mathphile

Before there were numerals there were notches. Then someone noticed that where a mark sits can matter as much as what it is.

The tally is the oldest: one mark for one thing. The Babylonians, about four thousand years ago, wrote in base 60 with two wedges, one for ones and one for tens, and we still divide hours and degrees by 60. The Romans had no place value at all, which made arithmetic hard enough that merchants used a counting board. In China, from about the fourth century BC, calculators laid bamboo counting rods on a board in columns, vertical rods in one place and horizontal in the next, with an empty space for zero. In India, by the seventh century, the same idea gained a written zero, and the Indian mathematician Brahmagupta gave rules for arithmetic with it. The Persian scholar Muhammad al-Khwarizmi's ninth-century book carried those numerals into the Arabic-speaking world, and the Italian mathematician Leonardo of Pisa, known as Fibonacci, carried them to Europe in 1202. The ten digits you are reading now are that long relay. Binary, the last row, is what every computer in this exhibit speaks.

a² + b² = c². Babylonian scribes knew it, Chinese and Indian geometers drew it, and Euclid proved it for every right triangle there will ever be.

A clay tablet called Plimpton 322, from about 1800 BC, lists whole-number triples such as 3-4-5 and 5-12-13. The Chinese Zhoubi Suanjing, compiled in the centuries around the turn of the era, shows the same relation in a diagram, and Indian geometers wrote it into the Sulba Sutras. What Euclid added, in Book I of the Elements around 300 BC, was something different: not a case, not a picture, but an argument that works for every right triangle at once. That is what the word proof has meant ever since, a chain of steps each forced by the last, ending somewhere you did not know you could go. Look for a right triangle where the squares disagree. Nobody has found one, and the reason is not that people stopped looking.

Throw darts at a square with a circle inside. The share that lands in the circle approaches π⁄4. Chance, patiently counted, measures a shape.

π is the ratio of a circle's circumference to its diameter. It is the same for every circle that has ever been drawn, which is why it deserves a name. It is irrational, so its decimal expansion never repeats; it is transcendental, so no finite equation with whole-number coefficients can pin it down. Archimedes trapped it between 3 ¹⁰⁄₇₁ and 3 ¹⁄₇ by squeezing a circle between polygons of 96 sides. We now know over a hundred trillion digits, and still nobody knows whether every digit appears equally often.

φ = 1.6180339887… is the most irrational number: the hardest to approximate by fractions. That is exactly why a plant uses it.

Divide a line so that the whole is to the larger part as the larger is to the smaller, and you have the golden ratio. Its continued fraction is all ones, 1 + 1⁄(1 + 1⁄(1 + …)), so it stays as far as possible from every rational number. A seed head that turns by a rational fraction of a circle would stack its seeds into spokes and waste the space between. Turning by 360°⁄φ² ≈ 137.5° fills the disc most evenly, and the spirals you count are consecutive Fibonacci numbers: 34 one way, 55 the other.

Multiplying by i is a quarter turn. Growing at rate i is therefore travel along a circle. After a half-turn of length π you stand at −1.

e is the number that makes growth honest: a quantity growing at 100 % per year, compounded continuously, multiplies by e = 2.71828… Euler's formula e^iθ = cos θ + i sin θ says that exponential growth in an imaginary direction does not grow at all. It rotates. Set θ = π and the formula collapses to e^iπ = −1, which the American physicist Richard Feynman called the most remarkable formula in mathematics. Every wave, every alternating current, every quantum state is written in this notation.

Stanisław Ulam drew this while bored in a 1963 lecture. The diagonal streaks are quadratic polynomials that happen to be unusually rich in primes.

A prime has no divisors except 1 and itself; every other whole number is a product of primes in exactly one way. There are infinitely many, and Euclid's proof is three lines long and 2,300 years old. They thin out slowly: around a large number N, roughly one in every ln N integers is prime. That is the Prime Number Theorem, guessed by a fifteen-year-old Gauss from tables and proved a century later. The fine structure of their distribution is the Riemann Hypothesis, unproved since 1859, with a million dollars waiting.

Mod 2, Pascal's triangle is the Sierpiński triangle again, named for the Polish mathematician Wacław Sierpiński, who described it in 1915. The same shape from counting as from chance as from contraction. That is what it means for mathematics to be discovered rather than made.

The triangle is older than Pascal. It is in Pingala's Sanskrit prosody around 200 BC, in Chinese texts of the 11th century, and in the Persian poet and mathematician Omar Khayyam, around 1100. Its rows are the binomial coefficients: the number of ways to choose k things from n, the coefficients of (a + b)ⁿ, the counts of paths through a Galton board. Reduced modulo a prime p it is self-similar with p copies per level, which the French mathematician Édouard Lucas proved in 1878: a binomial coefficient is divisible by p exactly when, in base p, some digit of k exceeds the digit of n above it.

Cut a square off a golden rectangle and what is left is a smaller golden rectangle. It never stops, and the ratio it keeps is φ = 1.618…

Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13 …, build the same spiral of squares, and the ratio of each to the one before closes in on φ. Add squares and watch the readout. Where it is real: the spirals of a sunflower or pine cone, Le Corbusier's Modulor, the design system he published in 1948 from human proportions and the Fibonacci numbers, and Dalí's The Sacrament of the Last Supper (1955), painted on a canvas close to those proportions with a dodecahedron floating over the table. Where it is mostly myth: the Parthenon, the pyramids, the Mona Lisa, and most company logos, which were fitted with circles after the fact. A credit card is 1.586 long to wide, not 1.618.

Fold an A4 sheet in half and you get an A5 sheet with exactly the same proportions. Only one shape can do that: its sides must be in the ratio √2.

Give a sheet sides 1 and r. Folding it across the long side gives a sheet with sides 1 and r ⁄ 2, and for that to be a copy of the original the ratio of its long to short side, 2 ⁄ r, must equal r. So r² = 2. The German physicist Georg Christoph Lichtenberg noticed this in 1786; Germany standardised it in 1922 and the ISO system followed. A0 is one square metre, 841 × 1189 mm, each fold halves the area, and so the A4 in front of you is a sixteenth of a square metre.

Put one euro in a bank that pays 100% a year. Compound it once and you have 2. Twice, 2.25. Every instant, and you reach 2.71828…, and never more.

The Swiss mathematician Jacob Bernoulli found this limit in 1683 by asking exactly that question. The same e governs anything whose growth is proportional to its size, and anything whose decay is: populations, radioactive atoms, a cooling cup of coffee, the half-life of the medicine in the dose chapter.

Take one rule, x → r·x·(1 − x), and turn up r. The answer settles, then splits in two, then four, then eight, faster each time, until it never settles at all.

