Geometry

Euclid of Alexandria: the author without a face

Timeline c. 530 BCPythagoras and his followers study numbers and figures in southern Italy. c. 387 BCPlato founds the Academy in Athens. Over the door, according to tradition: “Let no one ignorant of geometry enter”. 331 BCAlexander the Great founds Alexandria, in the Nile delta. c. 300 BCEuclid teaches in Alexandria, in the time of King Ptolemy I, and writes the Elements. c. 250 BCArchimedes, in Syracuse, cites the Elements. c. AD 100A papyrus found at Oxyrhynchus, in Egypt, preserves a passage from Book II. c. 370Theon of Alexandria prepares the edition that will be copied over the following centuries. c. 450Proclus writes his commentary on Book I, with the little news we have of Euclid. c. 800The Elements are translated into Arabic in Baghdad. c. 1120Adelard of Bath translates the Arabic text into Latin. 1482First printed edition, in Venice, by Erhard Ratdolt. ◀ Previous Follow the journey Next ▶ c. 387 a.C. AthensPlato founds the Academy. The geometry Euclid will organise grows up in the Greek schools. c. 300 a.C. AlexandriaEuclid teaches in the time of Ptolemy I and writes the thirteen books of the Elements. c. 250 a.C. SyracuseArchimedes already cites the Elements in his works. c. 100 OxyrhynchusA papyrus found in the town’s rubbish heaps preserves a passage from Book II, with a figure. c. 370 AlexandriaTheon of Alexandria prepares the edition that will be copied for a thousand years. c. 450 AthensProclus writes his commentary on Book I, with the little news we have about Euclid. c. 800 BaghdadIn the House of Wisdom the Elements are translated into Arabic. c. 1120 BathAdelard of Bath translates the Arabic text into Latin, and the book returns to Europe. 1482 VeniceErhard Ratdolt prints the first edition, with the figures in the margins. More than a thousand editions followed. Enable JavaScript to see the map. The stages are listed below. The journey of the Elements, from Athens and Alexandria to the first printed edition in Venice. Map: © OpenStreetMap contributors

October 14, 2026 Read

The life of John Nash

Timeline 1928Born on 13 June in Bluefield, West Virginia, in the United States. 1945Enters the Carnegie Institute of Technology, in Pittsburgh, to study chemical engineering. Switches to mathematics. 1948Arrives at Princeton. Invents the game his colleagues call “Nash”, known today as Hex. 1950Defends the thesis “Non-cooperative games”, with the Nash equilibrium. Publishes “The bargaining problem”. 1951Becomes an instructor at MIT. 1956Publishes the isometric embedding theorem. 1957Marries Alicia Lardé, a physics student at MIT. 1958Publishes the result on the regularity of parabolic equations. 1959First psychiatric hospitalisation. Three decades of illness begin. 1994Receives the Nobel Prize in Economics, with John Harsanyi and Reinhard Selten. 2001Remarries Alicia. The film A Beautiful Mind is released. 2015Receives the Abel Prize, with Louis Nirenberg. Dies in a car crash on 23 May, together with Alicia. ◀ Previous Follow the journey Next ▶ 1928–1945 BluefieldHe grows up in a railway and coal town. He reads Men of Mathematics and proves Fermat’s little theorem by himself. 1945–1948 PittsburghAt the Carnegie Institute of Technology he swaps chemical engineering for mathematics and has the bargaining idea. 1948–1951 PrincetonHe invents Hex and defends his 28-page thesis with the Nash equilibrium. 1951–1959 MITA professor at MIT. He proves the embedding theorem and the result on parabolic equations. In 1959 his illness begins. 1970–2015 PrincetonHe wanders the corridors as “the phantom of Fine Hall” and recovers at the end of the 1980s. 1994 StockholmHe receives the Nobel Prize in Economics, with John Harsanyi and Reinhard Selten. 2015 OsloHe receives the Abel Prize, with Louis Nirenberg. On the way home, on 23 May, he and Alicia die in a taxi crash in New Jersey. Enable JavaScript to see the map. The stages are listed below. John Nash’s places, from Bluefield to Princeton, and the journeys to receive the Nobel and the Abel. Map: © OpenStreetMap contributors

October 14, 2026 Read

The life of Blaise Pascal

Timeline 1623Born on 19 June in Clermont, in the Auvergne region of France. 1626His mother, Antoinette Begon, dies. Blaise is 3. 1631His father, Étienne Pascal, moves to Paris with his three children. 1635At 12, he rediscovers on his own that the angles of a triangle add up to two right angles. 1640At 16, he publishes the Essay on conics, with the hexagon theorem. 1642Starts building a calculating machine to help his father, a tax official in Rouen. 1647Publishes the New experiments concerning the vacuum. Descartes visits him. 1648Florin Périer, his brother-in-law, measures the weight of air by climbing the Puy de Dôme. 1654Exchanges letters with Fermat about games of chance and writes the Treatise on the arithmetic triangle. On 23 November he has his “night of fire”. 1656Begins publishing the Provincial letters under a pseudonym. 1658Solves the problems of the cycloid and launches a contest about them. 1662Opens a line of public coaches in Paris. Dies on 19 August, aged 39. 1670His friends publish the Pensées, put together from his notes. ◀ Previous Follow the journey Next ▶ 1623–1631 ClermontHe is born in Clermont, in Auvergne. His mother dies when he is three. 1631–1639 ParisHis father teaches him at home and takes him to Mersenne’s gatherings. At 16 he writes the Essay on conics. 1640–1647 RouenHis father is a tax collector in Normandy. To help him, Blaise builds the Pascaline and carries out his experiments on the vacuum. 1648 Puy de DômeAt Pascal’s request his brother-in-law Florin Périer climbs the mountain with a barometer and shows that the air weighs less at the top. 1647–1662 ParisThe letters to Fermat on games of chance, the night of fire, the Provincial Letters, the cycloid and the public coaches. He dies in 1662, aged 39. 1655 Port-Royal-des-ChampsAfter the night of fire he spends periods at the convent where his sister Jacqueline lives and defends Port-Royal against the Jesuits. Enable JavaScript to see the map. The stages are listed below. Pascal between Clermont, Paris and Rouen, and the mountain that weighed the air. Map: © OpenStreetMap contributors

