Matpic
mathematics, in pictures

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.

Pascal's triangle mod 2, 32 rowsOnly the odd entries are drawn; the pattern that appears is the Sierpiński triangle.
Fig. 1 Pascal’s triangle with the odd numbers shaded and the even ones left blank. Nothing here was designed to make a pattern; the pattern is a consequence of the arithmetic. See Pascal’s triangle, two colours.

Recently added

15 essays, 19 August 2026 — everything that has been added, in order

123412431324134214231432213421432314234124132431312431423214324134123421412341324213423143124321all 24 ways of relabelling the 4 corners, and the 8 that move no distancethe other 16 change at least one distance between corners, so no motion of the plane performs them Algebra

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

6 figures 1way4ways4ways2ways4ways1way16 colourings of 4 corners in 2 colours, pooled into 6 classestwo colourings share a row exactly when some motion carries one to the other Algebra

Colourings nobody can tell apart

Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.

7 figures 3112ad − bc = 6 − 1 = 5area of the drawn shape = 5positive: the corners keep their order1 Algebra

The number that says how much room is left

A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.

6 figures radius 0.40 turns0 roots insideradius 10 turns0 roots insideradius 1.42 turns2 roots insideradius 23 turns3 roots insidep(z) = z³ − 2z + 2 — as the circle grows, the number of turns steps up from 0 to 3the count can only change by the image crossing the origin, and a crossing is a root Algebra

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

7 figures 9 of the 24 arrangements put nothing back where it startedthe shaded diagonal is where an object stays put; a marked grid is one with no mark on it Probability

Nobody gets their own hat

Hand back a pile of hats at random and ask for the chance that not one person gets their own. The answer barely moves as the crowd grows — it is a third and a bit at four people, and a third and a bit at four thousand.

7 figures 10 → 11.201 → 21.502 → 323 → 434 → 565 → 614.70 draws expected in all6 kinds, and the last is the expensive onethe waits are 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.70 drawsthe last one alone costs 6 draws on average, which is why the total grows faster than the number of kinds Probability

How long until every one turns up

Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

7 figures

Start anywhere

Twelve of 135 essays — the whole collection is a click away, or search it.

c²b²a²same four trianglessame four triangles 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.

7 figures 1357911total 6² = 36 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.

7 figures 131385322 × 131 × 81 × 51 × 31 × 22 × 1gcd(34, 13) = 1 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.

7 figures tetrahedron4 trianglescube6 squaresoctahedron8 trianglesdodecahedron12 pentagonsicosahedron20 triangles 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.

8 figures circleplane levelellipsetilted a littleparabolaparallel to the sidehyperbolasteeper still 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.

6 figures π2πθ1−1sin 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.

7 figures 0.511.522.5301234xysum ≈ 5.790exact = 6.300 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.

7 figures -111 term-113 terms-117 terms-1121 terms 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.

7 figures -2-1.5-1-0.50.511.52123456xyheight 0.37slope 0.37height 1.00slope 1.00height 2.72slope 2.72height 4.95slope 4.95 Analysis

The curve that is its own slope

There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.

7 figures beforeafter · area × 2.50210.51.5 Algebra

A matrix is a picture of what happens to the grid

Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.

7 figures realimaginaryθφzwzw|z| = 1.49|w| = 1.08|zw| = 1.61θ + φ = 76° Algebra

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

7 figures Ndegree 3Idegree 5Edegree 3Sdegree 3 Discrete

Seven bridges, and the invention of throwing things away

Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.

6 figures

All 135 essays · by ladder · by named object · what the figures prove

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.

33 essays

Pi turns up uninvited

A constant defined by circles, appearing in places with no circle anywhere in sight, and what that tends to mean.

6 essays

Doing infinitely many things

Sums that never end, subdivisions that never stop, and the care required to make either of them mean something.

18 essays

The same thing twice

Two constructions that look unrelated and turn out to be the same object wearing different clothes.

49 essays

Throwing things away

Progress made by deleting detail: the map that becomes a graph, the shape that becomes a number.

17 essays

Order out of noise

Random processes that reliably produce the same shape, and the reason that is less mysterious than it looks.

8 essays

One point away

Constructions that work perfectly except at a single exceptional place, and what is done about it.

4 essays

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.

36 essays

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.

19 essays

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.

8 essays

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.

9 essays

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.

7 essays

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.

22 essays

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.

28 essays