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, 16 September 2026 — everything that has been added, in order

A matching of 4 and a cover of 4. A bipartite graph with its largest matching drawn thick and a smallest set of vertices meeting every edge ringed, the two having the same size.

Discrete

What the search has when it fails

A largest matching is easy to find and hard to certify: the claim that nothing larger exists is a claim about every arrangement not tried. The certificate turns out to be free — it is the wreckage of the search that failed.

7 figures

A graph whose matching misses 2 of its 10 vertices. A graph with its largest matching drawn thick, a set of vertices ringed, and the pieces left when that set is deleted marked by whether they hold an odd number of vertices.

Discrete

The piece that cannot pair off

Take the sides away and the obstruction to a matching changes character completely. It is no longer a shortage of partners; it is a parity, and the quantity that measures it counts pieces of odd size rather than vertices of any size.

6 figures

The payoffs repetition makes available in the prisoner's dilemma. A plot of the two choosers' average payoffs, with the stage game's four cells marked, their convex hull drawn, the two minmax values shown as lines, and the region above both shaded.

Applied

Patience instead of a contract

Commitment had to assume an announcement binds. Play the same game again tomorrow and the assumption is unnecessary — the future does the binding. What it costs is that nearly every outcome becomes an equilibrium, so a theory that could not choose between two now cannot choose between infinitely many.

7 figures

The band two equilibria occupy, and the point a noisy reading leaves. A line of values of the payoff parameter with three regions marked — staying out dominant, both actions equilibria, investing dominant — and a single threshold inside the middle region.

Applied

The reading that is almost right

Every account of simultaneous choice so far has assumed the payoffs are known to both choosers and known to be known. Replace that with each chooser seeing a private reading off by a little, and a band of equilibria closes to a single point — so the assumption nobody states decides the answer.

6 figures

n / 2ⁿ held under a geometric series. A bar for each term of the series n / 2ⁿ with a decaying geometric curve above them, the curve lying above every bar from term 2 onwards.

Analysis

The series everything else is measured against

A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.

5 figures

Runs of doubling length in 1/n^2, and the geometric series that bounds them. A bar for the total of each run of terms of 1/n to the 2, with an outlined bar above it for the bound obtained by replacing every term in the run with its largest, the bounds forming a geometric series.

Analysis

The repair at the boundary

Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.

5 figures

Start anywhere

Eight of 517 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