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

Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

Algebra

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

7 figures

36 3-tuples with product e, and the 3 that no turn moves. Every ordered choice of 3 elements of the 6 symmetries of a 3-gon whose product is the identity, in cards grouped by cyclic turning. 3 cards hold a single tuple repeating one element; the other 11 hold 3 each.

Algebra

Necklaces made of symmetries

Lagrange's theorem says a subgroup's size divides the group's, and the converse is false. One piece of the converse is true: every prime that divides the size is the order of some element. The proof threads the group's own elements onto a necklace whose product is nothing, turns it, and counts — the argument that proved Fermat's little theorem with beads, with the beads replaced by motions.

6 figures $Thomae's function as a limit of tents, at n = 3 and 8. Continuous functions built from narrow triangles over the fractions, drawn at two stages, with the limit shown as a dot at height 1/q over every fraction p/q: a function continuous at the irrationals and discontinuous at the rationals.$ Analysis

A limit can jump at every fraction

A sequence of continuous functions can settle, point by point, on a function that is discontinuous at every rational number. It cannot settle on one that is discontinuous everywhere — the indicator of the rationals needs two limits in a row, and Riemann's integral cannot follow the second. The line between the two is Baire's theorem, and it measures smallness by gaps rather than by length.

5 figures

xⁿ is uniform off a strip, with N = 14, 29, 59 for 3 strips. The functions x to the n on the unit interval with a band of half-width 0.05 around zero. For each of 3 strips next to 1, the member at which every later one stays in the band away from the strip.

Analysis

Uniform, except on a small set

The functions xⁿ settle on their limit at every point and never uniformly — the trouble is all in a strip next to 1. Throw the strip away and the convergence is uniform on what is left, however thin the strip. Egorov proved that this always happens on an interval, Lusin proved the matching fact about a single function, and a bump sliding off along the whole line shows why both need a set of finite length to start from.

6 figures

Applied

The nearest consistent verdict

When a court's majorities contradict each other, one repair is to announce the consistent verdict that disagrees with the judges least. It treats the premises and the conclusion alike, which neither of the two standard procedures does. On the classic case it returns a three-way tie; on five judges, with every question weighted equally, it never returns a single answer on a troubled profile at all — and what breaks the tie is a decision about which question matters more.

5 figures

Which agendas majority can vote on safely. A table of 7 agendas with the size of their largest minimally inconsistent set and the count of inconsistent majority outcomes over all profiles of three and five judges.

Applied

Agendas that cannot contradict themselves

A court voting on two unconnected questions never contradicts itself, and neither does one voting on a chain of thresholds. A court voting on two premises and their conjunction sometimes does. What separates them is the size of the smallest sets of judgements that cannot all be true: pairs are harmless, because two majorities always share a judge, and triples are not. The same count says exactly how large a supermajority has to be to stay consistent on any agenda.

6 figures

Start anywhere

Eight of 605 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