The splits arrive at r of about 3, 3.449, 3.544, 3.564, 3.569, and the gaps between them shrink by nearly the same factor every time: 4.6692… The American physicist Mitchell Feigenbaum found that number in 1975 with a pocket calculator, and then found it was the same for every system that breaks into chaos by doubling: dripping taps, electronic circuits, heart cells, animal populations. Different physics, one constant, which is why it earns a place beside π and e. It is the thread of the butterfly chapter, seen from the other end.

About 1⁄137.036: the strength of the electromagnetic force, with no units to blame and no theory that predicts it.

α, the fine-structure constant, sets how strongly light and charged particles interact, and so why atoms have the size and colour they do. Richard Feynman called it a magic number that nobody understands. It is a fitting place to end a chapter on constants: the numbers we can derive, and the ones we can only measure.

A speedometer shows how fast you are going right now, but speed is distance divided by time, and in a single instant you cover no distance in no time. Zero divided by zero. Calculus is the mathematics of making sense of that.

Pick two points on a curve and the line through them has a slope anyone can work out: rise over run. Now slide the second point toward the first. The gap h shrinks, the line swings, and its slope settles on one number, the slope of the curve at that point, its derivative. Never at h = 0, always as h heads there: the limit. The same trick works for area. Fill the space under a curve with rectangles; their total is easy to add up and always a little wrong. Use thinner rectangles and the error shrinks; in the limit it vanishes.

Isaac Newton in England and Gottfried Leibniz in Germany found this independently in the 1660s and 1670s and spent years arguing over who came first. Leibniz's notation, dy⁄dx and the long S of ∫, won. Then came the surprise that makes it one subject instead of two: the rate at which the area grows, as you push its right edge along, is exactly the height of the curve there. Slopes and areas undo each other. Every rocket in this journey, every half-life, every orbit was worked out with that sentence.

z → z² + c. For each point c of the plane, ask whether that iteration stays bounded. Black if it does. The boundary between yes and no is this.

Benoit Mandelbrot first saw a coarse printout of this set in 1980 at IBM, and assumed the specks around it were dust on the printer. They were baby copies of the whole, connected to the body by filaments too thin to print. The set is connected (Adrien Douady and John Hubbard, 1982), its boundary has fractal dimension exactly 2 (Mitsuhiro Shishikura, 1991), and its area is known only to a few decimal places. Every bulb hanging off the main cardioid corresponds to a rhythm: the biggest bulb is period 2, the next period 3, and the antenna of each bulb has as many spokes as its period.

The Sierpiński triangle has zero area and infinite perimeter. Its dimension is log 3 ⁄ log 2 ≈ 1.585: more than a line, less than a plane.

No individual point knows where the triangle is. Each jump is random. But every jump is a contraction toward one of three corners, and the only shape left unchanged by all three contractions at once is this one, so the wandering point is herded onto it no matter where it starts. The British mathematician Michael Barnsley, in the 1980s, called this an iterated function system; ferns, clouds and coastlines can be stored as a handful of such rules and regrown on demand.

Write a list of infinite strings of 0s and 1s. Now build a new string by flipping the diagonal. It differs from every string on the list, so the list was never complete.

The whole numbers can be listed one, two, three. Cantor showed that the real numbers cannot be listed at all, however cleverly you try, because the diagonal trick builds a number that every list misses. So the infinity of real numbers is strictly larger than the infinity of whole numbers. Many mathematicians of his day, Kronecker among them, thought this was madness, and Cantor suffered for it. Hilbert later said no one would drive us from the paradise Cantor had made. Whether any size lies between the two, the continuum hypothesis, was shown by Gödel in 1940 and the American mathematician Paul Cohen in 1963 to be undecidable from the usual axioms.

The German-American architect Ludwig Mies van der Rohe said less is more. Hilbert's curve, 1891, is the strictest version: one line, no crossing, every point reached in the limit.

Each order of the Hilbert curve replaces every segment with a smaller copy of the whole, rotated to connect. At order 1 it is a ∪; at order 8 it is 65,535 segments long and the eye reads it as a solid grey square. It is a continuous map from the line onto the plane, which Cantor had shown possible in 1878 and most mathematicians found monstrous: a one-dimensional thing that is also two-dimensional. Today it is how databases index maps: points near each other on the curve are near each other on the ground, so a single sorted list can answer "what is nearby?"

A point has 1 vertex. Drag it: a line, 2. Drag that: a square, 4. A cube, 8. A tesseract, 16, with 32 edges, 24 square faces and 8 cubic cells.

Nothing in the construction stops at three. Each new dimension is the old shape swept sideways in a direction perpendicular to everything so far; the vertex count doubles and the edges are the old edges, twice, plus one per old vertex. What you see is a 4-D rotation projected to 3-D and then to the screen, the way a wire cube's shadow on a wall is a 2-D drawing of a 3-D thing. The inner cube that seems to grow and swallow the outer one is not changing size; it is swinging toward and away from you along an axis you do not have.

Joseph Fourier proposed in 1807 that heat flow could be understood by splitting any function into sines. The Academy didn't believe him. He was right about more than heat.

A square wave is a sine, plus a third of a sine three times as fast, plus a fifth of one five times as fast, and so on through the odd harmonics. Each circle in the canvas is one term; the smaller, faster circles ride on the larger, slower ones, and their final point traces the sum. Your ear does this decomposition physically: the cochlea is a bank of resonators, which is why a clarinet and a violin playing the same A sound different: same fundamental, different recipe of circles.

Earlier on this page, circles added up to draw any shape. JPEG does the same thing with a picture: every eight-by-eight square of pixels is exactly a mix of 64 patterns of stripes, from flat grey to the finest checkerboard.

Most of a photograph changes slowly: sky, skin, a wall in shadow. In those squares almost all the weight sits in the first few, smoothest patterns, and the fine ones carry next to nothing. Your eye is also far less sensitive to fine detail than to broad shading. So JPEG stores the smooth patterns carefully, rounds the fine ones hard, most of them all the way to zero, and packs the long runs of zeros into a few bits. A typical photo shrinks to about a tenth of its raw size before most people can tell.

Push it too far and the trick shows. Sharp edges, like a horizon or a line of text, need the fine patterns, and without them they ring and blur, and the eight-by-eight grid starts to appear. That blockiness on an over-compressed image is the mathematics showing through.

A speaker is a magnet, a coil of wire and a paper cone. Current flows one way and the cone pushes the air; it flows back and the cone pulls. Sound is that push, travelling.

