第 1 章
Geometry: The Hidden Language of Everything Around Us
When Jordan Ellenberg was a child, he disliked geometry. This might seem surprising for someone who grew up to become a mathematics professor and author of a book celebrating the subject. Yet his initial aversion reveals something profound about geometry's peculiar status in our mathematical education. For many, geometry stands out as either the moment math finally clicked or when it became hopelessly abstract. Unlike other branches of mathematics, geometry is viscerally built into our bodies from birth-our brains immediately begin reckoning with where things are and what they look like. Even under the influence of psychedelics like ayahuasca, when higher reasoning dissolves, pure geometric forms emerge first in human consciousness. Despite its primal nature, geometry remains divisive-some find its visual reasoning intuitive, while others struggle with spatial concepts. Yet as NASA mathematician Katherine Johnson once noted when her geometric knowledge proved essential among colleagues who had forgotten theirs, geometry isn't just an academic exercise but a fundamental tool for understanding our world. From pandemic modeling to artificial intelligence, from democratic redistricting to quantum physics, geometry permeates every aspect of modern life-not as a dusty relic but as a living language evolving faster than ever before.
第 2 章
The Geometric Mind of Abraham Lincoln
In 1864, Abraham Lincoln revealed to Reverend J.P. Gulliver that geometry was the source of his persuasive rhetorical skill. Struggling to understand what "demonstrate" truly meant, Lincoln retreated to his father's house and mastered the six books of Euclid until he could recite any proposition at sight. This wasn't merely academic curiosity-Lincoln found in Euclid a model for building "a tall, rock-solid building of belief and agreement" through rigorous deductive steps based on self-evident axioms.
This Euclidean thinking echoes throughout Lincoln's most famous speeches. In the Gettysburg Address, he characterizes America as "dedicated to the proposition that all men are created equal"-using "proposition" in the mathematical sense of a fact that follows logically from self-evident axioms. Like Thomas Jefferson before him, Lincoln viewed geometric principles as fundamental to democracy itself. But where Jefferson approached geometry as part of classical education befitting a cultivated patrician, Lincoln-self-educated and rural-embraced geometry's mental habits: refusing to leave things half-understood, boiling down thoughts, and tracing back steps of reason.
For Lincoln, geometry taught intellectual honesty-"it was morally impossible for Lincoln to argue dishonestly," noted his friend Henry Clay Whitney. Lincoln's distinctive quality wasn't superior intelligence but integrity, the principle of only saying what you can justify. When someone outside mathematics claims to be "just being logical," they're often disguising opinions as theorems. Lincoln exemplified the opposite-the ability to "tear the mask off from a fallacy."
Unfortunately, geometry education often devolves into proving obvious facts through tedious steps. The true value isn't that students will need to prove geometric theorems later in life, but that they'll learn to recognize non-proofs. Geometry requires both formal deduction and intuition-what Poincare called the two pillars of mathematical thought. Without intuition, geometry becomes "a dead sponge," just an elegant framework without life. Yet intuition alone can't be trusted completely.
The parallel postulate illustrates this tension. Mathematicians spent centuries trying to prove Euclid's fifth axiom from his other four, only to discover non-Euclidean geometries where the axiom doesn't hold-geometries that proved crucial to understanding physical space. Modern geometry increasingly places symmetry at its center, moving beyond Euclid's approach while preserving mathematical rigor. This evolution shows how geometry isn't a static collection of ancient truths but a living language that continues to develop in response to new questions and challenges.
第 3 章
Holes, Pants, and Topology's Revolution
When mathematical questions go viral online, they often reveal deeper conceptual issues beneath seemingly simple questions. The debate over how many holes a straw has-zero, one, or two-exemplifies this perfectly, having sparked arguments from philosophy journals to bodybuilding forums. Each position has its flaws. The "zero holes" argument fails because objects like bagels clearly have holes without material being removed. The "two holes" position struggles to define where one hole ends and another begins. The "one hole" theory seems most reasonable but leads to counterintuitive conclusions about anatomy.
