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.
Recently added
22 essays, 21 October 2026 — everything that has been added, in order
Algebra
The nearest point is not the best guess
Measure ten unrelated quantities once each, with noise, and report the measurements. Nothing seems better. Yet pulling all ten towards nought by an amount the data choose gives a smaller total error whatever the true values are — in three dimensions or more, and never in one or two. The reason is a right angle: in many dimensions noise is nearly perpendicular to the truth, so a measurement is always too long, and the best correction is the shadow of the truth on it.
Algebra
How often a table of random signs is flat
Fill a square table with random plus and minus signs and ask whether its determinant is nought. For four rows the answer is yes two times in three. The chance then falls, and since 2020 it has been known to fall at exactly the rate of the dullest accident, two rows or columns equal or opposite. The conjecture is that this accident is, in the end, the whole story. Every size a computer can reach says otherwise, and the theory says the computer has not gone far enough.
Analysis
Where to take the samples
Every numerical integral is a weighted sum of samples of the curve. Spacing the samples evenly gives errors that shrink like a power of their number; choosing where they go, as Gauss did in 1814, makes six samples exact for every polynomial up to degree eleven and turns the power into a geometric rate. The surprise is that the humble trapezoid rule, evenly spaced and exact only for straight lines, does the same thing on any smooth curve that goes once round a circle — and that a single corner defeats every one of them.
Analysis
The sine of one degree
Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.
Applied
A core is a rare thing to find at random
A game has a core when the whole can be shared so that no group would do better on its own, and Bondareva and Shapley turned that into a finite list of checks. For three players there are five checks; for four, forty-one; for five, 1,291. Draw the coalitions' values at random and the list wins: a third of three-player games fail, nineteen four-player games in twenty, and not one five-player game in two thousand passes. The cores that theorems promise come from structure, and structure is exactly what a random game lacks.
Applied
The element the judges are worst at
A court can vote element by element or on the verdict, and the comparison so far assumed judges equally good at every element. Real judges are surer about the law than about a contested fact. Make one element hard and both procedures sink towards the majority on that element, but they sink differently: the verdict vote sees only the product of the judges' competences and cannot tell an even court from a lopsided one, while the premise vote is ruled by its weakest majority. And when a judge's errors on two elements come from the same misreading of the facts, the gap between the procedures closes.
Start anywhere
Eight of 1078 essays — the whole collection is a click away, or search it.
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.
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.
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.
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.
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.
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.
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.
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.
The eleven fields
the spine — every essay sits in exactly one
Geometry
Shapes, and the arguments that can be made by rearranging them.
Analysis
Limits, curves, and what happens when the going does not stop.
Algebra
Structure — what stays true when the numbers change.
Discrete
Counting, graphs, and things that come in whole pieces.
Topology
What survives bending, and what does not.
Probability
Randomness with a shape.
Number
The whole numbers, and how much structure they turn out to have.
Dynamics
One rule, applied over and over, and what the sequence does in the end.
Logic
What can be said in a system, what follows from it, and what it cannot settle about itself.
Computation
A fixed set of operations, and the exact question of what it can and cannot build.
Applied
A rule for choosing, stated exactly, and what it forces on whoever adopts it.
Longest series
one idea, several arguments — all 134 of them
Pseudorandomness
- 1 The planes a recurrence cannot leave
- 2 The test that ranks the generators
- 3 Four numbers and the rule is yours
- +12 more
Equilibrium
- 1 The value from both sides
- 2 The road that makes everyone later
- 3 A signal both can see
- +11 more
Finite fields
- 1 The field with four elements
- 2 Every element is a power of one of them
- 3 Solutions that come in multiples of p
- +10 more
Error-correcting codes
- 1 Distance is a picture
- 2 Sixteen spheres that fill a cube
- 3 Finding the error without reading the message
- +9 more
Euler characteristic
- 1 Every corner pays for itself
- 2 Two trees, and every edge in exactly one of them
- 3 Seven hundred and twenty degrees of gap
- +8 more
Homotopy
- 1 A loop that cannot be pulled tight
- 2 The group a space has at a point
- 3 Every cover is a subgroup
- +8 more
Threads running through
themes, not categories
Proof without words
Arguments that are complete once they have been looked at properly. Not illustrations of proofs — the proofs themselves.
Pi turns up uninvited
A constant defined by circles, appearing in places with no circle anywhere in sight, and what that tends to mean.
Doing infinitely many things
Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.
The same thing twice
Two constructions that look unrelated and turn out to be the same object wearing different clothes.
Throwing things away
Progress made by deleting detail: the map that becomes a graph, the shape that becomes a number.
Order out of noise
Random processes that reliably produce the same shape, and the reason that is less mysterious than it looks.
One point away
Constructions that work perfectly except at a single exceptional place, and what is done about it.
Things that cannot be done
Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
Counting the same thing twice
One collection, counted by two different methods, and an identity that falls out because both answers have to agree. The proof is the pair of counts.
Sensitive to everything
Systems where a difference too small to draw becomes the whole difference, and the reason that is a property of the rule rather than of the measurement.
Small rules, large behaviour
Rules short enough to write on one line, producing behaviour nobody can summarise — and the finding that the size of a rule predicts nothing about the difficulty of the questions it raises.
What a system cannot say
Rules asked a question about themselves, and an answer that is provably not available from inside — which is a different kind of limit from not knowing yet.
Decided by exhaustion
Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.
Small cases lie
Patterns that hold for every example anyone would check by hand, and then stop. The cases within reach are not a sample of the cases.
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