What reaches your ear is a single number changing in time: air pressure, rising and falling a few hundred to a few thousand times a second. A piano, a voice and an orchestra are all one wiggling line. Fourier showed that any such line is a sum of pure sine waves (see Circles that draw), and your inner ear does that sum in reverse, a coiled strip of hair cells each tuned to one frequency. When music is playing, the cone above moves with it; the dots are air molecules, crowding and spreading as the wave passes.

Pythagoras heard that a string stopped at ⅔ of its length sounds a fifth above the open string. Consonance is small whole numbers.

The figure plots one vibration against another: x = sin(a·t), y = sin(b·t + δ). At 1:1 it is a circle; 2:1, the octave, is a figure of eight; 3:2 the fifth; 4:3 the fourth; 5:4 the major third. Rational ratios close into a figure; irrational ones never do, and fill the square. Press the button to hear the two pitches the figure is drawing; the music behind you steps aside while it plays.

In the 1890s a Paris composer wrote one page of piano music and told the pianist to play it 840 times in a row. The first complete performance we know of, in 1963, took about eighteen hours.

The piece is Vexations, by Erik Satie (1866–1925). He left no explanation, only a direction to prepare in silence and serious stillness. Mathematically it is a single idea, a loop: take a short pattern and apply it again, and again. That is iteration, the same act that builds the Mandelbrot set from one rule. His better-known Gymnopédies are a gentler case: beneath the slow waltz, the accompaniment mostly swaps between two chords, a cycle of period two.

What repetition changes is not the music but the listener. By Shannon's measure a message you can predict carries almost no information, and after its first minute Vexations is almost perfectly predictable. Yet people who have sat through long performances describe hearing something new, which is the point Glass makes about focusing the ear. A pattern repeated enough stops being a sequence and becomes a place.

Press play. Over the soft two-chord figure, a low melody loops every so many repetitions and a high melody every so many others; the sliders set the two lengths. The grid marks each loop's first note, the low line in the bottom half of a square and the high line in the top, and outlines the squares where both begin together, with a bell. Pick 3 and 4 and you hear them slide apart and lock back into step every 12, the gears of minimalism.

1. 1 2. 1 2 3. 1 2 3 4. 1 2 3 4 5. Then back down. Glass called it additive process, and built some of the most recognisable music of the century from it.

In Two Pages (1968) a five-note figure is played, then repeated with one note added, then another, then subtracted, for twenty minutes; the ear stops hearing notes and starts hearing structure. Music in Twelve Parts (1971–74) layers several such processes at once. In Einstein on the Beach (1976) the chorus simply sings the numbers, one two three four, the counting made audible. And in the Opening of Glassworks (1982) the right hand plays in threes against the left in twos: the 3:2 ratio from the figure above, as rhythm instead of pitch. The canvas plays an additive process on any number of notes; watch the shape it makes, a triangle, the same shape as the sum 1 + 2 + … + n.

Improvisation looks like chance. Underneath it is a map of twelve keys, and the best players learned its geometry.

Arrange the twelve keys in a circle of fifths and every way of cutting the octave into equal steps becomes a regular shape. Jump a major third, four semitones, and three jumps bring you home: a triangle. A minor third gives a square, a whole tone a hexagon, a tritone a straight line through the middle. Those are the divisors of 12, and they are the only ways to slice an octave evenly. In 1959 the saxophonist John Coltrane recorded "Giant Steps" with its key centres on the triangle, at B, G and E♭, and later drew his own diagram of the circle to explain how he thought about it. Tap a key, choose a shape, and press Trace to hear what moving across the circle in equal strides sounds like.

Even the looseness has a number in it. When jazz players swing, the long note of a pair is not exactly twice the short one: the ratio drifts with the tempo, nearer three to one when slow and nearer even when fast. Improvisation is itself a balance of the predictable and the surprising, the trade Shannon measured, and researchers fit probability models to recorded solos. It is not random. It is chance with rules, which is also how a coin flipped many times builds a bell curve.

The Central Limit Theorem: add up enough independent small chances, of almost any kind, and their sum is bell-shaped. It is why the normal distribution is normal.

Francis Galton built this board in 1873 to make the point physically. A ball that bounces left or right twelve times lands in a bin given by the number of rights; the count of paths to each bin is a row of Pascal's triangle, and Pascal's triangle, scaled, converges to e^−x²⁄2. Heights, measurement errors, the noise in a photograph, the daily moves of a stock, exam marks: anything that is the sum of many small independent pushes fall under the same curve. Galton called it the supreme law of unreason.

Edward Lorenz, 1961: a weather model restarted from printed numbers rounded to three decimals diverged completely from the original run. Determinism without predictability.

These three equations are a cartoon of convection: warm fluid rising, cooling, falling. They have no randomness in them. Yet the trajectory never repeats and never settles, orbiting two lobes and switching between them at moments no formula can forecast far ahead. The shape it fills, the strange attractor, has fractal dimension about 2.06. Lorenz's title for the 1972 talk gave the idea its name: "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"

John Conway designed the rules in 1970 to be as simple as possible while remaining unpredictable. It is Turing-complete: Life can compute anything that can be computed, including Life.

The glider gun (the programmer Bill Gosper, 1970) fires a glider every thirty generations, settling the question of whether a pattern could grow without bound. The R-pentomino, five cells, runs for 1,103 generations before stabilising, throwing off six gliders. People have built in Life a digital clock, a Turing machine, and a pattern that prints the decimal digits of π. Conway, who died in 2020, came to resent that the game eclipsed his deeper work on groups and surreal numbers. It is still the clearest demonstration that complexity needs no complexity to come from.

If every sick person infects two more, one case becomes a thousand in ten rounds. If every sick person infects fewer than one, the chain breaks on its own.

In 1927 William Kermack and Anderson McKendrick, a chemist and a doctor in Edinburgh, wrote an outbreak as three numbers: the people who can still catch it, the people who have it, and the people who have recovered. Each day some of the first group catch it from the second, and some of the second move into the third. From those two rules comes the curve that rises, peaks and falls, and the single number that governs it, R, the average number of people each case infects. R is not fixed by the germ alone. Distance, ventilation, masks, handwashing and staying home all lower it, which is why, in 2020, the whole world spent months trying to push one number below one.

Two things fall out of the mathematics. The first is the threshold: once a fraction 1 − 1⁄R of people are immune, each case meets too few people it can infect and the outbreak shrinks by itself. For measles, with an R near 15, that is about 93 percent, which is why vaccination rates matter so much for it. The second is the shape. Lowering R does not just delay the outbreak, it squashes the whole curve, so fewer people are sick at once and fewer catch it at all. That is what flattening the curve meant: keeping the peak under the line of how many people the hospitals can care for at once.