Topology offers a way to think about shapes by focusing on essential properties while ignoring irrelevant details like exact dimensions or curvature. Henri Poincare captured topology's essence with his epigram: "Geometry is the art of reasoning well from badly drawn figures." In topology, circles needn't be perfect-even squares or squiggles count as circles so long as they're closed curves. What matters isn't precision but fundamental properties like whether a shape has holes.
When analyzing pants from a topological perspective, we discover they have two holes-not three-because the waist opening isn't a separate hole but the combination of the two leg holes. Emmy Noether formalized this in the 1920s with her concept of homology groups, seeing holes not as discrete objects but as dimensions or directions in space. Just as you need two independent directions to navigate a map, pants have two independent "hole dimensions." The three openings (two legs and waist) are related-traveling along all three brings you back to where you started, as they sum to zero.
What elevates topology beyond mere wordplay is its application to higher dimensions that defy visualization. The Euler characteristic provides a formal way to count holes, defined as one minus the number of holes or as the difference between even-dimensional and odd-dimensional holes. An unpopped balloon has Euler characteristic 2 with -1 holes, a popped balloon has characteristic 1 with 0 holes, a straw has characteristic 0 with 1 hole, and pants have characteristic -1 with 2 holes.
This system allows us to systematically track how shapes transform when cut or sewn together, even when our intuition about "holes" becomes strained. As Poincare declared, "Nobody doubts nowadays that the geometry of n dimensions is a real object." Even in the nineteenth century, studying celestial mechanics required tracking objects in six dimensions each. When analyzing multiple bodies simultaneously, the dimensions multiply-making formal mathematical language essential where visual intuition fails completely.
第 4 章
The Art of Giving the Same Name to Different Things
Symmetry forms the foundation of modern geometry, with different types of symmetry defining different geometric systems. In Euclidean geometry, rigid motions (translations, reflections, and rotations) are the fundamental symmetries, providing a more natural way to define congruence than Euclid's original approach. As Poincare noted, "Mathematics is the art of giving the same name to different things" - we collapse distinctions when they don't matter for our purposes.
Beyond rigid motions, we can define looser notions of sameness. When we allow magnification and shrinking, we get similar figures where angles remain invariant but areas change. We can invent entirely new transformations like the "scronch" - stretching vertically while shrinking horizontally by the same factor, preserving area but changing shape. Each notion of symmetry creates its own geometry with its own invariants - the properties that remain unchanged under transformation.
In 1904, Poincare addressed the International Congress at the Louisiana Purchase Exposition in St. Louis, warning of a "serious crisis" in physics. Maxwell's equations weren't invariant under the expected transformations. Poincare soon showed they were symmetric after all, but under novel "Lorentz transformations" that intermingled space and time. The scronch plane served as a simplified model for what would become relativistic spacetime - where the speed of light remains constant regardless of reference frame.
Though Poincare discovered this strange geometry in Maxwell's equations, it was Einstein who boldly reimagined physics according to these symmetries in 1905. Minkowski formalized the geometry, and Emmy Noether later established the fundamental connection between symmetries and conservation laws. Despite her groundbreaking work, Noether was expelled from Gottingen in 1933 for being Jewish, and died shortly after emigrating to America, remembered by Einstein as discovering "methods which have proved of enormous importance."
This principle extends beyond geometry - Lincoln himself used symmetry-based reasoning when arguing that if one person could enslave another, the reverse must also be permissible, suggesting moral permissibility should be invariant under reflection. The choice of which figures are "really" the same depends entirely on what properties we care about preserving, whether in mathematics, physics, or ethical reasoning.
第 5 章
Random Walks and Mosquito Mathematics
At the 1904 St. Louis exposition, Sir Ronald Ross, famous for discovering mosquitoes transmitted malaria, presented the first glimpse of random walk theory - a geometric concept that would revolutionize physics, finance, and literary analysis. Ross posed a fundamental question: if mosquitoes breed outside a protected circular zone, how large must that zone be to remain malaria-free?
The answer depends on how far mosquitoes typically wander in their short lifespans. Ross's mosquito model reveals a fundamental mathematical principle: a randomly moving mosquito travels on average about the square root of (2N/) kilometers in N days. A ten-day mosquito travels roughly 2.52km, while one living two hundred days would reach about 11.27km - only four times farther despite living twenty times longer.
