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

Never negative is a wider thing than a sum of squares. For x^4 y^2 + x^2 y^4 - c x^2 y^2 + 1: sum of squares exactly for c at most 0, non-negative exactly for c at most 3, least value 1 - c^3/27 for positive c.

Algebra

Never negative and no sum of squares

Any quadratic that never goes below nought can be completed into squares, and the squares are the proof. In 1888 Hilbert showed the same habit must fail at higher degree without exhibiting a single failure; the first one was written down 79 years later. Motzkin's x⁴y² + x²y⁴ − 3x²y² + 1 touches nought four times and never crosses it, and one cell of a four-by-four table shows that no squares can make it. Multiply it by 1 + x² + y² and five squares appear.

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The 7 × 7 Fourier matrix as arrows. Entries of the order 7 discrete Fourier matrix as unit arrows, with the minor on rows 1,3,4 and columns 0,2,5 highlighted; its determinant has modulus 2.645751.

Algebra

A prime leaves no room for a comb

Take the table of powers of a root of unity, pick any rows and any equal number of columns, and take the determinant. At order 12 about one such determinant in six is exactly nought. At order 11, among 184,756 of them, none is, and in 1926 Chebotarev proved that none ever is at a prime. It means a signal on p places and its transform together occupy at least p + 1, while a comb on twelve places and its transform take 4 and 3.

6 figures

Which powers can reach x. Square-integral distance from x to the span of chosen powers, by count of powers: even 5.607e-9; primes 1.165e-4; squares 9.053e-2; powers 3.497e-2; limit for squares 0.090525.

Analysis

Which powers of x are enough

Every continuous function on an interval is nearly a polynomial, and it seems that all the powers of x should be needed for that. They are not. The even powers will do, and so will the primes, though the primes run out at a rate of one number in ln n. The squares will not: however many are combined, they stay 0.0905 away from x itself. The dividing line, found by Müntz in 1914, is whether the reciprocals of the powers add up to infinity.

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Four walls that stop every line through a square. Opaque barriers for the unit square, all verified over 1440 directions: three sides 3.00000; two diagonals 2.82843; Steiner tree on all four 2.73205; three corners and a stub 2.63896.

Analysis

The shortest wall that stops every line

A straight cable runs somewhere under a square field, and a trench has to find it for certain. Three sides of the field will do, at length 3. Two diagonals cost 2.83, and a Steiner tree 2.73. The best trench known has length 2.6390 and comes in two pieces that do not touch. Averaging over lines proves nothing shorter than 2 can work, and in over a century the gap between the two numbers has hardly moved.

6 figures

An arrival order that punishes a plain rule. Upper triangular instance with 6 workers and 6 jobs: a rule that takes the last suitable job matches 3, the best matching in hindsight 6.

Applied

Matching workers who will not wait

Workers arrive one at a time, each says which jobs would suit, and each must be given a job at once or lost. Any sensible rule matches half of what hindsight could. Choosing at random at each arrival can still fall to half. Draw one random order of the jobs before anyone arrives and stick to it, and the share cannot fall below 1 − 1/e, 63.2%. The same number is a ceiling for every rule, since on one shuffled triangle of a market they all score alike.

6 figures

Whose vote counts in a two-tier election. Relative chance a voter decides a two-tier vote, districts 9,25,49,81,121,169,225. one vote per district: 2.319 1.367 0.971 0.754 0.616 0.521 0.452. weight in proportion to voters: 0.501 0.296 0.630 0.815 1.466 1.240 2.051. weight in proportion to √voters: 1.021 1.003 0.998 0.996 0.995 0.994 0.994.

Applied

A vote worth a square root

Give each country in a council votes in proportion to its people, and a citizen of the largest has about fifteen times the say of a citizen of the smallest. Give each country one vote and the same ratio runs the other way. Lionel Penrose saw in 1946 that the fair weight is the square root of the population, because one voter's chance of deciding an election falls as one over the square root of the crowd. Computed for twenty-seven states, the best quota is 61.5%, and it holds the ratio to 1.006.

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Start anywhere

Eight of 1144 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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