All ideas
48 ideas in 7 threads. The ones marked ▶ have something to play with.
Shape and space
- c. 1800 BCRight triangles in clayIn a right-angled triangle, the square on the longest side equals the two other squares added together.
- c. 300 BCEuclid’s ElementsStart from a few things everyone agrees on, and prove everything else from them.
- c. 150 BCThe first trigonometry table ▶The straight line across a circle depends only on the angle it spans, so you can list it once and look it up forever.
- 1637Shapes become equations ▶Every point on a circle is the same distance r from the centre, written as numbers.
- 1854Curved spaceDistance depends on where you are; the g tells you how the space stretches at each point.
- 1908Spacetime ▶Distance in spacetime mixes time and space, with time counting negatively.
Numbers and algebra
- AD 628Zero becomes a numberNothing is a number too, with its own rules.
- c. AD 820Algebra gets its nameDo the same thing to both sides until the unknown stands alone.
- 1614Logarithms ▶To multiply two numbers, add their logarithms, then look the answer back up.
- 1748Euler’s formula ▶Raising e to an imaginary power walks you around a circle.
- 1829Eigenvalues ▶Some special directions are only stretched by a matrix, never turned; λ says by how much.
Change and calculus
- c. 1400Infinite sums in Kerala ▶An endless list of ever-smaller fractions adds up to one exact number.
- 1665–1684The invention of calculus ▶Adding up a rate of change over time gives you the total change. Speed and distance are two views of one journey.
- 1701Newton’s law of cooling ▶The hotter something is compared with its surroundings, the faster it cools.
- 1715Taylor series ▶Near a point, any smooth curve looks like a straight line, then a parabola, then a better and better polynomial.
- 1812Laplace transform and poles ▶Turn a signal into a function of s; the points where it blows up (the poles) tell you how the system rings, fades or grows.
- 1822Everything is waves ▶Any repeating pattern is a stack of pure waves, each with its own size.
- 10 June 2000The swaying footbridge ▶A guitar string’s equation plus one term: the crowd’s push. When it beats the damping, sway grows.
Fields and vector calculus
- 1813–1826The divergence theorem ▶What flows out through the skin of a region equals what is produced inside it.
- 1828Green’s theorem ▶The total swirl around the edge equals all the tiny swirls inside, added up.
- 1850–1854Stokes’ theorem ▶Swirl around the rim of any surface equals the curl poking through the surface.
Earth, planets and orbits
- c. 240 BCMeasuring the Earth with a shadow ▶If walking a distance d turns the Sun’s angle by 7.2°, then a full 360° circle around the Earth is fifty times d.
- c. AD 150Circles on circles ▶Add a small spinning circle to a big spinning circle and the point on the end traces loops. (Written here in modern notation Ptolemy never had.)
- 1543The Sun in the centre ▶Change where you stand, and a complicated motion becomes a simple one.
- 1609–1619Kepler’s laws of planets ▶Square a planet’s year and it is proportional to the cube of its average distance from the Sun.
- 1610A telescope on Jupiter ▶Double the falling time and an object falls four times as far.
- 1687Universal gravitation ▶Every mass pulls on every other; double the distance and the pull drops to a quarter.
- 1705–1758A comet returns on schedule ▶A comet on a long ellipse comes back every 76 years or so, nudged a little by Jupiter and Saturn.
- 1772Balance points in space ▶Where gravity and the turning of the system balance, the push in every direction is zero.
- 1801Finding a lost worldChoose the orbit that makes the total squared error between prediction and observation as small as possible.
- 1846A planet found on paper ▶Uranus feels the Sun, plus a small extra pull. Work backwards from the extra pull to find what is pulling.
- 1859Mercury misbehaves ▶The orbit’s closest point creeps around the Sun slightly faster than Newton predicts.
- 1956The age of the Earth ▶In every half-life, half of what is left decays; so the amount left tells you the time passed.
- 1995A planet around another sun ▶Kepler’s third law, with Newton’s constant: measure the year, and you know the orbit’s size.
Light and waves
- 1865Light is electromagnetism ▶The speed of light follows from two numbers measured with magnets and electric charges.
- 1887The experiment that found nothingLight’s speed came out the same whichever way the instrument pointed.
- 1912A ruler for the universe ▶Light spreads over a sphere, so a star twice as far looks four times dimmer.
- 29 May 1919Starlight bendsLight grazing the Sun is bent by about 1.75 arcseconds, twice the simple Newtonian guess.
- 1965The afterglow of the Big BangThe universe’s oldest light has been stretched by expansion until it is cold microwaves.
- April 2019The first image of a black hole ▶Each pair of dishes measures one wave-pattern ingredient of the image; together they rebuild it.
Spacetime and the cosmos
- 1905Special relativity ▶The faster a clock moves past you, the slower you see it tick.
- 1905E = mc²A small amount of mass is a huge amount of energy.
- November 1915General relativity ▶The curving of spacetime (left) is set by the matter and energy in it (right).
- 1916The first black hole solutionSqueeze any mass inside this radius and it becomes a black hole. For the Sun, about 3 km.
- 1922–1929The expanding universe ▶A galaxy twice as far away is moving away twice as fast.
- 1970Galaxies spin too fast ▶An orbit’s speed depends only on the mass inside it; a flat speed curve means mass keeps growing outward.
- 1998The expansion speeds up ▶The rate at which the universe grows is itself growing.
- 14 September 2015Hearing black holes merge ▶After the merger, the signal is a single tone fading fast: the black hole ringing.