This random walk principle explains why political polls of just 200 people can reliably predict the preferences of millions of voters - just as a single spoonful reveals the flavor of an entire soup. In a perfectly balanced state like Wisconsin, a 200-person poll might show 106 Republicans and 94 Democrats without being suspicious, but finding 120 Republicans would be as unlikely as a mosquito wandering 40km from home - just 3 in 1000.
However, this reliability depends on truly random sampling with no bias. Even slight biases dramatically increase the likelihood of misleading results - explaining why real-world polls often miss their predicted margins of error. This "law of anti-averages" explains why Nate Silver gave Trump a 33% chance of winning in 2016 while other forecasters put it at 7% or less - Silver accounted for correlated polling errors across swing states.
When Ross's mosquito problem exceeded his mathematical abilities, he turned to Karl Pearson, the charismatic, unconventional professor at University College London. Pearson recognized the importance of Ross's problem, spending "more than a whole day" attempting to solve it before publishing the question in Nature.
In July 1905, the same month Pearson posed Ross's question, Einstein published his paper on Brownian motion-the mysterious jittering of small particles suspended in liquid. Einstein's breakthrough explained this motion as the result of countless invisible molecules randomly striking the particles-exactly the same mathematical concept as Ross's mosquito problem, just at a microscopic scale with trillions of movements per second instead of one per day.
In early 20th century Russia, two mathematical cliques feuded over probability's implications for free will. Pavel Nekrasov, a former Orthodox theologian and czarist conservative, battled with Andrei Markov, an atheist nicknamed "Andrei the Furious." Markov developed what we now call the Markov chain-a system where future states depend on present ones. Markov showed that even in such dependent systems, long-term averages still stabilize predictably. He later applied his theory to literature, analyzing the first twenty thousand letters of Pushkin's "Eugene Onegin" to discover statistical patterns of consonants and vowels. Modern language models like GPT-3 follow this same principle but at much larger scale, analyzing hundreds of words at once to produce text that mimics human writing.
第 6 章
The Geometry of Games and Perfect Play
Marion Franklin Tinsley, a mild-mannered math professor from Tallahassee, was arguably the greatest champion in any competitive field in human history. His dominance was in checkers, where he lost just two games in championship matches over forty years. Tinsley's final opponent was Chinook, a computer program developed at the University of Alberta that eventually became the unbeatable world champion. Though Tinsley initially defeated Chinook in 1992, he resigned their 1994 rematch due to illness. X-rays revealed pancreatic cancer, and he died eight months later, marking the end of human dominance in checkers.
How can you prove that a game is unwinnable? Through geometry. Games are geometric structures that can be fully mapped. In the simple game of Nim, we can diagram every possible move sequence as a tree structure, showing why the second player will always beat the first player with perfect play. This tree diagram reveals the complete geometry of the game, with the starting position as the root and each branch representing a possible move choice.
Tree structures appear everywhere in our world, not just in games. Family trees show the branching of descendants from ancestors. Our arterial system forms a tree with the aorta as the root. Rivers form trees when viewed against their flow. Classification systems like Linnaeus's taxonomy follow tree structures. Trees work perfectly for representing these systems because they contain no cycles or infinite regress.
The game of Nim, like organizational structures and human vices, can be represented as a tree. To determine who wins, we work backward from the end positions. Empty positions (no stones) are marked "L" because whoever faces them loses. Positions with one stone are "W" because taking it wins the game. For any position, if at least one move leads to an L-position for your opponent, your position is a W. If all moves lead to W-positions for your opponent, your position is an L.
Checkers was mathematically solved in 2007 by Jonathan Schaeffer's team, who proved it's a draw when played perfectly. Despite having nearly 500 trillion possible positions, checkers could be analyzed by working backward and strategically pruning branches of the game tree. Yet despite this mathematical certainty, people worldwide still passionately play checkers. As Tinsley himself said, "If we play a lot of beautiful games, that will be my reward. Checkers is such a beautiful game that I don't mind losing."
