells a graduate-school story. Overwhelmed by ignorance, as everyone in a PhD program is, he told Joe Harris, an algebraic geometer in the department, that he wanted a year of leave to read EGA, Alexander Grothendieck and Jean Dieudonné's 1,500-page *Éléments de géométrie algébrique*, cover to cover. Harris said that was a really stupid idea. Not because the book is bad, but because Ellenberg would not actually do it. He would learn a fact from EGA when he needed that fact to solve a problem he already wanted to solve. Without a problem to bring him in, he would not read.

Ellenberg generalizes. Mathematics is hard; anything you do right is hard. Because it is hard, an abstract curriculum someone told you you should learn will not stick if you do not care. Hook it to an engineering problem, a physics problem, a machine-learning problem. Let that, not an idea of what the syllabus is, drive the learning. Lex compares the frustration to running, which he hates in the moment and loves afterward: you hold incomplete pictures, bad notation, missing chunks, and keep going. Ellenberg corrects the word *enjoy*. He does not enjoy dead ends; they piss him off. You have to accept them as part of the process. Lex offers David Goggins's line that if you do not quit where most people would, you might see a garden others never reach. Ellenberg asks about the Nathan's hot-dog contest champion at hot dog 35, correctly noting that most people would stop there. They agree the slogan has no standard answer.

Advice for young people is not a thought experiment for Ellenberg; he has two children, even if they do not listen. Students at decision points, he says, are beset by self-doubt, unsure what they are capable of and what they really want. One of the options is usually the high-self-esteem choice. Take yourself out of it, figure out what the completely confident version of you would do, and do that, then see what happens. He would never tell someone to abolish doubt. People probably should have doubts. Sometimes it is still right to act as the person without them would act. He has not yet said this to his own kids. Lex tells him he should.

Can mathematics reach the meaning of life? Ellenberg wrote about Blaise Pascal in *How Not to Be Wrong*. People treat Pascal's wager as a proof that you should believe in God. Reading Pascal, Ellenberg found almost the opposite. Ask Pascal why he believes and he says he met God: a mystical experience, not a calculation. The math is about reasons for behaving a certain way once you already have to bet. Mathematics already underlies learning machines and physics. "You have mathematics, you have ways to explore the mind, you have incredible technology, you also have love and friendship, and what the hell do you want to know what the meaning of it all is? Just enjoy it."

The conversation ends on the chaotic map at the front of *Shape*, a picture of everything Ellenberg was trying to hold in his head while writing, drawn only at the end. Many of the day's topics are on it; the connections are still being dissected. God is in the picture, on the edge, not in the center. Lex closes with a line from *How Not to Be Wrong*: knowing mathematics is like wearing X-ray specs that reveal hidden structures under the messy surface of the world.