The same arithmetic shapes the germs themselves. A germ that harms its host more, adding a death rate α, ends each infection sooner, and R becomes β ⁄ (γ + μ + α), where μ is the ordinary death rate. Harm usually buys some extra spread, so β grows with α, but more slowly. The ratio is then largest in the middle: a germ that kills too quickly runs out of hosts before it can pass itself on. Roy Anderson and Robert May set this out in 1982, and it is why many long-lasting infections settle toward mildness rather than toward the most harm they could do.

A medicine is a number that falls. Every so often it halves, and from that one fact you can predict how long a drug lasts, how often to take it, and how it builds up.

The rule is exponential decay: what is left after time t is C₀ · e^−kt, where k = ln 2 ⁄ half-life. It is the same e as in the growth chapter, run backwards. The hormone GLP-1, released by the gut after a meal to tell the body it has eaten, lasts only a minute or two in the blood, because an enzyme cuts it almost as fast as it is made. Semaglutide, the molecule behind drugs such as Ozempic, is built so that enzyme cannot cut it, and so that it clings to a protein that carries it around: its half-life is about a week. Press the buttons. A two-minute half-life is gone before the next dose; a one-week half-life, dosed weekly, stacks up.

Give a little and nothing happens. Give a lot and nothing more happens. In between, the effect climbs along an S.

That S is the Hill equation, from the British physiologist Archibald Hill in 1910: effect = Cⁿ ⁄ (EC50ⁿ + Cⁿ), the share of receptors a drug has switched on. Plot it against the logarithm of the dose and it is exactly the logistic curve 1 ⁄ (1 + e^−x), the same bend the neuron used in the learning chapter, with n setting how steep it is. Steepness is a safety number. At n = 1 the effect needs an 81-fold change in dose to travel from 10% to 90%; at n = 4 it needs only 3-fold, so a small error in dose swings a result from almost nothing to almost everything. Drag the dose and the steepness and watch the window narrow.

You cannot photograph a molecule in three dimensions. You photograph its shadow from many sides and let arithmetic rebuild the rest.

On the left is a made-up molecule. On the right is what a computer rebuilds from nothing but its one-dimensional shadows, taken from evenly spaced angles. With a few views you get a blur with streaks; add views and the shape sharpens; tick the filter, a ramp that boosts fine detail, and it snaps into focus. The Austrian mathematician Johann Radon proved in 1917 that this is possible. The same idea powers the CT scanner, which won Godfrey Hounsfield and Allan Cormack the 1979 Nobel Prize in Medicine. Cryo-electron microscopy does it for molecules: it freezes thousands of copies in ice, images each from an unknown direction, works out the directions, and builds a 3-D map. It won the 2017 Chemistry Nobel for Jacques Dubochet, Joachim Frank and Richard Henderson, and it is how many drug targets are now seen.

Stretch the dough to twice its length, fold it in half, repeat. Two specks of flour that start side by side are twice as far apart after every fold.

Mathematicians call this the baker's map, and it is one of the simplest pictures of chaos there is. The rule is completely fixed: double, fold, nothing random anywhere. Yet a gap of a thousandth of an inch becomes an inch after ten kneads, because 2¹⁰ is 1,024. Any uncertainty in where a speck began, however small, is doubled every time, so after enough kneads nobody can say where it will be. This is the same sensitivity as the butterfly in the weather, made with flour and your hands. It is also exactly why kneading works: stretching and folding is the fastest way there is to mix, and a baker needs no spoon to spread the yeast evenly through the whole loaf.

Bend a floppy slice into a U along its length and the tip stops drooping. That is a theorem.

In 1827 Carl Friedrich Gauss proved what he called his Theorema Egregium, his remarkable theorem: a surface keeps its curvature, the product of how much it bends in two directions, unless you stretch or tear it. A flat slice has zero curvature, and without tearing the crust it must stay zero. So once you curve it across, it cannot also curve along: the droop is forbidden. The same mathematics stiffens corrugated roofing, a tape measure that stands out straight, the fluted edge of a leaf, and a sheet of paper you can hold out flat by giving it one gentle curl.

Every greengrocer stacks oranges the same way, and it took mathematics almost four hundred years to prove they are right.

In 1611 Johannes Kepler claimed that no way of packing equal spheres fills more space than the grocer's pyramid, each layer sitting in the hollows of the one below, which fills π ⁄ √18, about 74 percent. Gauss proved it for regular, repeating stacks in 1831, but an irregular arrangement might still have done better. In 1998 the American mathematician Thomas Hales announced a proof that reduced the question to thousands of cases and checked them by computer; it was published in 2005, but the referees said they were only 99 percent sure. So Hales led a project called Flyspeck to have a computer check every logical step, finished in 2014 and published in 2017. In 2016 Maryna Viazovska proved the same kind of result in eight and twenty-four dimensions, and won a Fields Medal for it.

A turkey twice as heavy does not take twice as long. It takes about one and a half times as long.

Heat comes in through the skin and has to travel to the centre. Make the bird twice as heavy and every length grows by the cube root of 2, about 1.26, and the time heat takes to soak through grows with the square of that length. So the cooking time grows with weight to the power two thirds. If a 12 lb turkey takes 3 hours at 325 °F, a 24 lb turkey takes about 3 × 2^2⁄3, nearer 4 hours 45 minutes than 6. The same square–cube law is why a big pot of soup stays hot longer than a small cup, why an elephant needs big ears to cool down, and why a shrew must eat close to its own weight in food every day to keep warm.

The screen never sees your finger. It sees a few crossings of wires where the electricity behaves slightly differently, and it does arithmetic.

One layer of wires runs across, another runs down, and where they cross they form tiny capacitors that hold a little electric charge. The controller pulses each row in turn and listens on every column, more than a hundred times a second. Your body conducts, so a fingertip near a crossing steals some of the field between the wires, and the signal there drops. Near the centre of your touch it drops a lot, at the edges a little.

No single crossing can say where you are; they are too far apart. But weigh each crossing's position by how much its signal changed and take the average, and the answer lands between the wires, almost exactly where your finger is. It is the same centre-of-mass sum that balances a seesaw. Because every touch shows up as its own bump in the grid, the screen can follow many fingers at once, which is what made pinching to zoom possible when the iPhone arrived in 2007.

To leave Earth you need about 7 miles a second (11.2 km/s). To stay in orbit, about 4.8 (7.8 km/s), sideways, fast enough that you fall around the planet instead of into it. To come home and land on your tail, you need the exact moment.