Similarly, chess champions like Kasparov and Carlsen see their games not merely as computational problems but as "psychological warfare" with inherent beauty that machines can't fully appreciate. Perfection isn't beauty - we have proof that perfect players never win or lose. Whatever interest remains in these games exists only because humans are imperfect, feeling something when our imperfections encounter those of another.
第 7 章
The Power and Limits of Trial and Error
In chess, we may never fully label the game tree with wins, losses, and draws - not because we lack cleverness, but because the number of positions exceeds what any physical process could analyze before the universe ends. Yet computers play chess extraordinarily well without computing every possible position. Like modern artificial intelligence, they don't aim for perfection but something entirely different.
Fermat's mathematical approach was not infallible. He mistakenly believed that numbers of the form 2^(2^n) + 1 were always prime, having checked only small cases and missing that 4,294,967,297 was actually composite. This demonstrates how even brilliant mathematicians can be misled when things simply "feel right."
The bracelet theorem provides a practical test for primality - rather than testing divisibility directly, we can check if 2^n equals 2 more than a multiple of n. This creates a "non-constructive proof" that can determine a number is composite without revealing its factors. This distinction between knowing something exists versus knowing what it is parallels the difference between knowing a car has a leak versus knowing where the leak is.
Computer Go programs arrived later than chess or checkers programs. Unlike checkers where every position can be labeled as a win or loss, Go's complexity requires approximation. Programs like AlphaGo assign scores to board positions, with higher scores indicating advantage. This represents a fundamental AI trade-off: perfect accuracy requires impossible computation time, while oversimplification provides no useful guidance.
The "drunk Go" approach offers a surprisingly effective middle ground. By simulating random play from a given position thousands of times and measuring how often each player wins, the program can estimate position quality without deep strategic understanding. This resembles how random walks have been studied for centuries. The geometry of tournament boundaries affects accuracy and speed trade-offs. Tennis sets have no game limit, potentially creating marathons like the 70-68 Isner-Mahut match at Wimbledon. By contrast, the World Series ends after at most seven games. Different boundary shapes represent different algorithm designs - tennis optimizes for accuracy in determining the better player, while series formats prioritize speed.
Finding the optimal strategy in a game like Go requires navigating the geometry of the space of strategies - an abstract, infinite-dimensional terrain far more complex than a simple tree. The challenge is finding a decision-making protocol superior to human intuition within this vast space. Despite its seeming impossibility, the solution comes down to the crude but powerful method of trial and error.
第 8 章
The Geometry of Artificial Intelligence
AI isn't about killer robots or emotionless superminds - it's fundamentally just math, as NYU professor Meredith Broussard explains. The core concept resembles mountaineering without a map, where you can't see the broader landscape but must still reach the summit.
The fundamental technique powering machine learning is gradient descent - essentially sophisticated trial and error. Like a mountaineer who constantly assesses which direction offers the steepest upward slope and takes steps accordingly, an AI evaluates small changes to its strategy and chooses the one that improves performance most.
For image recognition tasks like identifying cats, the computer receives training data - thousands of images labeled "cat" or "not-cat." To a computer, each image is simply a grid of numbers. The machine's strategy is a mathematical function that transforms these numbers into an output between 0 and 1, indicating its confidence that the image contains a cat.
Success is measured by "wrongness scores" across the training set - if an image is a cat and the strategy outputs 1, that's perfect; if it outputs 0, that's completely wrong; if it outputs 0.8, it's slightly wrong (0.2). The goal is minimizing total wrongness across all images by gradually adjusting the strategy through gradient descent - making small changes that reduce wrongness most effectively.
The challenge with gradient descent lies in two fundamental limitations. First, it can get trapped in "local optima" - like a mountaineer who reaches a small peak but can't see the higher mountain nearby. From this position, every small step leads downward, making it impossible to reach the true summit without first descending through a valley.
The second problem is navigating the infinite-dimensional space of all possible strategies. Unlike a mountaineer choosing from 360 degrees of direction, an AI faces limitless possibilities. To make this manageable, we restrict the search to specific "knobs" we can adjust.
Despite its mystique, deep learning isn't new magic - it's just math that's been around since the 1950s. Neural networks resemble the architecture of a synthesizer like the Yamaha DX21 from 1985. In a synthesizer, sound operators connect to each other in complex ways, creating a vast range of possible sounds by adjusting relatively few knobs.
