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

22 essays, 19 October 2026 — everything that has been added, in order

Algebra

The class number is a degree

The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.

6 figures

Algebra

How near nought a sum of roots can come

Five arrows of length one, each pointing at a 122nd root of unity, laid end to end, return to within fifteen millionths of where they started. No five of the 123rd roots come within a thousandth. Two, three and four arrows shrink their smallest miss in a tidy power of n; five scatter over four powers of ten from one n to the next, and nobody knows the law they follow.

6 figures

Four ways to a slope, and the one that subtracts. f'(1.5)=4.053427893898620; forward best 2.60e-9 at h=1.0e-8; central best 6.41e-12 at h=3.2e-6; complex step at 1e-200: 1.00e-17; dual: 1.00e-17.

Analysis

A slope without a subtraction

The definition of a slope subtracts two nearly equal numbers and divides by a tiny one, and a computer that follows it loses half its digits however carefully the step is chosen. Two pieces of arithmetic avoid the subtraction altogether. Step sideways into the complex numbers by 10⁻²⁰⁰ and read off the imaginary part; or calculate with numbers a + bε in which ε² = 0, and the slope rides along exactly. Run backwards, the second gives all thousand slopes of a function of a thousand variables for under five times the cost of the function.

5 figures

Zeros of the exponential's partial sums. n=10: 10 zeros, balance range 2.1289–2.1339; n=20: 20 zeros, balance range 2.4462–2.4516; n=40: 40 zeros, balance range 2.7760–2.7827; n=80: 80 zeros, balance range 3.1127–3.1207; tip -0.278465.

Analysis

The zeros a partial sum cannot avoid

The exponential is never zero, and every partial sum 1 + z + z²/2! + … + zⁿ/n! of its series is zero n times. Divide those zeros by n and they line up on a single egg-shaped curve, |z e^(1−z)| = 1, found by Szegő in 1924. The curve is where two sizes tie — the exponential the series is reaching for, and the terms it has not yet reached — and the same tie, carried one step further, places every zero to within a few hundredths.

6 figures

Applied

A relaxation that answers in halves

The cheapest set of points touching every edge of a graph is hard to find. Allow each point to be taken fractionally and the problem becomes a linear programme — and its best answers are never messier than halves. Every point gets 0, ½ or 1, the 0s and 1s can always be trusted, and the proof is a picture: double the graph into a bipartite one, where a matching decides everything exactly.

6 figures

A block of shares in an ocean of small holders. a=0.02: exact 0.0249, limit 0.0204; a=0.06: exact 0.0647, limit 0.0638; a=0.1: exact 0.1144, limit 0.1111; a=0.14: exact 0.1642, limit 0.1628; a=0.18: exact 0.2139, limit 0.2195; a=0.22: exact 0.2836, limit 0.2821; a=0.26: exact 0.3532, limit 0.3514; a=0.3: exact 0.4229, limit 0.4286; a=0.34: exact 0.5124, limit 0.5152; a=0.38: exact 0.6119, limit 0.6129; a=0.42: exact 0.7214, limit 0.7241; a=0.46: exact 0.8507, limit 0.8519; a=0.5: exact 0.9950, limit 1.0000; a=0.54: exact 1.0000, limit 1.0000; a=0.58: exact 1.0000, limit 1.0000

Applied

A block in an ocean of small holders

A shareholder with 30% of a company whose other shares are scattered among thousands of small holders has, by the average over every order of arrival, 43% of the power to decide — 3/7 exactly, in the limit. The reason is a window of moments: if everyone arrives at a random time, the dispersed shares trickle in at a steady rate, and the block decides every vote in which it turns up while the trickle sits within 30% of the quota. The same picture, run along a diagonal, turns the average over orders into an integral and puts the two classic power indices on one curve.

6 figures

Start anywhere

Eight of 1056 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