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
20 essays, 28 August 2026 — everything that has been added, in order
![The polynomial whose roots are the stretches. The determinant of A − λI plotted against λ for the map [2, 1, 1, 2], with its roots at 3 and 1 marked.](/f/the-polynomial-whose-roots-are-the-stretches-1b3407e2.svg)
Algebra
The polynomial whose roots are the stretches
Finding the directions a map leaves alone means finding the numbers at which it crushes something to nothing. Those numbers are the roots of one quadratic, and everything the map does is written in its two coefficients.
![The same map, written in the basis of its own eigenvectors. Three panels: the map [2, 1, 1, 2] on the standard grid, the diagonal stretch by 3 and 1 it becomes on the eigenvector grid, and the two put back together.](/f/the-same-map-written-in-the-basis-of-its-own-eigenvectors-20597990.svg)
Algebra
The same map in a better basis
Measured along its own invariant directions, a linear map stops shearing and becomes two independent stretches. Nothing about the map has changed; the grid it is described against has.
![The level curve, and the axes the matrix chooses. The curve xᵀAx = 1 for the matrix [2, 0.8, 0.8, 1.4], drawn by solving for the radius at each angle, with the two eigen-directions marked; they cross at a right angle and are the axes of the curve.](/f/the-level-curve-and-the-axes-the-matrix-chooses-db3c3a9d.svg)
Algebra
Symmetry forces a right angle
A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.
![What the map does to a circle. The unit circle with two perpendicular directions marked, and its image under [1.6, 1.2, −0.4, 1.1] — an ellipse whose axes are the images of those two directions, of lengths 2.04 and 1.10.](/f/what-the-map-does-to-a-circle-cd467b80.svg)
Algebra
What a map does to a circle
Every linear map sends the unit circle to an ellipse. Two perpendicular directions go to two perpendicular directions, whatever the map is — even a map with no invariant direction at all, and even one that is not square.
Analysis
The area that names the number
The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.
Analysis
The equation with only one answer
A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.
Start anywhere
Eight of 245 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.
Deepest ladders
one idea, several arguments — all 134 of them
Pythagoras
- 1 Two squares, four triangles, and no algebra
- 2 Euclid proves it without moving anything
- 3 Every triple, on one circle
- +5 more
Central limit
- 1 A bell curve assembled out of coin flips
- 2 The average settles and the wobble does not
- 3 The shape that averaging leaves alone
- +4 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
- +4 more
Constructible numbers
- 1 What two points can build
- 2 Every step is a square root
- 3 The cube that will not double
- +3 more
Eigenvectors
- 1 The directions a map leaves alone
- 2 The polynomial whose roots are the stretches
- 3 The same map in a better basis
- +2 more
Monte carlo
- 1 Getting pi by dropping needles on the floor
- 2 The error that does not care how many dimensions
- 3 Sampling where the answer lives
- +2 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 245 essays · by ladder · by named object · by generator · what the figures prove · what is new