The mystery is why these networks generalize so well to new examples. Despite having enough parameters to perfectly memorize training data, they somehow find solutions that work on unseen cases. We use neural networks not because we know they're optimal, but because their architecture makes gradient descent possible - we're searching "under the lamppost" where we can see, even if better solutions might exist elsewhere.
第 9 章
Different Ways of Measuring Distance
What makes a circle a circle? The standard definition - points equidistant from a center - changes dramatically when we redefine distance. In urban travel, distance might mean travel time, creating irregular "isochrone" maps with starfish-shaped circles showing points reachable in equal time from a center. In Manhattan's grid system, distance measured in city blocks creates square "circles." Even family relationships form a geometry, with distance measured by steps through a family tree - siblings are at distance two from each other, first cousins at distance four, and generally, nth cousins are at distance 2(n+1) from each other.
Just as geographic maps help us visualize physical space, we can create maps of abstract concepts by defining distances between them. Word2vec places words in 300-dimensional space, positioning words with similar contexts nearby, creating fascinating analogies: "he" to "she" applied to "king" yields "queen." These relationships expose our biases - "gorgeous" is the feminized version of "stunning," "sassiness" the feminine "swagger." Word2vec reveals unexpected connections: a lady bagel is a "muffin," a Hindu bagel is a "vada pav," and the Oconomowoc of New York is Saratoga Springs.
However, these examples are cherry-picked; Word2vec often produces mere synonyms, misspellings, or simply incorrect analogies. It's not a magic meaning machine but mirrors how we speak, revealing the unconscious patterns of our language.
The concept of difficulty isn't one-dimensional but multifaceted. Some problems like factoring large numbers or playing perfect checkers are easy for computers but hard for humans. Conversely, tasks like determining whether a sequence has an even or odd number of elements or extrapolating patterns beyond training data are intuitive for humans but challenging for machines. While computers have processing power, they lack taste - the ability to recognize the simplest, most natural extension of a pattern. This multidimensional geometry of difficulty reminds us that machines may excel in some areas while humans maintain advantages in others, particularly with outlier cases and creative insights.
第 10 章
Mathematical Models of Disease Spread
Writing during the COVID-19 pandemic, Ellenberg notes how the world received a crash course in disease mathematics. Ronald Ross, who pioneered mathematical approaches to epidemiology, attempted to bring quantitative rigor to medicine just as Newton had tamed the seemingly random appearances of comets through laws of motion. Ross's 1904 lecture on mathematical epidemiology failed to connect with his medical audience, who "did not understand a word" he said.
By the mid-1910s, Ross was determined to create a mathematical theory for epidemics comparable to Newton's work on celestial bodies. His ambition extended beyond disease to what he called "The Theory of Happenings" - a framework for quantifying how any condition spreads through populations. Recognizing his mathematical limitations, Ross hired mathematician Hilda Hudson, a specialist in algebraic geometry who had published her first mathematical proof at age ten.
Ross's approach to epidemic modeling rested on a fundamental mathematical principle: what happened today will happen tomorrow. In its simplest form, if one infected person transmits a disease to two others during their ten-day contagious period, then 1,000 infected people would create 2,000 new cases after ten days, followed by 4,000, then 8,000 - forming a geometric progression.
Simple exponential growth models can lead to absurd predictions. In 2020, with COVID-19 cases growing 7% daily (60% weekly), projections suggested 17 million new daily cases by July - quickly exceeding the entire US population. These models fail because R0 isn't constant - it decreases as more people become infected and immune. If a disease has R0=2, once 10% of people are infected or immune, the effective R0 drops to 1.8. At 50% immunity, R0 becomes 1, and the epidemic begins to decline.
John Horton Conway, who died of COVID-19 in April 2020, was a remarkable mathematician whose work frequently involved visual thinking. Conway approached mathematics as play and play as mathematics, inventing numerous games with quirky names like Col, Snort, and Philosopher's Football. His contributions included developing the theory that games like Nim are a kind of number, cataloging all knots with eleven or fewer crossings, and creating the famous Game of Life algorithm.
