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, 29 October 2026 — everything that has been added, in order

Algebra

A second empty square

With one empty square a sliding tray can reach exactly half its arrangements, and the half it misses is guarded by a sign. Leave a second square empty and the guard disappears — every arrangement is reachable, two tiles can be exchanged in four moves, and the far end of the three-by-three tray comes in from thirty-one moves to twenty-four. Searching every tray with every number of holes shows what each extra square of room is worth.

6 figures

The most rotations that can be kept a given angle apart. Linear programming bound on the number of rotations pairwise at least θ apart, θ from 54° to 180°, log scale. Groups: four-group 4 at 180° (bound 4.00); tetrahedron 12 at 120° (bound 12.00); cube 24 at 90° (bound 28.94); icosahedron 60 at 72° (bound 60.00).

Algebra

Sixty rotations as far apart as rotations can be

The sixty rotations of the icosahedron are each at least 72 degrees from every other. A single polynomial, with coefficients that are all positive and values that are nowhere positive where it matters, proves that no sixty rotations do better and no sixty-one do as well. The same calculation proves the tetrahedron's twelve best at 120 degrees, and leaves the cube's twenty-four short of what it allows.

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Three series for Catalan's constant and how fast they arrive. Log10 errors after 1 to 60 terms: plain series -4.46, Euler transform -18.53, Ramanujan series -39.46.

Analysis

The sum after Leibniz's, made geometric

Leibniz's series adds the reciprocals of the odd numbers with alternating signs and gives π/4. Square the odd numbers and the same pattern gives Catalan's constant, 0.915965…, about which almost nothing is known — not even whether it is a fraction. A formula of Ramanujan's turns its slow series into one whose terms shrink by a factor of four, which makes a thousand decimals cheap, and the decimals and continued fraction then say exactly how much digits can settle and where they stop.

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The cost of running a computation backwards with a few stored states. Minimal recomputation to reverse chains of length up to 200 with 1, 2, 3, 4, 6 slots; at length 200: 1: 19900, 2: 2470, 3: 1305, 4: 948, 6: 680.

Analysis

Running a computation backwards with little memory

The reverse sweep that gives a whole gradient for the price of a few evaluations needs the computation's intermediate states in reverse order, and storing them all can cost more memory than the machine has. Keep only a few and recompute the rest, and the cheapest schedule has an exact answer found by Andreas Griewank in 1992: with c stored states and no step run more than r times, the longest chain that can be reversed is the binomial coefficient C(c + r, c). Pascal's triangle is the table of what memory buys.

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Applied

Sellers in a square town

On a street, two sellers settle back to back in the middle, three never settle, and four settle in pairs at the quarter points. Spread the customers over a square town instead and the first fact survives while the third does not: the street's pairs leave a seller half a per cent of the town to gain by stepping off the line. Computing every seller's market exactly and searching for an arrangement where nobody wants to move finds one only for two sellers.

6 figures

An optimal way to divide 9 soldiers among three battlefields. Colonel Blotto with 9 soldiers each on three fields: 55 allocations, optimal mixture on 7: 6-3-0:0.143 5-0-4:0.143 4-4-1:0.143 3-1-5:0.143 2-5-2:0.143 1-2-6:0.143 0-6-3:0.143.

Applied

Two colonels and a linear programme

Two colonels divide their soldiers among three battlefields at once, and whoever sends more to a field wins it. Every division can be beaten by some other, so neither colonel has a safe plan — until they are allowed to choose at random. The best random choice is a linear programme, the opponent's best is its dual, and the theorem that the two meet is von Neumann's. With nine soldiers each, one optimal plan uses only seven divisions, and on every field it sends each number from nought to six exactly once.

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Start anywhere

Eight of 1122 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.

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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.

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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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