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, 10 October 2026 — everything that has been added, in order
Algebra
The fraction written on every bulb
Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.
Algebra
Two rays for every fraction
Seen from far away, every point of the Mandelbrot set's outside has an angle, and the curves of constant angle come in to land on the set. At the root of the p/q bulb exactly two of them land, and their angles are written by the staircase under a line of slope p/q. The limbs those pairs fence off share out the whole circle of angles between them, by the identity Σ φ(q)/(2^q − 1) = 1.
Analysis
What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
Analysis
Where a random walk leaves the snowflake
Start a random walk at the centre of the Koch snowflake and record where it first touches the edge. The edge has dimension 1.26, but the places the walks arrive form a set of dimension one: 60,000 walks spread their entropy by ln 3 a level, not ln 4, and the busiest fifth of the edge takes nine tenths of them. Makarov proved in 1985 that it must be so for every simply connected region in the plane.
Applied
One, plus a quarter, plus a ninth
Give n workers n jobs with every cost drawn at random, average one, and find the cheapest way to pair them. However large n is, the cheapest total averages less than π²/6, and for every n it is exactly 1 + 1/4 + 1/9 + … + 1/n². Parisi guessed the formula in 1998 from n = 1, 2 and 3; thirteen sizes of random tables, solved exactly, land on it within sampling error.
Applied
A best reply to the past
Let two players each answer the whole history of the other's play as though it were a fixed plan. In a zero-sum game the history closes in on the equilibrium, as Julia Robinson proved in 1951. In Lloyd Shapley's three-by-three game it never does: play runs round six profiles for ever, each run the one before plus the one three before plus two, so the runs grow by the real root of x³ = x² + 1 and every run is a fixed share of all the history.
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Eight of 968 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
Equilibrium
- 1 The value from both sides
- 2 The road that makes everyone later
- 3 A signal both can see
- +10 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
- +9 more
Pseudorandomness
- 1 The planes a recurrence cannot leave
- 2 The test that ranks the generators
- 3 Four numbers and the rule is yours
- +9 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
Fixed points
- 1 Something always stays put
- 2 Nothing on a sphere can be combed flat
- 3 A point that pulls, and a point that pushes
- +7 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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