Examining COVID-19 statistics reveals a paradox: while white Americans made up only 35% of confirmed cases, they accounted for 49.5% of deaths, suggesting higher mortality among white patients. The explanation lies in age demographics - white Americans are, on average, older than other racial groups, and COVID-19 is significantly more dangerous for the elderly. This demonstrates Simpson's paradox - a statistical phenomenon where a trend appears in combined data but disappears or reverses when the data is separated into groups.
The SIR model embodies the Newtonian principle that what happens today will happen tomorrow, just under more complex rules. In arithmetic progressions, each day's increase is constant; in geometric progressions, it's proportional to the current value. With the SIR model, tomorrow's values come from running today's values through the same mathematical "machine." When measured at increasingly smaller time intervals, these difference equations become differential equations-the mathematical language that describes continuous change in physical systems, from planetary orbits to pandemic spread.
第 11 章
The Dangers of Mathematical Extrapolation
In May 2020, the White House released a "cubic fit" model predicting COVID-19 deaths would drop to zero within weeks-a prediction that proved wildly optimistic. This mathematical misstep echoes historical debates about epidemic patterns, particularly the 1865-66 British rinderpest outbreak. While politician Robert Lowe warned Parliament about a "terrible law of increase" that would see cattle deaths explode exponentially, physician William Farr correctly predicted the epidemic would soon subside, despite skepticism from the medical establishment.
The author explains how the mathematical techniques behind his square root party trick have ancient origins. Around 600 CE, mathematicians like Brahmagupta in India and Liu Zhuo in China developed sophisticated interpolation methods for astronomical calculations. This method allows complex functions to be approximated using only basic arithmetic operations. The technique was later transmitted to the Arab world and rediscovered in England, notably by Henry Briggs who used it to compile logarithm tables.
The author discusses the dangers of mathematical extrapolation through Farr's rinderpest predictions. While Farr correctly predicted the epidemic would soon peak, his timing was premature and his long-term projections were off. Farr's approach was largely forgotten until John Brownlee revived it in the early 20th century, noting that Farr's constant-ratio method produces a symmetrical bell curve. However, real epidemics rarely follow perfect bell curves-they typically rise quickly but decline slowly, "coming up the elevator and going down the stairs."
There exist two approaches to prediction: reverse engineering (understanding mechanisms to derive equations) and curve fitting (observing patterns without necessarily understanding why). Ronald Ross championed the first approach, seeking to understand disease transmission mechanics, while William Farr exemplified the second, finding patterns without explaining mechanisms.
Most scientists naturally prefer reverse engineering, but curve fitting has seen a resurgence through machine learning. Google's language translation works not by understanding grammar rules as Chomsky would prefer, but by observing billions of sentences to detect statistical patterns-much like Farr's approach.
The most successful COVID-19 modelers, like MIT graduate Youyang Gu, combined both approaches-using mechanistic equations while tuning parameters with machine learning. We need both historical patterns and underlying laws, especially when preparing for novel pandemics without historical precedent.
第 12 章
How Geometry Shapes Our Democratic Systems
The 2018 Wisconsin elections revealed a stark democratic paradox: Democrats swept statewide offices including the governorship, yet Republicans maintained overwhelming control of the state legislature. This wasn't accidental - it was the deliberate result of partisan redistricting. Despite Democrats winning more votes statewide, Republicans maintained a 63-36 assembly majority because district lines had been precisely engineered to produce that outcome.
Wisconsin's partisan district map was created in secret in 2011 by Republican operatives including Joseph Handrick, a political strategist who had been plotting electoral strategies since his college days. The mapmakers classified their work as either "assertive" or "aggressive" based on how much they favored Republicans. The final map, dubbed "Joe Aggressive," was designed to maintain Republican control even when they lost the popular vote.
Democratic governments fundamentally require citizen representation, but implementing this principle fairly proves nearly impossible. Modern governments are too large for direct democracy, so we elect representatives - but how we choose them matters tremendously. The problem with geographic districts is that someone must draw the boundaries, giving enormous power to those wielding the "scissors" - typically legislators themselves.