The rocket equation, written by the Russian schoolteacher Konstantin Tsiolkovsky in 1903, is the reason rockets are mostly fuel: Δv = I_sp g₀ ln(m₀ ⁄ m_f). The speed you can gain grows only with the logarithm of how much heavier you are full than empty, so each extra mile a second (1.6 km/s) costs exponentially more fuel. That is why rockets drop stages, and why bringing a booster back, which means keeping fuel in reserve just to land, was long thought not worth it. The gauge on the right is that logarithm, draining.

A nearly empty Falcon 9 booster weighs about 56,000 pounds (25,600 kg); one Merlin engine at its lowest setting pushes harder than that. So the booster cannot hover. It must light its engine at precisely the altitude where constant deceleration brings it to zero speed at zero height: h = v² ⁄ 2a. Pilots call it a suicide burn; engineers call it a hoverslam. The dashed line in the canvas is that altitude, recalculated every frame.

An engine that pushes straight through a vehicle's centre of mass only moves it. Move the push off centre and it also turns it. Flying is the art of never letting the turning win.

The turning effect is torque, τ = r × F: the force times how far off centre it acts, measured at right angles. A single landing booster has one engine at the bottom, so any sideways push would tip it over; that is why its engine swivels a few degrees to steer while the rocket balances on its own exhaust. A car hovering on thrusters has a harder job. It must hold up its whole weight, F = m g, every second it is in the air: a 3,000 lb car needs 3,000 pounds of thrust (13,300 newtons) just to stay where it is. Split that between a jet at the front and one at the back and the slightest mismatch tips the nose up or down, so the thrust has to be rebalanced many times a second. A drone does the same with its four rotors, speeding one up and another down to hold itself level.

In 2018 Elon Musk said that a SpaceX option package for the next Tesla Roadster, the sports car from the electric-car maker he leads, would add about ten small cold-gas thrusters, jets of compressed air with no flame; he has since said it might even fly. The arithmetic of hovering explains why that is hard. Cold gas is a weak propellant, and the rocket equation sets how long it lasts.

A convolution: a 3 × 3 grid of numbers, laid over every pixel in turn, each neighbour multiplied by its weight and summed. Change the nine numbers above and you change what the machine can see.

A convolutional neural network does not have its kernels written by hand. It starts with random numbers and learns them from millions of labelled pictures. The first layer always ends up finding edges, much like the ones above, and much like the cells David Hubel and Torsten Wiesel found in a cat's visual cortex in 1959, which earned them a Nobel prize. Deeper layers combine edges into corners and curves, then wheels and faces, then "cyclist, moving left." A car's cameras run networks like this many times a second, on every camera at once. If you use your camera, the processing happens entirely in this page; no image leaves your device.

Before a machine can say "car", people have to show it, millions of times: here is a car, and here is exactly where it ends.

To a computer a photograph is a block of numbers. Every pixel is three of them, how much red, green and blue, each from 0 to 255, so one of these frames, 356 pixels tall and 720 wide, is a tensor of shape 356 × 720 × 3: 768,960 numbers. Nothing in those numbers says car.

A recognition network reads the block through layers of small filters, the same kind of sliding 3 × 3 weights as the edge detectors above, each layer turning one tensor into another: edges first, then corners and wheels, then whole shapes. At the end it gives boxes and, for each box, a score for every class it knows, turned into probabilities by the softmax function so that car, truck and person add up to 1. None of those filters is written by hand. They are learned, by the gradient descent of the section on learning, from examples that people have drawn.

That is the labour behind machine vision. In 2009 the computer scientist Fei-Fei Li and her team, at Princeton and then Stanford, released ImageNet: about 14 million photographs, each labelled by hand, much of it by paid online workers. In 2012 a convolutional network built by Alex Krizhevsky and Ilya Sutskever with their adviser, the British-Canadian computer scientist Geoffrey Hinton, cut the error on the ImageNet challenge so sharply that the whole field changed course. Microsoft's COCO dataset, in 2014, added about 1.5 million outlined objects in everyday scenes. Cars on the road now record far more video than people could ever label, so larger networks propose the boxes and people check them.

Drawing a box is itself a measurement, and two people never draw exactly the same one. The standard score for how well two boxes agree is the intersection over union: the area they share divided by the area they cover together, 1 for identical and 0 for no overlap. By the convention of the PASCAL Visual Object Classes challenge, which ran from 2005 to 2012, a guess counts as a correct find at 0.5 or more.

A camera divides depth away. Every photograph is the world flattened: x on the picture is f·X ⁄ Z, so a near small thing and a far large thing can land on exactly the same pixels.

Nature's answer is a second eye. Yours sit about 6.5 centimetres apart, so each sees a nearby object at a slightly different angle, and your brain turns that difference, the disparity, into distance by similar triangles: Z = f·B ⁄ d. It is why predators, from owls to cats to us, have eyes in front, giving up the wide view of prey animals with eyes on the sides of the head. But the triangle gets thin fast. Because Z is divided by the disparity, the error in Z grows with the square of the distance: twice as far is four times as blurred. Past a few dozen metres your two eyes see almost the same picture, and you judge distance from other clues instead: how big things ought to be, which object hides which, how things slide past as you move.

Machines face the same arithmetic. Lidar escapes it by measuring directly: a laser pulse goes out and back at the speed of light, and distance = c·t ⁄ 2, accurate to centimetres whether the object is near or far. Tesla, the carmaker Elon Musk has led since 2008, took the other path, the one nature took, at Musk's insistence: cameras only, and neural networks trained to infer depth from the same clues a person uses. The mathematics says what each bet costs. Triangulation is cheap and gets blurry with distance; timing light is exact and costs a spinning laser; inference is learned, and is only as good as its examples.

Double your speed and the braking distance quadruples. Look away for one second at 30 mph (48 km/h) and you travel 44 feet (13 m) blind.

Stopping has two parts. First the reaction: the car keeps moving while a person, or a computer, notices and decides, a distance of speed × reaction time. Then the braking: friction removes the car's kinetic energy, which takes v² ⁄ (2μg) feet. Because the second term is squared, speed matters more than anything else, which is the physics behind school-zone speed limits. It is also why engineers keep shaving the reaction time: a system that notices a pedestrian 0.2 seconds sooner than a driver buys about 9 feet (2.8 m) at 30 mph (48 km/h). Slide the pedestrian toward the car and watch the impact speed.

A number without a unit is not a number. In 1999 a $125 million spacecraft was lost because two teams disagreed about one.

