Geometry

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 8, 2026 Read

The hypotheses on which geometry is founded

The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. — Bernhard Riemann, On the Hypotheses which lie at the Bases of Geometry (1854) On 10 June 1854, Riemann presented to the Göttingen faculty the lecture On the Hypotheses which lie at the Bases of Geometry. He wrote it for an audience that included philosophers and historians, almost without formulas. The text was only published in 1868, after his death, and it changed geometry.

October 31, 2026 Read

The Nash bargaining solution

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. — John Nash, The Bargaining Problem (1950) In 1948, still an undergraduate at the Carnegie Institute of Technology, Nash took a course in international economics, the only economics course of his life. He came out of it with a question: when two people negotiate, how do they share the gains from a deal? Economists thought the answer depended on each side’s psychology, skill and bluffing, and that there was nothing to say in general. Nash showed that, if the deal must satisfy a few reasonable requirements, there is only one possible answer. The paper, written before he arrived at Princeton, was published in 1950 in the journal Econometrica.

October 15, 2026 Read

Algebra done with squares and rectangles

If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. — Euclid, Elements, Proposition II.4 The Greeks had no letters for unknowns, no plus or minus signs, no equals sign. When they wanted to speak of $a\cdot b$, they spoke of a rectangle with sides $a$ and $b$; when they wanted $a^2$, they spoke of a square with side $a$. The words “square” and “cube” that we use for powers come from this. Book II of the Elements is made entirely in this way: it consists of identities of algebra, stated and proved as facts about areas. Historians call it geometric algebra.

October 12, 2026 Read

The Pascaline: a sum made of gears

In 1640 Étienne Pascal was appointed tax commissioner for Normandy. The job meant adding up column after column of figures late into the night, and his son Blaise, then 17, helped. To make matters worse, French money was not decimal: $$1\ \text{livre}=20\ \text{sols},\qquad 1\ \text{sol}=12\ \text{deniers}.$$Adding $3$ livres, $15$ sols and $8$ deniers to $2$ livres, $7$ sols and $9$ deniers needs three kinds of carry: deniers: $8+9=17$, which is $1$ sol and $5$ deniers; sols: $15+7+1=23$, which is $1$ livre and $3$ sols; livres: $3+2+1=6$. Result: $6$ livres, $3$ sols and $5$ deniers. Do that a thousand times a night and you will make mistakes. In 1642 Blaise began building a machine that would not.

October 8, 2026 Read

The sphere and the cylinder

Greek geometers knew how to compute the volume of prisms, pyramids, cylinders and cones. The sphere resisted. In the two-book treatise On the Sphere and Cylinder, Archimedes solved the problem and found relations of surprising simplicity. The theorems Call a great circle the circle that passes through the centre of the sphere, like the equator. Archimedes proved that: the surface of the sphere is four times the area of the great circle: $S = 4\pi r^2$; the volume of the sphere is four times that of a cone whose base is the great circle and whose height is the radius: $V = 4 \cdot \frac{1}{3}\pi r^2 \cdot r = \frac{4}{3}\pi r^3$. And from this he drew the consequence that delighted him most. Consider the cylinder that exactly encloses the sphere, with radius $r$ and height $2r$. Its volume is $\pi r^2 \cdot 2r = 2\pi r^3$, and its total surface, counting both lids, is $2\pi r \cdot 2r + 2\pi r^2 = 6\pi r^2$. So

October 29, 2026 Read

Rotating in space with quaternions

Complex numbers of modulus 1 rotate the plane. Hamilton hoped his numbers would do the same in space, and they do, but in a subtler way than he first imagined. The recipe Write a point of space $(x, y, z)$ as the pure quaternion $v = xi + yj + zk$. To rotate by an angle $\theta$ about an axis given by a unit vector $u = u_1 i + u_2 j + u_3 k$, take

October 27, 2026 Read

Crumpling without stretching: the embedding theorem

Take a sheet of paper. You can roll it into a cylinder without tearing or stretching it: distances measured on the sheet, with a flexible ruler, stay the same. But you cannot wrap a ball with it without making folds. The sheet and the sphere have different intrinsic geometries, and Gauss proved in 1827 that this difference, the curvature, does not change when the surface is bent without stretching. In the nineteenth century Bernhard Riemann had an even bolder idea: to describe a surface, or a space of any dimension, by its internal measurements alone, without drawing it anywhere. All it takes is a rule, the metric, that says how to measure the length of any curve. This is how general relativity describes spacetime: from inside, without a larger space around it.

October 15, 2026 Read

Euclid's algorithm: the greatest common divisor

The Euclidean algorithm is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. — Donald Knuth, The Art of Computer Programming, vol. 2 (1969) You have two rods, one 252 cm long and the other 105 cm, and you want the longest ruler that measures both exactly, a whole number of times. In today’s language you want the greatest common divisor, $\gcd(252,105)$. You could list all the divisors of each number and look for the largest common one, but with big numbers that is impractical. In Book VII of the Elements, Euclid teaches a much faster way.

October 12, 2026 Read

The mystic hexagram: six points on a conic

In February 1640 a sheet printed on one side only, like a poster, circulated in Paris under the title Essay on conics. It had no proofs, only definitions, three lemmas and a list of results. The author was 16 years old and his name was Blaise Pascal. When Father Mersenne showed the sheet to Descartes, the reaction was suspicion: Descartes thought the young man had taken everything from Girard Desargues, the architect and geometer Pascal admired. Desargues’s influence was real, and Pascal himself acknowledged it. But the central result on the sheet was new.

October 8, 2026 Read

The heat equation and the regularity of De Giorgi and Nash

In 1900, at the International Congress of Mathematicians in Paris, David Hilbert presented a list of 23 problems that, in his view, would guide twentieth-century mathematics. Problem 19 asked, in simple terms: are the solutions of the “well-behaved” problems of the calculus of variations always smooth functions? Half a century later, the question was still open for an important class of equations. In 1956 Nash heard about it from a colleague and decided to attack it.

October 15, 2026 Read

The primes never run out

Prime numbers are more than any assigned multitude of prime numbers. — Euclid, Elements, Proposition IX.20 A prime number is an integer greater than 1 that can only be divided by 1 and by itself: 2, 3, 5, 7, 11, 13, … All the others, such as $12=2\cdot2\cdot3$, are products of primes. The primes are the atoms of arithmetic. The further you go, the rarer they become. Between 1 and 100 there are 25 primes; between 1,000,001 and 1,000,100, only 6. Do they run out at some point? Euclid showed that they do not, with a four-line argument that the English mathematician G. H. Hardy considered one of the most beautiful in all of mathematics.

October 12, 2026 Read