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.

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22 essays, 18 September 2026 — everything that has been added, in order

Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Algebra

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

6 figures

A quadratic in the exponent, completed. Two panels sharing an x-axis. Above, the parabola −x² + 2x with its top at x = 1 marked. Below, e raised to that parabola: a bell centred at the same x = 1, with peak height e^1, beside the faint unmoved bell e^(−x²).

Algebra

One number under every bell

The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.

6 figures

Area out to infinity, for three powers. Left: the curves 1/√x, 1/x, 1/x² from x = 0 to 10, with the region beyond x = 1 shaded under the lowest. Right: the area from 1 to T for each, on logarithmic scales, for T up to 10^6. 1/√x keeps growing, 1/x keeps growing, 1/x² levels off at 1.

Analysis

An endless region with a finite area

A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.

5 figures

Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

Analysis

A rectangle cut by a curve

Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

6 figures

Every stable matching leaves out the same people. 4 stable matchings of a market with short lists, drawn as two columns joined by edges; every one leaves the same letter and number unmatched.

Applied

The people every stable answer leaves out

Let the lists be short and let one side take several partners. Stable matchings still exist and there can be many of them — but every one leaves out exactly the same people, and a member who is left with an empty place holds exactly the same partners in every one. A three-line count proves it.

5 figures

The odd ring that forbids a stable pairing. The ranked lists of 6 people and a stable partition of them drawn on a circle: pairs as plain chords, a ring of three or more as arrows from each person to the one they hold. An odd ring is present, as it is in every stable partition of this instance.

Applied

A ring that no pairing can break

Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.

7 figures

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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