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

10 essays, 14 November 2026 — everything that has been added, in order

Hilbert's polygons, stage by stage. Hilbert's polygons at stages 1, 2, 3 and 6, each visiting the centres of 4, 16, 64 and 4,096 squares in order, from the bottom-left corner to the bottom-right.

Analysis

A path that fills a square

A pen moving continuously for one unit of time can pass through every point of a square. Hilbert's polygons, each threading four times as many squares as the last, move no point more than 0.4507 × 2⁻ⁿ from where the previous one put it — and that one estimate, through the theorem that uniform limits stay continuous, is the whole proof. What the limit pays is exactness of a kind: it repeats some points four times, and its points can never be closer than the square root of their parameters allows.

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Seven numbers whose 128 subset sums all differ. The 128 subset sums of 20, 31, 37, 40, 42, 43, 44, spread from 0 to 257 with no two equal, above those of the powers of two 1 to 64.

Discrete

Numbers whose subset sums all differ

The powers of two 1, 2, 4, …, 64 have 128 subset sums and no two are equal. So do 20, 31, 37, 40, 42, 43 and 44, whose largest member is smaller, and an exhaustive search shows no seven numbers do it with less. A pigeonhole on the sums says the largest member must grow like 2ⁿ/n; a variance argument improves that to 2ⁿ/√n; the best known sets sit near 0.22 × 2ⁿ. Whether the truth is a constant times 2ⁿ is Erdős's five-hundred-dollar question, and the pigeonhole cannot reach it.

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Dynamics

Fifty-four numbers funnel into 6174

Take four digits, not all the same, arrange them largest first and smallest first, and subtract. Repeat. Every one of the 9,990 starts reaches 6174, and none needs more than seven steps. The reason is that the first subtraction forgets almost everything: its result depends only on two differences of digits, so ten thousand starts collapse to fifty-four values, and a map on fifty-four points can be drawn whole. Change the number of digits or the base, and the same collapse ends in cycles instead.

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Every prime's Fibonacci period lies on one of two lines or below. The period of the Fibonacci numbers modulo each prime p up to 1,000: for p ≡ ±1 mod 5 it divides p − 1, for p ≡ ±2 mod 5 it divides 2(p + 1); 116 of the 165 primes from 7 to 997 lie on their line.

Algebra

The period the golden ratio sets

The Fibonacci numbers, taken modulo any whole number, eventually repeat — the last digits every sixty terms. At a prime the period is not arbitrary: it divides p − 1 when the prime ends in 1 or 9, and 2(p + 1) when it ends in 3 or 7. The reason is where the golden ratio lives. Modulo the first kind of prime it is an ordinary remainder; modulo the second it has to be adjoined, as √5 is adjoined to the rationals, and in that larger field the p-th power swaps φ with its conjugate.

6 figures

A list of sentences each denying all the later ones. Fourteen sentences, each saying that every later sentence is false, drawn as a grid of which sentences each one mentions — every cell above the diagonal and none on or below it — beside the only consistent assignment of truth values, in which the last sentence alone is true.

Logic

A liar without a loop

Every paradox of truth seemed to need a sentence that talks about itself, directly or round a loop. Stephen Yablo found one that does not: an infinite list in which each sentence says that all the later ones are false. Cut it off anywhere and exactly one assignment of truth values works — all false but the last; keep it infinite and none does. A finite system with no loop always has exactly one, so the paradox lives in the one place a finite picture cannot draw: a list with no last sentence.

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Number

The largest piece that still fits

Fill a unit with unit fractions, always taking the largest one that still fits: 1/2, 1/3, 1/7, 1/43, 1/1807. The denominators are Sylvester's sequence, each one more than the product of all before it, and after k of them what is left is exactly one over the next term minus one. Curtiss proved in 1922 that no k unit fractions get closer to 1 from below; an exact search confirms it to six terms. Applied to almost any other fraction, the same greedy rule is wasteful — 8/97 ends with a denominator of 150 digits.

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Start anywhere

Eight of 1198 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.

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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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