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, 14 October 2026 — everything that has been added, in order
Probability
What not knowing the coin costs
Kelly's rule stakes the edge, and the edge is the one thing a bettor never has. Estimate it from twenty tosses and stake the estimate, and a lucky start can turn a favourable coin into certain ruin. Learn it toss by toss instead, with Laplace's rule, and the cost of ignorance shrinks to a factor of about the square root of the number of tosses — because a bettor who learns is exactly a bettor who split the money among every fixed fraction at once.
Probability
Evidence that can be multiplied
A p-value from one study and a p-value from the next cannot simply be multiplied, and a researcher who decides whether to run the next study by looking at the last one quietly breaks the rule that combines them. An e-value — a number whose average under the null is at most one — can be multiplied, study after study, under any rule for continuing, and never passes twenty to one more than one time in twenty. What it costs is tosses, and what it demands is a decision about which alternatives to be good at finding.
Topology
How many ways a curve can cross itself
A closed curve that crosses itself three times can be drawn in exactly six essentially different ways on a sphere, and twenty in the plane. Four crossings give nineteen and eighty-two; six give 376 and 2,645. Every one of them is a word — the order in which the crossings are met — and a choice of direction at each crossing, and the whole census comes from asking which of those choices close up on a sphere.
Topology
The lists no covering can have
A covering of the sphere by itself with three branch points is three permutations of the sheets whose product is the identity. The Riemann–Hurwitz count says which lists of cycle shapes might work, and a search through every permutation says which do. Up to thirteen sheets, 150 lists pass the count and fail the search — and not one of them has a prime number of sheets.
Number
Two neighbours share out their small primes
A number and the next one have no prime in common, and the natural guess is that their largest primes are independent. In the limit that is believed and nearly proved. Below ten million it is visibly false: neighbours are both made of small primes only nine-tenths as often as independence predicts, a run of four only two-thirds as often, and the correlation between their largest primes is negative at distance one and positive at distance six — because a small prime that divides one of two neighbours is a small prime the other cannot have.
Number
The product raised to the twenty-fourth power
Euler's product (1 − q)(1 − q²)(1 − q³)… has the partition numbers for its reciprocal. Raise it to the twenty-fourth power and multiply by q, and the coefficients that come out — 1, −24, 252, −1472, 4830, −6048 — multiply like the primes they are indexed by, stay inside a bound Ramanujan guessed and Deligne proved fifty-eight years later, agree with a divisor sum modulo 691, and know, modulo 23, how a cubic equation factors.
Start anywhere
Eight of 1034 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
- +8 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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