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

44 essays, 11 October 2026 — everything that has been added, in order

Algebra

Three coordinates and a fourth power

The Heisenberg group is made of triples of whole numbers, and two moves generate all of it. Count the triples within r moves of the start and the count grows like r⁴, not r³: at forty-four moves there are 1,600,703 of them, against 117,569 for the cubic lattice. The fourth power comes from one coordinate that the moves reach by enclosing area, so that height costs only the square root of itself — and seen from far away, the ball of reachable triples is a curved solid with a dimple at each pole.

6 figures

Every magic square of order three is a point in a space of three. Basis J, U = (1,-1,0,-1,0,1,0,1,-1), V = (0,-1,1,1,0,-1,-1,1,0); Lo Shu = 5J − 3U + V; dimension of the order-3 magic space 3.

Algebra

The magic squares are a space

A magic square is a grid whose rows, columns and diagonals share one sum. The conditions are linear, so the squares form a space: three-dimensional for order three, eight-dimensional for order four, n² − 2n in general. Inside the space of order three the eight classical squares are eight points with coordinates ±1 and ±3. Of the 880 squares of order four, 640 are singular matrices, and the search finds the reason: a symmetry of the rows and columns that carries every number to the one it adds to seventeen with.

6 figures

One length and three ways of guessing it. Perimeter of the ellipse with semi-axes 1, b: b=1 6.283185307, b=0.5 4.844224110, b=0.1 4.063974180, b→0 4; guesses at b=0.5: Kepler 4.442883, π(a+b) 4.712389, Euler 4.967294.

Analysis

The length round an ellipse

An ellipse has an area anyone can write down, πab, and a perimeter nobody can: it is an integral with no expression in the functions of school mathematics. Kepler guessed it, Euler guessed better, and in 1914 Ramanujan wrote down a formula whose error falls like the fifth power of how far the ellipse is from a circle — off by four hundredths of a per cent even for a segment. Gauss had already found the way to compute it exactly, doubling the correct digits at every step.

5 figures

Forty terms, two digits or thirteen. Errors after 40 terms: series -6.249e-3, Aitken -1.053e-6, Euler -1.250e-13; Wynn after 17 terms 1.250e-13.

Analysis

Turning a slow series geometric

Leibniz's series 1 − 1/3 + 1/5 − 1/7 + … adds up to π/4, and it takes five billion terms to give ten decimal places. Euler found a rule that rewrites it, using exactly the same terms, as a series whose terms halve at every step — and forty of those give thirteen places. Aitken's and Shanks's later rules do better still. None of them works on every slow series, and the reason is that each one is a guess about the shape of the error.

5 figures

Where sellers stand when nobody wants to move. 2 sellers: 1 equilibria (e.g. 12,12 of 24); 3 sellers: 0 equilibria; 4 sellers: 3 equilibria (e.g. 6,6,17,18 of 24); 5 sellers: 4 equilibria (e.g. 4,4,12,19,20 of 24); 6 sellers: 25 equilibria (e.g. 3,3,9,14,20,21 of 24).

Applied

Sellers on a street stand back to back

Put two ice-cream sellers on a beach and they end up side by side in the middle, though customers would walk half as far if they stood at the quarter marks. Three sellers never settle at all: whoever is squeezed in the middle leaps outside, for ever. Four stand in pairs, five in pairs with one alone in the centre, six in endlessly many ways. Let them set prices too, with travel costing the square of the distance, and the conclusion reverses: the two sellers move as far apart as the beach allows.

6 figures

A party that loses every issue wins the vote. Groups A: 110, B: 101, C: 011, D: 000, E: 000; X wins 3–2; issues won by X: 0 of 3.

Applied

A party that loses every issue can win

Two parties take opposite sides on three issues, and every voter backs the party they agree with on more of them. The party that wins the election can be on the losing side of all three issues — in a fifth of random electorates it loses most of them, and in about one in a hundred it loses every one. A majority of voters can even find themselves outvoted on most of the questions decided. Only when issues are settled by three-to-one majorities on average is that ruled out.

5 figures

Start anywhere

Eight of 1012 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