Mathematics, in pictures.

Some mathematical ideas are hard because they are genuinely hard. Others are hard only because nobody drew them properly. This is a collection of essays about the second kind — one idea at a time, illustrated to the point where the argument becomes visible.

Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle. — Fig. 1 Pascal’s triangle with the odd numbers shaded and the even ones left blank. Nothing here was designed to make a pattern; the pattern is a consequence of the arithmetic. See Pascal’s triangle, two colours.

Recently added

22 essays, 9 October 2026 — everything that has been added, in order

Every cubic x³ + bx + c up to 30, sorted by its Galois group. A 61 by 61 grid of cubics x³ + bx + c: 309 reducible, lying on lines c = −r³ − rb; 20 cyclic with square discriminant; 3392 with group S3.

Algebra

Twenty-seven million cubics, sorted by their symmetry

Write down every cubic x³ + ax² + bx + c with coefficients up to 150 and sort each by its Galois group. 98.7% have all six symmetries of three roots. The rest are reducible at a rate of fifteen in every H², or cyclic at five in every H to the three-halves — and the cyclic ones turn up four times as often as chance would allow, because discriminants are not random numbers.

6 figures

Even quartics never have the full group, and one step away almost all do. Even quartics x^4 + bx^2 + d to 20: reducible 199, S4 0, A4 0, D4 1338, C4 20, V4 124. With +x: reducible 157, S4 1524, A4 0, D4 0, C4 0, V4 0.

Algebra

The quartics whose roots stay paired

A quartic can have five different Galois groups, and in a box of nearly fourteen million quartics they do not come in order of size. The eight-element group D₄ outnumbers the twelve-element A₄ thirty-five to one. Every even quartic has a smaller group than S₄, and moving one step off that plane restores the full group almost everywhere.

6 figures

The sum of all factorials in base 5: digits that stop changing. Partial sums of n! for N = 0 to 21, last 14 base-5 digits; the 5-adic limit ends …13010230042224.

Analysis

A sum of factorials that converges at every prime

1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.

6 figures

Riemann's function on one period, with its one kind of smooth point. R(x) = Σ sin(n²x)/n² on [0, 2π]; R(π) = 0 with derivative −1/2; R(π/2) = 1.2336, a cusp.

Analysis

The slope a Gauss sum leaves behind

Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.

6 figures

Three random games and the cells where best replies meet. Three random 5 by 5 games with best replies marked; pure equilibria (cells best for both players): 0, 1 and 2.

Applied

A third of random games have no pure equilibrium

Fill a game's payoff table with random numbers and ask whether some cell is each player's best reply to the other. Exactly one cell is expected to be, at every size, and the chance that none is climbs from one in eight to 1/e — the same constant that counts the shuffles in which nobody gets their own hat back, and for the same reason.

6 figures

Where the equilibria of a random four-by-four game are found. 7 equilibria with supports (1, 1), (3, 2), (4, 3), (13, 12), (14, 13), (34, 23), (134, 123).

Applied

Every random game has an odd number of equilibria

Find every equilibrium of eleven thousand random games — pure and mixed, by trying every pair of strategy sets the players might mix over — and the totals are 1, 3, 5, 7, … and never an even number. The count is a sum of signs: equilibria of index +1 outnumber those of −1 by exactly one in every game. The average grows about 28% with each strategy added, and searching for the game with the most turns up the coordination game's 2ⁿ − 1, a pattern that holds only up to five strategies.

6 figures

Start anywhere

Eight of 946 essays — the whole collection is a click away, or search it.

The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Geometry

Two squares, four triangles, and no algebra

The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.

7 figures

Odd numbers as square shells. Nested L-shaped shells of 1, 3, 5 … 11 cells stack into a 6 by 6 square.

Geometry

Every square is a stack of odd numbers

Add up the odd numbers in order and the running totals are 1, 4, 9, 16, 25. This is not a coincidence, and the reason fits in a single picture.

7 figures

Euclid's algorithm on a 34 by 13 rectangle. The rectangle is tiled by peeling off the largest square that fits, again and again, until nothing is left.

Geometry

The oldest algorithm, drawn as a tiling

Euclid's method for finding a greatest common divisor is usually presented as a loop. It is also a way of tiling a rectangle with squares, and the tiling explains why it works.

7 figures

The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale.

Geometry

Why the list of perfect solids stops at five

There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.

8 figures

Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

Geometry

One cone, four curves

The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.

6 figures

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

Analysis

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

7 figures

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Analysis

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

7 figures

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

Analysis

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

7 figures