Shape

Jordan Ellenberg argues that geometry is not a dusty classroom relic but a living, rapidly advancing discipline embedded in problems ranging from pandemic modeling to artificial intelligence to the mechanics of democratic representation. Drawing on history, biography, and his own experience as a public mathematician, he contends that geometric thinking shapes how people reason about the world, whether or not they recognize it as geometry.

Ellenberg opens by tracing a centuries-long tradition of finding consolation and power in geometric reasoning. He recounts how Abraham Lincoln, the self-taught lawyer who would become the 16th president, studied the ancient Greek geometer Euclid's Elements, a classical text on geometry and deductive proof, by candlelight to understand the concept of "demonstration." Lincoln came to see rigorous deduction as the basis of honest argument. Ellenberg contrasts Lincoln's democratic relationship with geometry to that of Thomas Jefferson, the college-educated founding father who studied Euclid at William and Mary. He argues that the real purpose of teaching proof is not to establish facts about triangles but to train students to distinguish genuine reasoning from arguments dressed up to look logical. He illustrates this through the pons asinorum, the classical theorem that an isosceles triangle has two equal angles, showing how Pappus of Alexandria, a late-antique mathematician, offered a proof that treats the triangle as congruent to its own mirror image, capturing the idea of symmetry that Euclid's original proof obscures.

From classical proof, Ellenberg moves to topology, the branch of geometry concerned with properties that survive stretching and bending. He uses the viral internet debate over how many holes a straw has to introduce the French mathematician Henri Poincaré's insight that geometry can operate at a level of abstraction where size, angle, and distance are irrelevant. Poincaré's epigram that geometry is "the art of reasoning well from badly drawn figures" (40) captures this spirit: In topology, a square and a circle are the same shape, and a straw, shortened and flattened, reveals itself to have exactly one hole. Ellenberg credits Emmy Noether, the German mathematician, with formalizing hole-counting in the 1920s through her concept of the homology group, which treats holes not as discrete objects but as directions in a space of possibilities.

Ellenberg then argues that the choice of which transformations count as symmetries determines what kind of geometry one is doing. Euclidean geometry treats rigid motions as its symmetries; topology allows any continuous deformation. He connects this idea to physics through Poincaré's discovery that Maxwell's equations of electromagnetism were symmetric under Lorentz transformations, a novel family of transformations intermingling space and time. The physicist Albert Einstein drew the radical conclusion, remaking physics by accepting that the geometry of the universe was not what Euclid had supposed. Noether later proved a sweeping theorem connecting every type of symmetry to a conservation law, unifying a tangle of separate physical principles.

The middle chapters trace the random walk, a single mathematical idea—a model of how something moves through a sequence of random steps—that surfaced independently around 1900 across several fields. Ronald Ross, the British doctor who discovered that mosquitoes transmit malaria, asked how far a randomly wandering mosquito would travel from its birthplace. The statistician Karl Pearson posed the same question in the journal Nature. Louis Bachelier, a student of Poincaré, applied random-walk reasoning to stock prices, concluding that "the expected gain of a speculator is zero" (82). Einstein used the same framework to explain Brownian motion, the jittering of particles suspended in fluid, as evidence that matter consists of molecules. In Russia, the mathematician Andrei Markov invented the Markov chain, a sequence of random variables where each depends only on the one before it, to disprove a rival's claim that the Law of Large Numbers, the principle that averages stabilize over many trials, constituted proof of free will. Markov applied his chains to Pushkin's Eugene Onegin, analyzing patterns of consonants and vowels, a line of work Ellenberg traces forward to modern language-generation algorithms like GPT-3.

Ellenberg devotes several chapters to artificial intelligence, framing it as a geometric problem. A machine learning algorithm searches a vast landscape of possible strategies using gradient descent: It assesses the slope in every direction, takes a step along the steepest upward path, and repeats. Neural networks, whose architecture dates to the perceptron, an early model of a simple learning machine devised in 1957 by Frank Rosenblatt, a psychologist, define a tractable region of this landscape. Modern deep networks like GPT-3 have 175 billion adjustable parameters. Ellenberg acknowledges that no one fully understands why gradient descent on neural networks generalizes so well to new data.

The book also explores epidemic modeling. Ellenberg describes the SIR model, which divides a population into susceptible, infected, and recovered groups and uses difference equations to trace how an outbreak rises and falls. The critical quantity is R₀, the average number of new infections each infected person causes: Above one, the epidemic grows exponentially; below one, it decays. He shows how naive curve-fitting can produce misleading projections, as when the White House Council of Economic Advisers used a cubic polynomial in May 2020 to predict that COVID-19 deaths would drop to near zero within weeks. Ellenberg contrasts such extrapolation with models built on understood mechanisms, arguing that the best forecasting blends both approaches.

A major chapter addresses gerrymandering, the practice of drawing legislative district boundaries to entrench partisan advantage. Ellenberg opens with the 2018 Wisconsin midterm elections, in which Democrats swept every statewide office but Republicans retained a 63-36 assembly majority, a result engineered by operatives who had spent months testing computer-generated maps. He traces gerrymandering's history from colonial Pennsylvania through the Virginia politician Patrick Henry's attempt to keep the statesman James Madison out of Congress, and argues that modern computing has transformed gerrymandering from an imprecise art into a precise science. After dismantling several proposed fairness standards, including proportional representation and the "efficiency gap," Ellenberg advocates the ensemble method: A computer generates thousands of district maps using a neutral algorithm, and the contested map's outcomes are compared against this distribution. The mathematician Moon Duchin developed the ReCom algorithm for generating these maps, which works by merging adjacent districts, finding a random spanning tree (a loop-free network connecting all wards), and cutting one edge to split the merged region in two. Analysis of Wisconsin's map showed it to be an extreme outlier. In the 2019 Supreme Court case Rucho v. Common Cause, however, the Court declared partisan gerrymandering beyond federal judicial reach in a five-to-four decision, with Chief Justice John Roberts acknowledging the practice is "incompatible with democratic principles" (406) but ruling the courts cannot intervene. Reform efforts subsequently shifted to state courts and ballot initiatives.

Ellenberg closes by contrasting geometry as a tool of authority with geometry as a source of liberation. He juxtaposes two poems by Rita Dove, a Pulitzer Prize-winning poet: "Flash Cards," in which arithmetic is an imposed discipline, and "Geometry," in which proving a theorem causes the house to expand. He poses a thought experiment: If the Poincaré Conjecture, one of the most famous problems in topology, had been resolved not by the reclusive Russian mathematician Grigori Perelman but by a machine in a form humans could verify but not understand, the result would settle the question but yield no insight. Insight, Ellenberg argues, is the real goal. He ends with a story from the Talmud, a central text of Rabbinic Judaism, about the Oven of Akhnai: Rabbi Joshua overrules even the voice of God by insisting that authority resides in human reasoning, and God laughs with delight. The story captures what Ellenberg sees as geometry's deepest promise: not the imposition of certainty from above, but the expansion of understanding from within.