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Science

1,585 cards. Each one is a 40-second story with a five-minute read behind it. Page 7 of 27.

361. Flipping a matrix across its diagonal reveals surprisingly deep propertiesTake a grid of numbers and mirror it across the line running from top left to bottom right, so rows become columns. That simple move, the transpose, was introduced by Arthur Cayle…
362. Two orbiting bodies can be solved exactly, but adding a third breaks everythingPredicting how two stars or a planet and its moon move around each other is a solved problem with a neat, complete answer. Add just one more body and no general solution exists ex…
363. In taxicab geometry, circles are squares and pi equals exactly 4Measure distance the way a cab crosses Manhattan, block by block along the grid, and geometry turns strange. A circle becomes a tilted square. The ratio of its circumference to it…
364. Some rectangles cross themselves, and some have no right angles at allEveryone knows a rectangle has four right angles. Geometers are less sure. A crossed rectangle folds over itself like a bow tie, and on curved surfaces the shapes called rectangle…
365. The oldest trick in logic: assume the opposite and watch it collapseTo prove something is true, you can pretend it is false and follow the consequences until they fall apart. This move, reductio ad absurdum, runs from a satirical Greek poem about …
366. The rhombus is named after an ancient spinning toyDiamond, lozenge, calisson: the four-sided shape with equal edges has a menu of names, one of them a French sweet. The word rhombus itself comes from a Greek term for something th…
367. One infinite sum quietly encodes the pattern of every prime numberAdd one, a half squared, a third squared and so on forever, and you have touched the Riemann zeta function. Euler found it was secretly built from the primes. Riemann then pushed …
368. The universe measures flat, yet it could still loop back on itselfDraw a giant triangle across space and add up its angles. Measurements so far say they total 180 degrees, meaning space is flat to within a fraction of a percent. That settles the…
369. Shing-Tung Yau won the Fields Medal while holding no passport at allIn 1982 Shing-Tung Yau became the first ethnic Chinese mathematician to win the Fields Medal. He accepted it officially stateless, his Hong Kong residency revoked and American cit…
370. The sinc function is the shape hiding inside every digital recordingDivide the sine of x by x and you get a wave that swells to a peak of 1 and then ripples away in both directions. That humble curve, the sinc function, is how engineers rebuild a …
371. Squaring the circle took two thousand years to prove impossibleBuild a square with exactly the area of a given circle, using only compass and straightedge, in a finite number of steps. Greek geometers tried. So did prisoners, Jesuits and legi…
372. A teenager on a ship to England worked out how heavy dead stars can getSailing to Cambridge in 1930, a young Indian graduate spent the voyage adding relativity to the physics of collapsed stars. The upshot, now called the Chandrasekhar limit, says no…
373. A slanted box hides one of geometry's most stubborn unsolved puzzlesTake a cube and push it sideways so every face leans into a parallelogram. The result is a parallelepiped. In 2009 mathematicians found dozens whose edges and diagonals are all wh…
374. The humble parallelogram depends on Euclid's most controversial assumptionEveryone learns that a parallelogram's opposite sides match in length. What schoolbooks rarely mention is that this cannot be proved without Euclid's contested parallel postulate.…
375. Three lengths of chain in the ratio 3, 4, 5 make a right angleGardeners and surveyors have a low-tech way to mark out a square corner across a large field: lay three chains in the ratio 3:4:5 into a triangle. The corner opposite the longest …
376. Piero della Francesca painted like a geometer because he was oneToday Piero della Francesca is celebrated for calm, luminous frescoes. In his own time he was also a serious mathematician who wrote treatises on perspective and on the five regul…
377. From Pythagorean triples to Gauss, quadratic forms connect ancient puzzles and modern mathsWhich whole numbers can be written as the sum of two squares? That question, answered by Fermat, belongs to the theory of quadratic forms, polynomials where every term has degree …
378. Give a paper strip one half-twist and it loses a sideTape the ends of a paper strip together after a single half-twist and you get a surface with just one side and one edge. Mathematicians described it in 1858, but Roman mosaic make…
379. How many misses before the third six? The negative binomial knowsRoll a die and count every non-six until the third six appears. That tally follows the negative binomial distribution, a workhorse of statistics that also describes why a few infe…
380. Why December means ten and a centipede never has a hundred legsDecember is the twelfth month, yet its name says tenth. Centipedes rarely have anywhere near a hundred legs. English is packed with number prefixes borrowed from Latin and Greek, …
