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, 25 September 2026 — everything that has been added, in order
Algebra
What is left when the middle is taken out
Cut a finite piece out of a group's picture and count the parts of what remains that run off forever. The integers leave two, the plane one, a tree more with every cut — and no group anywhere leaves exactly three, because a third end is always the first of infinitely many.
Algebra
The polygon a lattice becomes from far away
Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.
Analysis
A coin in front of every power
Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.
Analysis
Two sign patterns that land together
At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.
Applied
One extra person on one side
In a random market of a thousand a side, whoever proposes gets about their seventh choice and whoever receives gets about their hundred-and-fortieth. Add one person to one side and the advantage of proposing all but disappears: the shorter side does well and the longer side badly, whichever side proposes, and most people are left with exactly one stable partner.
Applied
The matching in the middle
List every stable matching of a market, give each member their stable partners sorted from best to worst, and hand each the one in the middle. Nothing says the result should even be a matching — two people might pick the same partner — and yet it always is one, it is always stable, and the other side gets its median partners too.
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Eight of 671 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
- +6 more
Apportionment
- 1 The seat that vanishes when the house grows
- 2 Five rules and one dial
- 3 The rule with no favourites
- +5 more
Covering spaces
- 1 The same loop, unrolled
- 2 The subgroup that is freer than the group
- 3 The symmetries a cover has of its own
- +5 more
Error-correcting codes
- 1 Distance is a picture
- 2 Sixteen spheres that fill a cube
- 3 Finding the error without reading the message
- +5 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
- +5 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
- +5 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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