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

0.518 on each side0.508 on each sidethe cut runs at 162.8° and leaves 50.00% of the first shape and 50.00% of the second on the nearsidethe two shapes were not arranged to make this possible; a single line halving both exists forany two of them, and would exist for two hundred Topology

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

7 figures 0 times round1 time round2 times roundthe number under each loop is counted by walking it and adding the angle turned through, as seen from the holea loop can be slid and stretched at will inside the ring without changing that number, and there is no way tochange it without leaving the ring Topology

A loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

7 figures 3 colours5 colours7 coloursdeterminantthe unknot3 — all one colour5 — all one colour7 — all one colour1the trefoil95 — all one colour7 — all one colour3the figure-eight knot3 — all one colour257 — all one colour5every count is taken by trying every assignment of colours to arcs and keeping the ones the rule allowsa knot takes p colours in more than one way exactly when p divides its determinant, and the determinant is computedhere from the crossings by exact whole-number elimination Topology

Colours that count more than three

Three colours prove the trefoil is knotted and say nothing at all about the figure-eight, which refuses them exactly as an unknotted loop does. The repair is to stop colouring and start counting — with five colours, or seven, and with the arithmetic done modulo the number of them.

6 figures the subgrouper²r · the subgrouprr³m₁ · the subgroupm₁m₃m₂ · the subgroupm₂m₄2 elements in the subgroup, 4 blocks, 2 × 4 = 8 in the group — which is Lagrange'stheorem, and it is arithmetic about the picture rather than a theorem quoted at iteach block is the subgroup with one element composed onto the front of it, so it is thesame size; two blocks that share an element are the same block, which is why theycannot overlap Algebra

The blocks a subgroup cuts out

Take any part of a group that is closed under composition, and it slices the whole group into blocks of its own size that do not overlap. Everything Lagrange's theorem says is arithmetic about that picture — and whether the blocks can be multiplied is a separate question with a surprising answer.

7 figures before — the line that will vanishafter — all of it on one linethe matrix sends every point of one line — the direction (-0.447, 0.894) — to the origin, and everything else onto a singleline through itone dimension is lost and one survives; that they add to the two the plane started with is the rank-nullity theorem, and it isthe whole content of the picture Algebra

What a map throws away

A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.

7 figures average 2.0001.0003.0002.000the 3 roots add to 6.000 and multiply to 6.000, both of which are coefficients divided bythe leading onetheir average is at 2.000, marked — moving the constant term moves the roots about andcannot move that point at all Algebra

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

6 figures

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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 150 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.

36 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.

20 essays

The same thing twice

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

56 essays

Throwing things away

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

18 essays

Order out of noise

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

9 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.

39 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.

24 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.

29 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.

29 essays