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

18 essays, 30 September 2026 — everything that has been added, in order

A fair coin judged by someone who thinks it is biased. Posterior probabilities of three wrong coin biases over two thousand tosses of a fair coin, for five runs; the bias 0.6 wins every run.

Probability

Certain of a coin that is not there

Bayes' theorem is exact, and it is only as good as the list of hypotheses it is given. Hand it a list that leaves out the truth and it does not hesitate: it becomes certain of the entry that is least wrong, in a precise sense — the one closest in Kullback–Leibler divergence — and if two entries are equally wrong it never settles at all. Hand it a model that assumes independence where there is none, and its intervals shrink as fast as they would for honest data while covering the truth less and less often.

6 figures

The Flint Hills series, summed to ten million terms. A staircase of partial sums on a logarithmic scale of n, jumping at 1, 3, 22 and 355 and flat afterwards near 30.3.

Analysis

A series that waits on π

Add 1/(n³ sin² n) for n = 1, 2, 3, … and the terms are mostly tiny, except where n is almost a multiple of π and sin n is almost nought. Ten million terms add to 30.3145, four-fifths of it from the single term at n = 355. Whether the sum is finite depends on how closely fractions can approach π — on a number called its irrationality measure — and the best proof available says only that the measure is below 7.1, where the series needs it below 2.5.

6 figures

The Cantor function inside a funnel of exponent log 2/log 3. The Cantor staircase with two curves of the form plus or minus the distance to one quarter raised to the power log 2 over log 3, forming a funnel the staircase stays inside.

Analysis

The exponent a staircase shares with its set

The Cantor function rises from nought to one on a set of length nought, and it is Hölder continuous with exponent log 2/log 3 — the same number as the dimension of that set. It is not a coincidence: both numbers say that an interval of width r carries mass r to the power 0.6309, and that one inequality proves the dimension and the smoothness at once. Tilt the weights of the construction and the two numbers separate, which shows exactly what the coincidence was measuring.

6 figures

A ball of the square grid with random travel times. A ragged roughly round region of grid cells shaded in bands by the time a signal from the centre first reaches them.

Algebra

The shape a random ball grows into

Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.

6 figures

Which pairs of the square's symmetries commute: 40 of 64. A 8-by-8 grid over the elements of the 8 symmetries of a 4-gon, rows and columns grouped into 5 conjugacy classes, with a filled square wherever the two elements commute: 40 filled squares, and each row's count of filled squares written at its end.

Algebra

Five-eighths of the pairs, and no more

Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.

9 figures

Applied

What the agenda leaves standing

Ask for a rule that settles each question from the votes on that question, follows a unanimous court and never contradicts itself, and search every such rule for three judges. On a ranking of three options, three survive: one dictator per judge. On two premises and their conjunction, seven survive: every rule in which a fixed set of judges must all agree. On a chain of thresholds, a hundred and twenty-nine, majority among them. The difference is not in the rules. It is in which answers force which, and whether that forcing ever runs back.

7 figures

Start anywhere

Eight of 820 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