381. Right angles escaped geometry and now describe Wi-Fi, DNA and vinyl recordsOrthogonal started life as a Greek word for a rectangle. Today mathematicians use it for vectors that meet at right angles, but engineers, chemists, statisticians and philosophers…
382. For two thousand years, mathematicians tried to prove Euclid's awkward fifth ruleEuclid's first four postulates are almost obviously true. The fifth, about when two lines must eventually meet, is long and clumsy, and generations of brilliant thinkers were sure…
383. John Conway grew tired of being famous for the Game of LifeIn October 1970 Martin Gardner's Scientific American column described a pen-and-paper pastime in which cells on a grid live or die by simple rules. It became Gardner's most-read c…
384. From body cells to chess games, real numbers get astonishingly bigYour body holds an estimated 37.2 trillion cells, and your brain around 100 trillion neural connections. Those figures are tiny next to the possible games of chess, estimated at a…
385. The law of cosines is Pythagoras stretched to fit every trianglePythagoras's theorem works only when a triangle has a right angle. The law of cosines fixes that by adding a correction term that depends on the angle between two sides, and it be…
386. The law of sines hides a circle behind every triangleDivide any side of a triangle by the sine of the angle facing it and you get the same answer for all three sides. That shared number is no coincidence: it equals the diameter of t…
387. The world's oldest dated four-by-four magic square was a perfume recipeAround 550 CE the Indian scholar Varahamihira drew a four-by-four grid for blending perfumes. Each cell named an ingredient and its proportion, and any four picked along a row, co…
388. Calculus took two thousand years and many countries before Newton and LeibnizThe word calculus means small pebble in Latin, after the stones once used for counting. The ideas it names were gathered almost as slowly as pebbles: Greek geometers, Chinese and …
389. A horoscope is a geometric sky map, even though its predictions failStrip away the fortune-telling and a horoscope is a piece of geometry: a stylised map of the sky above one spot at one moment, with the planets and key angles plotted on a wheel. …
390. Change one rule of Euclid and parallel lines multiply without endEuclid's geometry says that through a point off a line, exactly one parallel can be drawn. Hyperbolic geometry keeps every other rule but allows at least two, and therefore infini…
391. Why arcsin is called arc, and why sin to the minus one confuses peopleSine turns an angle into a ratio; its inverse turns the ratio back into an angle. Naming that reverse step has caused two centuries of muddle, from John Herschel's 1813 superscrip…
392. John Wheeler named black holes, wormholes and quantum foam, and taught FeynmanJohn Archibald Wheeler had a gift for names. He popularised black hole, coined wormhole, quantum foam and it from bit, and even renamed Enrico Fermi's slower downer the neutron mo…
393. Free software is about freedom of speech, not free beerFree software can cost money and can even be sold for profit. What makes it free, in Richard Stallman's sense, is that anyone who receives it may run it, study it, change it and p…
394. One theorem about differential forms swallows four famous results of calculusStokes' theorem, Green's theorem, the divergence theorem and even the fundamental theorem of calculus look like separate facts in a textbook. Written in the language of differenti…
395. A Hiroshima regret helped inspire the idea of the digital libraryVannevar Bush had backed research behind the bomb dropped on Hiroshima. After seeing the destruction, he imagined the Memex, a desk with two screens and a keyboard giving rapid ac…
396. The first clickable banner ad got clicked by 44 percent of viewersIn 1994 AT&T ran the first clickable web banner, its You Will campaign, and within four months 44 percent of the people who saw it clicked. That banner belongs to the early years …
397. Sample a sound at twice its highest pitch and nothing is lostThe Nyquist–Shannon sampling theorem makes a startling promise: a signal can be rebuilt exactly from snapshots, provided they come faster than twice its highest frequency. That ru…
398. Divergence measures how much a flow swells or shrinks at a pointWarm a pocket of air and it expands in every direction, so arrows describing its motion all point outward. Mathematicians capture that outward push with a single number called div…
399. Count every faucet and drain inside a box to know its total outflowPicture a closed surface floating in a tank of water. The divergence theorem, also named after Gauss and Ostrogradsky, says the net amount of water crossing that surface equals th…
400. A beam theory from around 1750 waited for the Eiffel TowerEngineers can predict how far a beam will sag and how much it can carry thanks to Euler–Bernoulli beam theory. It was first stated around 1750, but it saw no large-scale use until…
