Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
1. The Woman Who Outlived Her Buyer
In 1965 a lawyer in Arles bought the flat of Jeanne Calment, 90, for 2,500 francs a month until she died. The life table expected a bill of about 144,600 francs. He died thirty years later, having paid about 900,000. Gompertz's law puts her 122 years at 1 in 81 billion or worse; the plateau measured in Italians over 105, at 1 in 15 million.
Read more →2. The Shock That Refuses to Break
A weak tidal bore does not arrive as a wall but as a train of waves, and its first wave grows to twice the height of the step. The Korteweg–de Vries equation explains both, and one more thing: the energy a breaking bore burns, an undular bore keeps, at exactly the same rate, in a wave train that never stops growing.
Read more →3. The Rule That Fits One Ward
Hospitals are told to keep their beds at most 85% full. At that occupancy a 100-bed ward is full about once a month, a 20-bed ward within a week and a 500-bed hospital about once in six years. The Ornstein–Uhlenbeck process explains why: what matters is the distance to a full ward in standard deviations, and it grows with the square root of the size.
Read more →4. The Birds That Turn as One
A starling watches its six or seven nearest neighbours, however far away they are. Put that rule and its obvious rival into Vicsek's flock of 400 birds and the rival wins when the birds are packed close, then collapses when the flock spreads out. Counting costs a little agreement and buys a flock that does not break.
Read more →5. The IQ Map That Filled Its Own Blanks
The world IQ map gives 200 countries a colour, and they all look equally sure. Behind the United States stand 62,296 people; behind Angola, nineteen; behind fifty countries, nobody, their colour copied from next door, and five sit at a floor of 60 that no test produced. Yet the measured part holds up: it tracks the World Bank's school tests at r = 0.81. The copies track nothing.
Read more →6. The Wave That Nobody Ordered
In the Beer Game, customer demand steps once from 4 cases a week to 8, yet the factory's orders swing to 33. Written as 27 differential equations and fed the ordering rule Sterman fitted to real players, the model reproduces their waves, and shows the one number that stops them: what was already on its way.
Read more →7. The Equation That Paints a Jaguar
In 1952 Alan Turing wrote down two chemicals and a race, and showed that spots can appear from nothing. Seventy years on, a pair of his equations paints a jaguar: a spotted cub first, then broken rosettes with dark centres, from three changes of rate. And in fetal house-cat skin, biologists have now caught the real thing: an activator and an inhibitor laying down the stripes before birth.
Read more →8. The Battleship That Solved a Differential Equation
The 16-inch guns of USS Iowa threw a 1,225 kg shell 38.7 km, and aiming them is a boundary-value problem for an ordinary differential equation. Fitted to the Navy's range tables, the model finds the gun's best angle beyond its turret's 45-degree stop, a bunker that breaks only from farther away, and a weaker charge that spares a town.
Read more →9. The Head That Was Written in Harmonics
Marcus Agrippa had the Roman empire surveyed and its map engraved on marble. This post maps his head onto a globe: a scanned portrait relaxed conformally onto the sphere, then written as a series of spherical harmonics, 786,432 numbers to degree 511. The series closes in on the stone as the degree rises, and the last place it arrives is the sharpest edge on the head.
Read more →10. The Motor That Counted Its Own Steps
Kinesin walks along the rails of a nerve cell 8 nm at a time, and only the steps can be seen. In 1994 the noise of those steps showed that two or more hidden processes drive each one. The theorem behind that inference, proved by Aldous and Shepp in 1987, says no cycle of n hidden states can keep better time than n equal stages in a line.
Read more →11. The Third Walker
Einstein and Gödel walked home together for years, and on one question they never settled: Gödel's rotating universe, where a traveller can loop back into the past, and his claim that it proves the passing of time is unreal, here too. Einstein called it important and left it open. Hermann Weyl worked down the corridor. This is the walk where he joins them.
