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, 23 October 2026 — everything that has been added, in order
Algebra
Two fields that every prime confuses
Reduce x⁸ − 3 and x⁸ − 48 modulo any prime other than 2 and 3 and factor them: the two lists of factor degrees are the same, at all 9,590 such primes below 100,000. The fields the two equations define are still different. The reason is a pair of ways for one group to permute eight roots, with the same cycle lengths element by element and different fixed points — the same coincidence, in degree seven, as the points and lines of the Fano plane.
![Three roots that one map rotates, for every t. Shanks cubic roots for t in [-9,6]; at t=2: 3.507019, -0.221876, -1.285142; map permutation 1,2,0.](/f/three-roots-that-one-map-rotates-for-every-t-bf803443.svg)
Algebra
A line of cubics that are all cyclic
In a box of cubics the full group of six symmetries is almost universal, and the three rotations are a rare exception. Daniel Shanks's x³ − tx² − (t + 3)x − 1 has the rotations at every whole t, and at every rational one. Its discriminant is the square of t² + 3t + 9, its roots are turned into one another by x ↦ −1/(1 + x), and on a suitable circle every member is an equilateral triangle.
Analysis
A rule that counts lets the sum escape
Strike out every number containing a 9 and the harmonic series converges, to 22.92. Keep only the numbers with as many 7s as 3s and it diverges — slowly, like the square root of the number of digits. Ask for two such balances and it still diverges; ask for three and it converges. The dividing line is Pólya's theorem that a random walk comes home in one and two dimensions and not in three.
Analysis
No set fills half of every interval
A fat Cantor set has length one half and contains no interval, so every interval holds some of it and some of its gaps. Look closely at any one point, though, and the window around it ends up almost all set or almost all gap. Lebesgue proved in 1910 that this is forced for every set with a length: the share of a shrinking window is 0 or 1 almost everywhere. So no set can fill exactly half of every interval, however it is built.
Applied
How often the rules find the majority favourite
When one candidate beats every rival head to head, a voting rule ought to elect it. Plurality does about three times in four when every ranking is equally likely, the runoff between the top two 96 times in a hundred, Borda's count 90. Every profile of up to 51 voters is counted exactly, the limit is sampled, and adding candidates pulls plurality down below one half.
Applied
How many stable matchings a market has
A market with random preferences on both sides has not one stable matching but many — a few at ten a side, a few dozen at a hundred, about a thousand at a thousand — and all of them are reached from the proposers' best by a chain of rotations. Boris Pittel proved that the average grows like n ln n / e. The averages measured here sit at a third of that at every size a computer can reach.
Start anywhere
Eight of 1100 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
- +11 more
Error-correcting codes
- 1 Distance is a picture
- 2 Sixteen spheres that fill a cube
- 3 Finding the error without reading the message
- +10 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
- +9 more
Unique factorisation
- 1 One way to factor, and no other
- 2 The shape of a number's divisors
- 3 A factorisation that hides its primes
- +9 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.
All 1100 essays · by series · by named object · by figure · what the figures prove · what is new