401. Quake III's mysterious constant 0x5F3759DF sparked a hunt for its authorBuried in the code of Quake III Arena was a baffling line that subtracted a number from the constant 0x5F3759DF and somehow produced an inverse square root. Programmers guessed Jo…
402. Leibniz used the chain rule first, and fumbled a sign doing itIf a car is twice as fast as a bicycle and the bicycle four times faster than a walker, the car beats the walker eightfold. That multiplication of rates is the chain rule of calcu…
403. Christoffel symbols track how coordinates twist across a curved spaceOn a curved surface, the directions you call north and east quietly rotate as you move. Christoffel symbols are the bookkeeping for that drift, a three-dimensional grid of numbers…
404. Before microchips, computers stored bits as sound waves in mercuryEarly computers remembered things in some strange ways. One design sent pulses of sound rippling through tubes of mercury; another painted dots of charge on a cathode ray screen. …
405. Solving the cubic sparked secret formulas, betrayal and a maths duelRenaissance Italy settled mathematical rivalries with public contests, money on the table. The prize problem was the cubic equation. A method guarded as a secret, a broken promise…
406. Curl measures how much a flowing field spins at each pointDrop a tiny paddle wheel into moving water and it may start to turn. The curl of a vector field captures that spin as an arrow: its length gives the strength of the swirl, its dir…
407. Two vertical bars around a number trace back to Weierstrass in 1841The absolute value strips a number of its sign, so both 3 and minus 3 become 3. Think of it as distance from zero. That simple idea, written with the twin bars Karl Weierstrass in…
408. Some infinities are bigger than others, and Cantor labelled them with alephGeorg Cantor realised that infinite sets come in different sizes and named those sizes after the Hebrew letter aleph. The smallest, aleph-null, counts the natural numbers. Whether…
409. Every crystal fits one of just 14 repeating point patternsSalt, diamond and quartz look nothing alike, yet the scaffolding beneath every crystal belongs to one of only 14 point arrangements in three dimensions. Auguste Bravais catalogued…
410. The Cauchy–Schwarz inequality caps how much two vectors can agreeMultiply two vectors together with a dot product and the result can never exceed the product of their lengths. That modest-sounding cap, the Cauchy–Schwarz inequality, took three …
411. The charity behind Wikipedia writes none of its articlesThe Wikimedia Foundation keeps Wikipedia and 14 sister projects online, yet it neither writes nor edits their content; volunteers do all of that. The foundation was created in 200…
412. A total order means any two things can be ranked against each otherPick any two real numbers and one is always less than or equal to the other. Pick any two letters and the dictionary settles which comes first. Mathematicians call an arrangement …
413. Almost every number is transcendental, yet proving any single one is hardPi and e are the celebrities, but transcendental numbers, those that are not the root of any polynomial with integer coefficients, make up almost all real numbers. The strange par…
414. Triangular numbers, the stacked dots behind a famous schoolboy legendArrange dots in rows of one, two, three and so on until they form a neat equilateral triangle, and the total is a triangular number. A popular tale says young Gauss found their sh…
415. U and V were once the same letter, split apart by medieval scribesFor centuries printers wrote upon as vpon and have as haue. U and V were a single letter with two shapes, pointed at the start of a word and rounded elsewhere, whatever sound it m…
416. Surreal numbers grew out of Go endgames and contain infinity itselfWhile studying endgames in the board game Go, John Horton Conway stumbled onto a number system that holds every real number plus infinite and infinitesimal ones. Donald Knuth gave…
417. Aristotle's syllogism dominated logic for many centuries until Frege came alongAll men are mortal; Socrates is a man; so Socrates is mortal. Aristotle set out this pattern of reasoning around 350 BC, and it proved so durable that Immanuel Kant declared logic…
418. The symmetric group: every way of shuffling a set, treated as algebraTake a handful of objects and list every possible rearrangement. Treat each rearrangement as an action you can perform one after another, and you have the symmetric group. It soun…
419. Why a set of linear equations has one answer, none, or infinitely manyTwo straight lines on a page can cross once, run parallel forever, or lie exactly on top of each other. That picture explains every system of linear equations: any such system has…
420. In logic, a tautology is a statement that simply cannot be falseEither it is raining right now or it is not. Whatever the weather, the sentence holds. Logicians call such statements tautologies: formulas true under every possible assignment of…

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