Read more →12. The Cable That Whispered Too Slowly
In 1858 the first Atlantic cable carried about a tenth of a word a minute. William Thomson had predicted why in 1855: a long cable carries a signal the way a bar carries heat, and its delay grows with the square of its length. The chief electrician called that "a fiction of the schools" and pushed up to 2,000 volts down the cable. The equation was linear; the volts changed no time, and within weeks the cable was dead.
Read more →13. The Lease That Was as Good as Forever
In 1898 Britain leased Hong Kong's New Territories for 99 years, a term its negotiator thought "as good as forever". In money he was right: at 5% a year, 99 years hold 99.2% of forever. A slow walk from a project's value forever to its value for T years, first by subtracting forever from forever, then by making the end random, which turns out to be compound interest run backwards.
Read more →14. The Fleet That Lost by Squaring
At Trafalgar 27 British ships of the line met 33. Lanchester's square law says that is 729 against 1,089: fought line against line, the British fleet is gone and 19 allied ships remain. Nelson's memorandum, written 111 years before the law, does its arithmetic, and solving the equations shows where the advantage really comes from: the enemy's idle guns.
Read more →15. The Needle That Knew Too Much
In 1901 an Italian teacher threw a stick onto a grid of wires 3,408 times and found π to six decimal places. An honest run that size misses by about 0.05. Buffon's needle says how wrong it should have been, a stopping rule says how 355/113 could come out anyway, and the test Fisher used on Mendel's peas says why being too right is itself the evidence.
Read more →16. The Tanks That Counted Themselves
Before D-Day the Allies needed to know how many tanks Germany built each month. The spies said about 1,400. A few statisticians read the serial numbers off wrecked gearboxes and said 246. The records, opened after the war, said 245 — and the best estimate throws away every serial number but one.
Read more →17. The Forest That Burns All at Once
Over forty fire seasons in British Columbia, the number of fires varied six-fold and the area burned nearly a thousand-fold. Fewer fires now burn thirty-five times the area they did in the 1990s. A percolation threshold explains why, and a fire PDE on a patchy forest shows fuel 12.5% drier turning the same lightning strike into a fire a hundred times larger.
Read more →18. One Hundred Prisoners, Fifty Opens
A hundred prisoners, a hundred boxes, fifty opens each, and all must succeed. Opening at random gives one chance in 1.27 x 10^30. One sentence of strategy gives 31 per cent, and it never decays with the size of the room.
Read more →19. A Coin With No Memory Still Looks Cold
For thirty years the hot hand was the textbook case of seeing pattern in noise. The test itself was biased: in four fair flips, the share of heads following a head averages 0.4048, not 0.5.
Read more →20. The Body That Is Always Falling
Stand still for a photograph and you are falling the whole time: an upside-down pendulum whose lean doubles every 0.22 seconds, caught by a brain that sees it late. The equation of standing is a second-order Langevin equation turned upside down, and it has a hard limit: with reflexes slower than 0.46 seconds, nothing can hold you up. Try it yourself.
Read more →21. Cholesky vs Eigendecomposition: Two Ways to Sample Correlated Gaussians
Two matrix square roots for sampling correlated normals: Cholesky is several times faster, but the eigendecomposition survives a correlation matrix that isn't positive definite.
Read more →22. The 1/√N Wall: Why Monte Carlo Is Slow, and What Beats It
Monte Carlo error falls like 1/sqrt(N) no matter how clever your sampler. Quasi-Monte Carlo bends the exponent, not just the constant.
Read more →23. Integrating Oscillations: Filon vs Levin
Two classical methods for oscillatory integrals that beat the O(omega)-points curse — compared against an exact solution, with the log-error plot to prove it.
Read more →24. The Average That Drowned a City
In 1997 Grand Forks built its dikes to a forecast crest of 49 feet. The river came in at 54.35. The forecast was a median, the damage was convex, and the city evaluated the damage at the average crest instead of averaging the damage. The forecaster's own record put the chance of topping the dikes at one in four.
Read more →25. Markov Chains in Supermarket Management
How Markov chains model checkout queues and customer flow — and what the stationary distribution tells a supermarket manager about staffing and store layout.
Read more →