October 11, 2026 Read

The Elements: definitions, postulates and common notions

A point is that which has no part. A line is breadthless length. — Euclid, Elements, Book I, Definitions 1 and 2 Imagine you want to convince someone, beyond any possible doubt, that the angles of a triangle always add up to 180°. Measuring is not enough: the protractor makes errors, and you can only measure a few triangles. You need an argument. But every argument starts from something. Where should you start? If every claim must be justified by another, and that one by another, the chain never ends.

October 14, 2026 Read

The game of Hex: why the first player wins

In 1948, in the common room of the Princeton mathematics building, where professors and students played chess and Go in the afternoons, John Nash presented a new game. The board is a rhombus covered with hexagonal cells, like a honeycomb. Two players, red and blue, take turns placing pieces, one at a time, on empty cells. The pieces never move. Red wants to form a chain of red pieces joining the top edge to the bottom one; blue, a blue chain joining the left edge to the right one.

October 14, 2026 Read

Pascal's triangle: build it by adding

Write a $1$. Below it, two $1$s. From then on, each row starts and ends with $1$, and each number in between is the sum of the two numbers just above it: $$\begin{array}{ccccccccccc} &&&&&1&&&&&\\ &&&&1&&1&&&&\\ &&&1&&2&&1&&&\\ &&1&&3&&3&&1&&\\ &1&&4&&6&&4&&1&\\ 1&&5&&10&&10&&5&&1 \end{array}$$That is all. A rule a child learns in a minute produces one of the richest tables in mathematics. Today it is called Pascal’s triangle, but Pascal was one of the last to find it.

October 11, 2026 Read

The first proposition: the equilateral triangle

On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, Proposition I.1 After the definitions, postulates and common notions, the Elements begin with a small and beautiful problem. You have a segment $AB$. You want to construct, with straightedge and compass only, a triangle whose three sides are equal to $AB$. It is the first of the book’s 465 propositions, and it shows in a few lines how all the others will work.

October 14, 2026 Read

The Nash equilibrium: nobody does better alone

An equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. — John Nash, Proceedings of the National Academy of Sciences (1950) In 1944 John von Neumann and the economist Oskar Morgenstern published Theory of Games and Economic Behavior, a huge book that turned strategic decisions into mathematics: card games, wars, markets. Von Neumann’s great result applied to two-player zero-sum games, in which what one wins the other loses, like chess or poker. But most of life is not like that: in a trade deal, a negotiation or traffic, both sides can win together, or lose together.

October 14, 2026 Read

Combinations: counting without listing

In Brazil’s Mega-Sena lottery you choose 6 numbers out of 60. How many different tickets are there? Listing them all would take a lifetime: there are more than fifty million. But you can count without listing, and Pascal showed how. Choosing without order Start with a small case: in how many ways can you choose 2 people out of a group of 5 to form a committee? If order mattered, there would be $5$ options for the first person and $4$ for the second: $5\cdot4=20$. But on a committee “Ana and Bruno” is the same as “Bruno and Ana”, and every pair has been counted twice. There are $20/2=10$ committees.

October 11, 2026 Read

Mixed strategies: the penalty kick and odds and evens

Every finite game has an equilibrium point. — John Nash, Non-Cooperative Games, doctoral thesis (1950) Some games have no equilibrium at all if each player picks a fixed move. The simplest is odds and evens. Ana calls “evens”, Bruno calls “odds”, and each shows one or two fingers. If the sum is even, Ana wins a point; if it is odd, Bruno wins. Reducing each choice to “one” or “two” fingers, the payoffs are:

October 14, 2026 Read

Pythagoras's theorem, Euclid's way

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. — Euclid, Elements, Proposition I.47 Long before the Greeks, the surveyors of Egypt, who re-marked the fields after each Nile flood, were called rope-stretchers. The story goes that they used a rope with 12 equally spaced knots: stretched into a triangle with sides 3, 4 and 5, it gave a right angle. The Babylonians knew much more: a clay tablet from around 1800 BC, Plimpton 322, lists right triangles with whole-number sides, some of them huge, such as 3,367, 3,456 and 4,825.

October 14, 2026 Read

The Chevalier de Méré's dice and the fair split

Antoine Gombaud, who styled himself the Chevalier de Méré, was a writer at the French court and a passionate gambler. Around 1654 he brought Pascal two questions about bets. Pascal found them so good that he wrote to Pierre de Fermat, and probability theory was born from those letters. About his friend, Pascal remarked to Fermat: He has a very good mind, but he is not a geometer; and that, as you know, is a great defect. — Blaise Pascal, letter to Fermat (29 July 1654)

October 11, 2026 Read