NASA's Mars Climate Orbiter came in far too low and was lost as it reached Mars. One team's software had reported the push of its small thruster firings in pound-force seconds, while the navigation software expected newton-seconds, a factor of 4.45 apart. Every correction over nine months was off by that factor, and the errors added up. In 1983 an Air Canada Boeing 767 ran out of fuel at 41,000 feet (12,500 m) because the ground crew worked in pounds when the calculation needed kilograms; it carried less than half the fuel it needed and glided to a safe landing at a disused airfield in Gimli, Manitoba. Nothing was wrong with the arithmetic in either case. The units, the grammar of numbers, were wrong.

A neuron is a weighted sum and a bend: f(w₁x₁ + w₂x₂ + b). Learning is calculus: measure the error, ask which way each weight should move to make it smaller, take a small step, repeat.

The network above has two inputs (where you clicked), two hidden layers of eight neurons, and one output (white or black), 97 weights in all. Each frame it looks at your points, computes how wrong it is, and uses the chain rule to send that error backwards through every layer: backpropagation, popularised in 1986 by David Rumelhart, Geoffrey Hinton and Ronald Williams. Each weight moves a little in the direction that reduces the error. That is gradient descent, and it is the whole of the trick. The large language models that write and talk use the same step, on a few hundred billion weights instead of 97, with text instead of dots.

A memory is a valley. Give the network a damaged picture and it rolls downhill into the nearest thing it knows.

Each of the 100 squares is a neuron, on or off, and every pair is joined by a connection that grows stronger when the two usually agree. Store three shapes that way and the network has three valleys, one for each. Start it anywhere, say a shape with a quarter of its squares flipped, and let one neuron at a time switch to whatever its neighbours vote for. The total disagreement, the energy, can only go down, so the picture settles at the bottom of the nearest valley. That is the American physicist John Hopfield's 1982 model of memory, and it shared the 2024 Physics Nobel Prize with Geoffrey Hinton.

Press Dream and the network starts from pure noise. It rolls downhill all the same and lands wherever the noise happens to fall: on one of the three shapes, or just as often on the photographic negative of one, since a flipped picture is a valley too, or on a blend it was never taught. In the 1980s Francis Crick, co-discoverer of the structure of DNA, and Graeme Mitchison suggested that this is what dreams are for. Waking learning digs the valleys, and dreaming visits the false ones so they can be filled in. Hopfield and colleagues showed the idea can work in the mathematics. Nobody has shown it is why we dream.

A worm a millimetre long finds its dinner with a rule a computer would recognise: if things are getting better, keep going; if they are getting worse, turn.

Caenorhabditis elegans, a soil roundworm with exactly 302 neurons and the first animal whose complete wiring diagram was mapped, in 1986, cannot see, and its head is too small to smell in stereo. It compares the scent now with the scent a moment ago, the rate of change dC⁄dt. While it rises, the worm swims straight; when it falls, the worm turns more often, in a new random direction. That bias is enough. A random walk that turns less when things improve drifts up the gradient, and bacteria find food the same way, a strategy called run and tumble.

It is the training of a neural network, run uphill. Neither the worm nor the network can see where the best place is. Each only measures whether the last step helped, and leans the next one that way.

A language model is a machine for guessing the next word. Everything else it does, answering, translating, writing code, comes from guessing very, very well.

In 1948 Claude Shannon counted which letters and words follow which in English, and showed that those counts alone produce text that looks eerily like language. The probability of the next word is just a fraction: how often it followed these words, divided by how often these words appeared. In Lewis Carroll's Alice's Adventures in Wonderland (1865), "very" is followed by "tired" and "sleepy", so after "very" those are the likeliest guesses. Attention turns the same idea into geometry: every word becomes a list of numbers, a point in many dimensions, and how much one word attends to another is the angle between them, measured by a dot product and turned into weights that add up to 1.

GPT does the same thing at a scale that changes what it is. In 2017 a team at Google led by Ashish Vaswani published "Attention Is All You Need", the transformer: every word looks at every earlier word and decides how much each one matters to it, so "it" can find the noun it refers to, ten words back. OpenAI, founded in 2015 by Sam Altman, Elon Musk, Ilya Sutskever, Greg Brockman and others, with Musk among its first and largest funders, built GPT on the transformer; its GPT-3, in 2020, had 175 billion numbers, its parameters, and learned them from about 300 billion words of text. Training is the calculus from the section on learning: guess the next word, measure how wrong the guess was, and nudge every one of the 175 billion numbers a tiny step in the direction that makes it less wrong. Repeat a few hundred billion times. ChatGPT added one more step in 2022: people ranked its answers, and it was tuned toward the ones they preferred.

Read the input. Update the world. Draw it. Repeat, sixty times a second. Every game, from Pong to the largest open world, is that loop.

The view above is drawn the way Wolfenstein 3D drew it in 1992: for each column of pixels, cast one ray from your eye across a flat map until it hits a wall, measure the distance, and draw a vertical line whose height is inversely proportional to it. Near walls are tall; far walls are short. Multiply the distance by the cosine of the ray's angle and the fish-eye bulge disappears. Turn on the map to see the rays fan out from you. Modern games do the same thing with millions of triangles: every point in the world is multiplied by a matrix that moves it into the camera's view, then divided by its depth, the same perspective divide Brunelleschi found with a mirror in 1413.

Pick a wall. Everything is either in front of it or behind it. Do the same on each side, and again, and you have a tree that answers the hardest question in drawing a world: what is in front of what.

That is binary space partitioning. Its first use was in flight simulators: Robert Schumacker and colleagues at General Electric split scenes this way for pilot trainers in 1969. Henry Fuchs, Zvi Kedem and Bruce Naylor gave it its modern form in 1980, a tree built once, in advance, from the walls of a scene. The programmer John Carmack, at id Software, built Doom (1993) on it, which is how a home computer could draw rooms of any shape at speed, and Quake (1996) took it into full three dimensions, cutting space with planes instead of lines. It is the next step after the raycaster above: instead of searching for walls one ray at a time, the world is sorted once, in advance. Before each frame the engine walks the tree from wherever you stand: the side of each wall you are on first, then the other side. Out come the walls in order from nearest to farthest, with no sorting. Move the dot and the numbers on the walls re-order themselves.

If a machine has a thousand parts and each one works 999 times in 1,000, the whole machine works only about 37 times in 100. Make each part ten times more reliable and it works 90 times in 100. Almost all of the work is in the last nines.

Chances of independent events multiply, so a system of n steps that each succeed with probability p succeeds with probability pⁿ. That one line explains why ideas are cheap and building is hard. In 1837 the English mathematician Charles Babbage designed the Analytical Engine, a mechanical computer with a memory, a processor and punched-card programs; in 1843 Ada Lovelace published the first program for it. The idea was complete. The engine needed tens of thousands of parts machined to tolerances that strained the workshops of the time; the money ran out, and it was never finished. General-purpose computers were built a century later, once electronics made each part dependable enough.

