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, 12 November 2026 — everything that has been added, in order

Two sign sequences laid on themselves. Length 41: random signs (seed 41) with E = 552 and merit factor 1.523; the rotated Legendre sequence of 41 with E = 176 and merit factor 4.776.

Algebra

A quarter turn quiets the echoes

Lay a sequence of signs on a shifted copy of itself and add up the squares of the overlaps. Random signs leave the total near n²/2, and the squares modulo a prime, started a quarter of the way along, cut it to a twelfth. Exhaustive search finds far quieter short sequences, and no infinite family is proved to do better than 6.342.

6 figures

Algebra

Multiply until every sign is positive

A form with a negative coefficient can still be positive wherever its variables are, and proving it usually takes an argument. Pólya found a certificate that needs none: multiply by a power of x + y and wait until every coefficient is positive. The power needed is finite exactly when the form never touches zero, and it grows like the reciprocal of how close it comes.

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Rebuilding a ball at its own centre. Fourier inversion of the indicator of the unit segment, disc and ball, keeping frequencies below R, evaluated at the centre: at R = 8 the values are 0.9873, 0.9206, 0.9873; in three dimensions at R = 7.75 it is 1.6369.

Analysis

A ball rebuilt everywhere but its centre

Rebuild a solid ball from its frequencies, keeping those below a cut-off, and read the result at the centre, where the ball is perfectly smooth. In the plane the answer settles on 1. In space it swings between 0.36 and 1.64 for ever, because the whole surface of the ball is the same distance from the centre and its jump arrives there in step.

6 figures

The curve whose bending grows with its length. Euler's spiral, curvature equal to arc length, for s from −7 to 7, winding into ±(0.88623, 0.88623).

Analysis

The curve that bends as far as it has gone

Say how sharply a curve bends at each distance along it, and the curve is fixed. Bending in proportion to distance gives Euler's spiral, whose arms are infinitely long and end a finite distance away. Bending that repeats decides at once whether the curve ever closes, and taken in steps the spiral is a Gauss sum.

6 figures

Three widows and the Mishna's table. The Mishna's division of estates of 100, 200 and 300 among claims of 100, 200 and 300: 33⅓ each; 50, 75, 75; 50, 100, 150.

Applied

The debts the Mishna divided

A man dies owing his three wives 100, 200 and 300. The Mishna divides an estate of 100 equally, of 300 in proportion, and of 200 as 50, 75 and 75, and for nearly two thousand years nobody could say why. The table is the nucleolus of a cooperative game, and the only division in which every pair splits its share as two people split a contested garment.

6 figures

Six goods, three people, and no envy. Three people's random values for six goods, with an envy-free allocation shaded: Ada values the bundles at 0.86, 0.85, 0.34; Ben values the bundles at 0.64, 1.83, 0.75; Cy values the bundles at 1.05, 1.24, 1.29.

Applied

Twice as many goods as people

Indivisible goods with random values cannot always be shared so that nobody envies anybody. With one good each it works only when everyone's favourite is different; with twice as many goods as people it almost always works. Deciding each case exactly shows the chance climbing steeply in between — and a simple draft getting there only when the goods divide evenly.

6 figures

Start anywhere

Eight of 1188 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