Today's computational power has transformed gerrymandering from art to science. Wisconsin's mapmakers ran countless simulations testing each potential district configuration across various political scenarios until finding one optimized for Republican control in all but extreme circumstances. This computational advantage, combined with increasingly predictable partisan voting patterns, creates maps that both preserve partisan majorities and protect incumbents through an entire census cycle.
The efficiency gap measures wasted votes - those cast for losing candidates or beyond what's needed to win. In our Crayola example, Purple wastes 400,000 votes while Orange wastes only 100,000, creating a massive 30% efficiency gap. Real elections typically show single-digit gaps, with some lawyers suggesting anything over 7% warrants court scrutiny.
The fundamental mistake in analyzing gerrymandering is searching for a numerical standard of fairness. The opposite of gerrymandering isn't proportional representation or zero efficiency gap - it's simply not gerrymandering. The real question is: "Does this districting produce maps similar to what a neutral party would draw?"
Political scientists Jowei Chen and Jonathan Rodden pioneered a solution: generate maps automatically and in large numbers using computers programmed with no partisan preference. This approach reframes the question: "Does this districting produce outcomes similar to a map randomly selected from all legally permissible maps?"
Despite acknowledging gerrymandering as "unjust and incompatible with democratic principles," Justice Roberts' decision suggests the Court can't intervene. He expresses hope that state constitutions, ballot initiatives, or Congress might address the problem instead. Despite this setback, reform efforts continue. Within a year, North Carolina's districts were invalidated by state judges for violating the state constitution, following Pennsylvania's similar ruling in 2018.
第 13 章
The Liberating Power of Geometric Thinking
In this conclusion, Ellenberg explores how geometry can either reinforce or challenge established power structures. He begins with British architect Herbert Baker's view that neoclassical geometry in colonial architecture embodied "law and order" imposed by British administration in India. Similarly, French monarchs used perfectly aligned formal gardens to represent unchanging hierarchical order.
Ellenberg analyzes Edwin Abbott's 1884 novel "Flatland," where geometric shapes form a rigid social hierarchy with circles at the top, triangles as the masses, and women as mere line segments. The Square protagonist glimpses a higher dimension when visited by a Sphere, but is imprisoned for spreading this revelation. Though initially dismissed by critics, the book became beloved by geometry enthusiasts.
Ellenberg reveals that Abbott, far from endorsing this rigid hierarchy, was actually satirizing it. Abbott was an Anglican priest who advocated for women's education. The novel's Christian allegory suggests geometry offers liberation from oppressive social orders by allowing beings to infer higher dimensions through pure thought.
Ellenberg contrasts two mathematical experiences through Pulitzer Prize-winning poet Rita Dove's childhood memories. Where arithmetic appears as an imposed authority in "Flash Cards," geometry in her poem "Geometry" creates liberation - walls become transparent, windows "hinged into butterflies," everything moving toward "some point true and unproven."
Geometry holds special power because it empowers students to create their own knowledge rather than defer to external authority. This independence makes it both dangerous to authorities and uniquely worth "writing poems about." As Ellenberg explains, the Pythagorean Theorem isn't true because Pythagoras said so, but because we can prove it ourselves.
Yet truth and proof aren't identical. Poincare insists on intuition's necessary role, arguing that machines might process facts but "the soul of the fact will always escape it." Formal proof extends intuition's reach, but understanding is the true goal - when "walls go transparent" and "the ceiling flies off." Mathematician David Blackwell put it directly: "I'm interested in understanding, which is quite a different thing" from mere research. Geometry, though universal across human cultures, exists to teach us things and "make the house expand."
Ellenberg recounts the Talmudic story of the oven of Akhnai, where Rabbi Eliezer defends his minority opinion against fellow rabbis with increasingly dramatic demonstrations. When even the voice of God supports Eliezer, Rabbi Joshua declares "The voice of God isn't a proof!" since the Torah is now on earth, and majority opinion rules. God laughs delightedly: "My children have defeated me!"
The paradox of geometric thinking is that we are simultaneously alone in our deepest thoughts yet connected to everyone on earth. "Everyone does geometry differently, but everyone does it" - it's how we measure both the world and ourselves.