In 1637 Pierre de Fermat wrote in a book's margin that he had a proof too large to fit there. The statement took 357 years to prove: Andrew Wiles finished it in 1994, after seven years of work, the last of them spent with Richard Taylor closing a gap found in his first proof. Many people had the idea of powered flight; in 1903 Orville and Wilbur Wright flew because they had built their own wind tunnel, measured about two hundred wing shapes and corrected the published tables everyone else relied on. The mathematics for landing a rocket upright was flown in tests by 2012. Turning it into a booster that came home took a run of crashes and tipped-over landings at sea before the first one landed upright, on land, in December 2015.

A hash is a one-way street: easy to compute, impossible to reverse. Change one letter and every bit changes. That is how a password is checked without ever being stored.

RSA is the other half. Pick two primes, p and q. Their product n is public; p and q are not. Anyone can lock a message with n; only someone who knows p and q can unlock it, because unlocking needs φ(n) = (p − 1)(q − 1), and finding that from n alone means factoring it. With the primes above, your phone factors n instantly. With 300-digit primes, every computer on Earth would need longer than the universe has existed. Every padlock in your browser bar is this arithmetic.

When you typed the address, your request was cut into packets, each one given a number, and sent by whatever route was free. They arrived out of order and were reassembled. Nothing on the way knew what they said.

The internet is a graph: about 75,000 networks, joined by cables and agreements, with no centre. Every router runs a shortest-path algorithm (the Dutch computer scientist Edsger Dijkstra's, 1956, written in twenty minutes at a café in Amsterdam) to decide where to send your packet next. The page you are reading came through roughly a dozen of them. Along the way, bits flip: cosmic rays, bad solder, a noisy radio. Richard Hamming, tired of his weekend computer runs failing at Bell Labs in 1950, designed codes that add a few check bits so that an error can be not only detected but located and reversed. Every QR code, every CD, every message from a Mars rover carries its own repair instructions. Claude Shannon had proved two years earlier that this was possible up to a hard limit, and told nobody how; Hamming showed how.

try · press Flip a random bit and watch the receiver repair it

Two strangers, shouting across a crowded room, can agree on a secret that nobody in the room can learn. That sentence sounded impossible until 1976. It is the reason you can buy anything online.

Whitfield Diffie and Martin Hellman's idea: pick a public number g and a public prime p. You choose a secret a and shout g^a mod p. I choose b and shout g^b mod p. You raise what I shouted to your a; I raise what you shouted to my b; we both hold g^ab mod p. Everyone heard everything, and nobody can get ab back from g^a and g^b, because undoing a modular power, the discrete logarithm, is as hard as factoring. This handshake happens, with 256-bit numbers on an elliptic curve, every time a padlock appears in your address bar; it happened when you opened this page.

Privacy also has a mathematics of its own. In 2000 Latanya Sweeney showed that 87 % of Americans are uniquely identified by three "anonymous" facts: ZIP code, birth date, sex, and re-identified the governor of Massachusetts in a released hospital dataset. Cynthia Dwork's answer, differential privacy (2006), adds precisely calibrated noise so that a dataset says the same thing whether or not you are in it. Apple and the US Census use it now. Privacy is not secrecy; it is a bound on how much any one person moves the answer.

try · move both secrets: the shared key always matches

Press a key and an electric current runs through three wired wheels, bounces off a fourth, and comes back through all three as a different letter. Then the first wheel turns, and the same key gives a different letter next time.

The German forces' Enigma machine turned every message into a new scramble, letter by letter. Each rotor is a fixed shuffle of the alphabet, a permutation; stacking three and turning them after every key makes a new permutation for every letter of a message. The settings changed every day: which three of five rotors, in what order, where each started, and ten cables on a plugboard swapping pairs of letters. That is 158,962,555,217,826,360,000 possible starting settings, and the operators believed no one could search them.

But the reflector that sends the current back through the rotors has a price: a letter can never encrypt to itself. That single impossibility let codebreakers rule out most guesses at once. In 1932 the Polish mathematician Marian Rejewski worked out the rotor wiring using the theory of permutation groups, with Jerzy Różycki and Henryk Zygalski. At Bletchley Park, Alan Turing and Gordon Welchman built the bombe, an electromechanical machine that tested thousands of settings an hour against a guessed word in the message. Historians estimate the breaking of Enigma shortened the Second World War by about two years.

There are about eight billion people. To pick out one of them takes thirty-three yes-or-no questions, because 2³³ ≈ 8.6 billion. Every small fact about a device is worth some fraction of a question. A screen size is worth about four, a time zone three; fonts, graphics card and settings add the rest.

This is Shannon's entropy, the same measure that sets how far a file can be compressed. In 2010 the Electronic Frontier Foundation measured the browsers of half a million volunteers and found that 84 % were unique: no cookie, no login, just the sum of small choices. Combined, those small facts behave like a hash: change one and the whole fingerprint changes, keep them and it stays the same from site to site.

Mathematics has a more hopeful answer, too. In 1985 Shafi Goldwasser, Silvio Micali and Charles Rackoff described proofs of knowledge that reveal nothing but the fact that you know: that you are over eighteen without saying your birthday, that you have the password without sending it, that a sum is correct without showing the numbers. Zero-knowledge proofs are no longer a curiosity; they settle transactions and verify identities today. The oldest question about identity (are you the same person you were, with all your parts replaced?) turns out to have an engineering form: what is the smallest thing you must reveal to be believed?

To recognise you, a phone throws a pattern of invisible dots at your face and measures how far each one moves.

Face ID's projector casts more than 30,000 infrared dots in a fixed, random-looking pattern. A camera set a short distance to one side photographs them. On a flat wall every dot lands where the phone expects it; on a face, each dot is pushed sideways by an amount that depends on how near that patch of skin is, the same triangulation the self-driving car uses with two cameras, and the same geometry Eratosthenes used to measure the Earth. From those shifts the phone builds a depth map of your face, turns it into a list of numbers, and compares the list with the one it stored. Apple puts the chance that a random stranger's face would unlock your phone at about one in a million.

Three kinds of fingerprint run through this journey: the small facts a device gives away, a face as a cloud of dots, and, at the end, the rhythm of hands typing a message. Each is identity written as numbers, and each is protected by the same mathematics.

Each satellite broadcasts the time. Your phone notes when the signal arrives, multiplies the delay by the speed of light, and knows how far away that satellite is. Light covers about a foot in a billionth of a second, so a clock that is off by one millionth of a second puts you nearly a thousand feet out.

