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

12 essays, 3 October 2026 — everything that has been added, in order

The Mandelbrot set counted cell by cell. A 96-cell-wide grid over the Mandelbrot set: 2034 cells certainly inside (1.3794), 290 undecided after 60 steps (0.1967).

Algebra

The area of the Mandelbrot set, from two sides

The Mandelbrot set has an area, and nobody knows it. Counting grid cells closes in on 1.5066 from both sides, but every count judges a cell by a single point, and points on the boundary never declare themselves. An exact formula exists — π times one minus a sum over the coefficients of the map onto the set's outside — and every partial sum is a guaranteed upper bound; after sixteen thousand coefficients the bound is still 1.786, and at the rate it falls it would need something like 10²⁵ more.

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The factorials, and three smooth curves through all of them. Γ(x+1), Γ(x+1)(1 + 0.5 sin² πx) and the degree-six interpolating polynomial through 0!…6!, on a log scale over 0 ≤ x ≤ 5.4; at x = 1/2: 0.88623, 1.32934, -3.58301.

Analysis

Only one curve through the factorials bends the right way

Infinitely many smooth curves pass through 1, 1, 2, 6, 24, 120, and some of them even obey the factorial's own rule, f(x + 1) = (x + 1)·f(x), at every x. Ask that the logarithm of the curve bend upwards everywhere and exactly one survives — Euler's integral. A wiggle of any size breaks the condition somewhere, the smaller the wiggle the further out, and the proof that nothing else survives is a squeeze that computes √π along the way.

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What a seller collects under first-price and second-price rules. Histograms of 20000 simulated revenues with 3 uniform bidders: second-price mean 0.4988, variance 0.0497; first-price mean 0.4995, variance 0.0167.

Applied

Two auctions that earn the same

In one sealed-bid auction the winner pays its own bid; in the other it pays the second-highest bid. Bidders behave completely differently — in the second they bid what the object is worth to them, in the first they shade their bids down by exactly a fraction — and the seller's revenue is spread differently. Yet the seller expects to collect precisely the same amount, (n − 1)/(n + 1) for n bidders with values spread evenly, and so does an auction in which everybody pays. The equality breaks the moment bidders dislike risk, and it says nothing about how much a reserve price can add.

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Langford pairings of order 3, 4 and 7. Rows of boxes with arcs joining equal numbers: order 3: 3 1 2 1 3 2; order 4: 4 1 3 1 2 4 3 2; order 7: 7 3 6 2 5 3 2 4 7 6 5 1 4 1.

Discrete

A sum that forbids half the pairings

Put two 1s, two 2s, …, two ns in a row so that between the two ks there are exactly k other numbers. For n = 3 there is one way, 312132; for n = 4 one way; for n = 5 and 6 none at all, and the reason is a single sum — adding up the positions of every entry in two ways forces n(3n − 1)/2 to be even. A search confirms the count forbids nothing that exists and finds 26 arrangements at n = 7 and 108,144 at n = 12, but the sum is all anyone knows about why.

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Where the Rössler flow's peaks fall, as one parameter changes. Peak values of x against c from 2.5 to 6.2 for the Rössler system with a = b = 0.2: period 1, 2 and 4 at c = 2.5, 3.5 and 4, chaos beyond about 4.2, a period-3 window near 5.3.

Dynamics

A flow that doubles like a parabola

Otto Rössler's three equations have one nonlinear term and a single knob. Turn it and the orbit closes after one loop, then two, then four, then never — the logistic map's cascade, in a flow in three dimensions. Record each peak of the orbit against the one before and the reason appears: the points lie on a single rounded hump. The gaps between doublings shrink by ratios 3.49, 4.54, 4.59, climbing towards Feigenbaum's 4.669, and the chaos contains a period-3 window that repeats the whole cascade.

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Malfatti's three circles against the three largest, one at a time. A triangle with angles 40, 70, 70 degrees: Malfatti's circles cover 69.1% of it, the greedy choice of incircle then largest remaining circles covers 73.7%.

Geometry

Three circles that touch and are not the largest

In 1803 Gian Francesco Malfatti asked how to cut three round columns from a triangular block of marble with as little waste as possible, and answered with three circles each touching the other two and two sides. The circles exist in every triangle and are found by solving three equations. They are never the answer to his question: the plain greedy rule — the inscribed circle first, then the largest circle that still fits, twice — always does better, by 1.4 per cent on the equilateral triangle and by almost double on a thin one.

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Eight of 858 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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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