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 rows. Only 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

20 essays, 29 August 2026 — everything that has been added, in order

The Klein bottle, drawn where it does not fit. A closed one-sided surface in three dimensions, drawn as a tube with a figure-eight cross-section that turns over once on the way round, with the circle where the drawing passes through itself marked. Topology

The bottle that needs a fourth dimension

Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.

6 figures The gluing abab makes a projective plane. A polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become. Topology

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

7 figures A sphere over a projective plane, cell by cell. An icosahedron with opposite faces drawn in matching colours, beside the count of cells it has and the count the quotient by the antipodal map has — every number halved, including the Euler characteristic. Topology

Two sheets over a one-sided surface

Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.

6 figures A frame carried round a Möbius band. A flat rectangle whose ends are about to be joined, with a pair of arrows carried along it — one along the band and one across it — and the sign of the frame at each station. Topology

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

6 figures Every candidate for a rational square root, tried. A column for each of 2, 3, 4, 5, 6, 7, 8, 9, listing the whole numbers that divide it with their squares, and the verdict the search returns. Number

Which roots refuse to be fractions

The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.

6 figures The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it. Number

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

6 figures

Start anywhere

Eight of 265 essays — the whole collection is a click away, or search it.

The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right. 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 Odd numbers as square shells. Nested L-shaped shells of 1, 3, 5 … 11 cells stack into a 6 by 6 square. 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 Euclid's algorithm on a 34 by 13 rectangle. The rectangle is tiled by peeling off the largest square that fits, again and again, until nothing is left. 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 The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale. 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 Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further. 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 A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve. 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 8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve. 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 Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains. 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

The eleven fields

the spine — every essay sits in exactly one

Geometry

Shapes, and the arguments that can be made by rearranging them.

25 essays

Analysis

Limits, curves, and what happens when the going does not stop.

26 essays

Algebra

Structure — what stays true when the numbers change.

24 essays

Discrete

Counting, graphs, and things that come in whole pieces.

25 essays

Topology

What survives bending, and what does not.

24 essays

Probability

Randomness with a shape.

26 essays

Number

The whole numbers, and how much structure they turn out to have.

24 essays

Dynamics

One rule, applied over and over, and what the sequence does in the end.

24 essays

Logic

What can be said in a system, what follows from it, and what it cannot settle about itself.

23 essays

Computation

A fixed set of operations, and the exact question of what it can and cannot build.

20 essays

Applied

A rule for choosing, stated exactly, and what it forces on whoever adopts it.

24 essays

Deepest ladders

one idea, several arguments — all 134 of them

Pythagoras

  1. 1 Two squares, four triangles, and no algebra
  2. 2 Euclid proves it without moving anything
  3. 3 Every triple, on one circle
  4. +5 more
8 rungs · geometry

Central limit

  1. 1 A bell curve assembled out of coin flips
  2. 2 The average settles and the wobble does not
  3. 3 The shape that averaging leaves alone
  4. +4 more
7 rungs · probability

Fixed points

  1. 1 Something always stays put
  2. 2 Nothing on a sphere can be combed flat
  3. 3 A point that pulls, and a point that pushes
  4. +4 more
7 rungs · topology

Constructible numbers

  1. 1 What two points can build
  2. 2 Every step is a square root
  3. 3 The cube that will not double
  4. +3 more
6 rungs · computation

Cardinality

  1. 1 Two injections make a bijection
  2. 2 The arithmetic that loses subtraction
  3. 3 A line with as many points as a square
  4. +2 more
5 rungs · logic

Eigenvectors

  1. 1 The directions a map leaves alone
  2. 2 The polynomial whose roots are the stretches
  3. 3 The same map in a better basis
  4. +2 more
5 rungs · algebra

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.

54 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.

12 essays

Doing infinitely many things

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

32 essays

The same thing twice

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

92 essays

Throwing things away

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

36 essays

Order out of noise

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

26 essays

One point away

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

7 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.

75 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.

59 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.

9 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.

12 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.

13 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.

53 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.

50 essays

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