One distance puts you somewhere on a sphere. Three spheres meet at a point, and a fourth satellite lets the phone correct its own cheap clock, which is why it needs four. The satellites carry atomic clocks, and here relativity comes in twice. They move at 8,700 mph, and Einstein's special relativity says a moving clock runs slow: about 7 microseconds a day. They sit 12,550 miles up, where gravity is weaker, and general relativity says a clock higher in a gravity well runs fast: about 46 microseconds a day. Together the clocks gain about 38 microseconds a day.

That sounds like nothing. At the speed of light it is about 7 miles of range error a day, growing every day. So the satellite clocks are built to tick slightly slow before launch, 10.22999999543 million times a second instead of 10.23 million, so that in orbit they keep time with clocks on the ground. Every map on every phone is a daily test of relativity, and relativity passes.

Imagine someone clicking links at random, forever. Most of the time they follow a link from the page they are on; now and then they get bored and jump to any page at all. The share of time they spend on each page is its rank.

In 1998 two Stanford students, Sergey Brin and Larry Page, used that imaginary surfer to rank the web. Search engines of the time mostly counted how often a page used your words, which anyone could fake. Brin and Page counted links instead, and weighted each link by the rank of the page it came from, so a single link from an important page outweighs a hundred from pages nobody links to. They called it PageRank, and founded Google on it that year.

The mathematics is older than the web. The surfer is a Markov chain, named for Andrey Markov, who in 1906 studied processes where the next step depends only on where you are now. Here the boredom matters: the surfer jumps to a random page 15% of the time, so they can never get stuck in a corner of the web, and the long-run shares are guaranteed to exist and to be unique.

A QR code still scans with a coffee stain on it. A scratched CD still plays. The same idea brought back the Voyager pictures of Uranus and Neptune across nearly three billion miles (4.5 billion kilometres) of noise.

In 1960 Irving Reed and Gustave Solomon, at MIT's Lincoln Laboratory, turned a message into a polynomial. Its bytes are the coefficients; send not just the message but extra values of the polynomial, and the receiver has more points than it needs to pin the curve down. With twelve spare bytes, as here, it can find and fix any six that were damaged, without being told which ones. Reed–Solomon codes went on to compact discs, DVDs, Blu-ray, deep-space probes, hard drives and barcodes.

When Masahiro Hara at Denso Wave designed the QR code in 1994 to track car parts in a factory, he built Reed–Solomon in from the start. A QR code's error-correction level is chosen when it is made: L, M, Q or H restores about 7, 15, 25 or 30% of the code. Some companies use level H to print their logo over the middle of the code; the maths quietly repairs the hole.

If you both cooperate, you each get 3. If you both defect, you each get 1. If one defects while the other cooperates, the defector gets 5 and the trusting one gets nothing.

Whatever the other player does, you score more by defecting, so two perfectly logical players both defect and each walk away with 1 when they could have had 3. Merrill Flood and Melvin Dresher found the game at the RAND Corporation in 1950, Albert Tucker gave it its story of two prisoners, and it showed that John Nash's equilibrium, the point where neither player gains by changing alone, can be the worst result for both. RAND was studying the arms race at the time, and the same logic, two sides each made safer alone by building more weapons and both made poorer together, ran through Cold War strategy from John von Neumann to the economist Thomas Schelling.

But people rarely meet only once. In 1980 the political scientist Robert Axelrod invited game theorists to send computer programs to play the game against each other for 200 rounds. The winner was the shortest program entered, four lines from the psychologist Anatol Rapoport: Tit for Tat. Cooperate first, then do whatever the other player did last. He ran the tournament again with more entrants who knew the result, and it won again. Axelrod summed up why: it is nice, never the first to defect; retaliatory, so it cannot be exploited; forgiving, returning to cooperation the moment the other side does; and clear, so the other side soon learns what it will do. Tit for Tat never beats the player in front of it. It wins by doing well with everyone.

For three thousand years the record was held by a person with a pen. For the last seventy it has doubled roughly every two years, held by a machine.

The Babylonians had 3⅛. Archimedes squeezed π between polygons and found two decimals. Zu Chongzhi in fifth-century China found seven, a record that stood for nine hundred years. The Dutch mathematician Ludolph van Ceulen, around 1600, spent most of his life on 35 digits and had them carved on his gravestone. The English amateur William Shanks spent twenty years computing 707 digits by hand in 1873; the last 180 were wrong, and nobody noticed until 1944. ENIAC found 2,037 digits in seventy hours in 1949. We now have over a hundred trillion. No one needs more than forty to measure the observable universe to the width of an atom. We compute them anyway, for the same reason we climb.

In 1976 a computer checked the Four Colour Theorem and mathematicians asked whether a proof no one could read was a proof. In 2025, machines reached gold-medal level at the International Mathematical Olympiad, and the question changed: whether a proof no one thought of is a discovery.

Kepler's sphere-packing conjecture, open since 1611, was proved in 1998 and took a decade and a computer to verify. AlphaProof solved Olympiad problems in 2024. Within a year, systems from several labs were producing proofs for Erdős problems that had sat unsolved for decades, not all of them correct, some rediscoveries, a few genuinely new. The pace is the point. The limiting step is becoming what to ask, not how to answer.

What does not change: a proof is still a proof only when it can be followed. Gödel showed in 1931 that no system, human or machine, can prove all the truths it can state, and Turing showed in 1936 that no machine can settle every question. The unknown is not a shrinking territory we will someday finish mapping. It is a property of the map.

Seen from the Earth, Mars drifts east for most of two years, stops, runs backwards for about ten weeks, stops again and carries on. Seen from the Sun, it just goes round.

Around 150 AD Ptolemy put the Earth at the centre and explained the backward loops by putting each planet on a small circle, an epicycle, whose centre rode on a large one. It worked well enough to predict the sky for fourteen hundred years, but every planet needed its own circles, adjusted and readjusted as the predictions drifted. In 1543 the Polish astronomer Nicolaus Copernicus moved the centre to the Sun, and the loops became an illusion: just the Earth overtaking a slower planet on the inside lane, the way a car you pass seems to slide backwards. In 1609 Kepler, working from the Danish astronomer Tycho Brahe's measurements of Mars, replaced the circles with ellipses, and in 1687 Newton showed that ellipses follow from a single rule that also makes an apple fall and keeps the Moon in its orbit.

The planets did not change. The point of view did, and with it a page of circles turned into one line. That is what the mathematics in this journey keeps doing: the complicated thing is real, but underneath it is something simple, waiting for the right place to stand.