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

21 essays, 8 September 2026 — everything that has been added, in order

A 2×3 sliding puzzle: 360 arrangements of 720 can be reached. Two arrangements of a small sliding puzzle side by side, the solved one and the one with two tiles exchanged, with the count of positions reachable by sliding found by walking every move.

Algebra

The puzzle that is exactly half solvable

A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.

6 figures

The character table of the permutations of 4 places, computed from traces. A table with one row per irreducible representation and one column per conjugacy class, giving the trace of the matrix each representation assigns, with the dimension column marked.

Algebra

When the label may be a matrix

A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

6 figures

Every point of a hull, as a mixture of three of 11 points. A scatter of points with its convex hull outlined, and several interior points each shown inside a triangle of three of the scattered points, found by trying every triple.

Analysis

Three points, however many there are

A point inside the hull of a thousand points is inside the hull of three of them. Any four points split into two groups whose hulls meet. And a family of convex sets, every three of which have a common point, has one common to all — three, in each case, being one more than the dimension.

6 figures

Two convex sets 1.50 apart, and the line that separates them. Two convex polygons with a straight line drawn between them, together with the shortest segment joining the two sets, whose perpendicular bisector the line is.

Analysis

A wall between two bodies

Two convex sets that do not meet can be told apart by a single straight line, and the line is a certificate — one object, checkable in a moment, proving something about every point of both. Remove convexity from either and no line exists, which is what the hypothesis was for.

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Applied

Four ways out, and what each costs

An impossibility theorem lists conditions and says no rule has them all. That leaves exactly as many escapes as there are conditions, each of them a real institution — a dictator, a two-stage procedure, a supermajority, a restricted agenda — and each escape's price can be counted rather than argued about.

6 figures

The largest code at each length, distance 3, against four bounds. A table with one row per word length, giving the exact size of the largest code of that length at the stated minimum distance and the values of the Singleton, Hamming, Plotkin and Gilbert-Varshamov bounds.

Computation

The best a code can be

A code is a set of words chosen far apart, and every construction answers "here is one" rather than "here is the best". The best can be computed at small lengths, and putting four classical bounds beside the exact answer shows which of them is doing the work and where none of them is.

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Start anywhere

Eight of 396 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.

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