4,000 years · 48 ideas · 37 you can play with

The universe keeps reusing the same few ideas.

The maths that makes a guitar ring also rings a black hole. The rule that found Neptune broke at Mercury, and Einstein fixed it. Follow the threads, play with every idea, and see how it all connects.

Start a journey: from an apple to a black holeWander the sky of ideas

A question to start with

How do you make a perfectly square corner with nothing but a rope?

Find out: Right triangles in clay (c. 1800 BC) →

Things that sound impossible (and are true)

Where the planets are right now

Today’s positions, worked out from the same orbital elements astronomers publish. Tap a planet to see the ideas it inspired: Mercury broke Newton’s gravity, Uranus led to Neptune, and Jupiter’s moons showed that not everything circles the Earth.

Zoom in on every world and mission →
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The sky of ideas

Every star is an idea; every line says how one led to another. Tap a star, play through history, or follow a journey.

Ancient world1900 BC – AD 500India and the Islamic worldAD 500 – 1500Scientific revolution1500 – 1700Enlightenment1700 – 1800Nineteenth century1800 – 1900Relativity1900 – 1935Modern cosmos1935 – todayShape and spaceNumbers and algebraChange and calculusFields and vector calculusEarth, planets and orbitsLight and wavesSpacetime and the cosmosBuilt on: Euclid proved, from axioms, the right-angle rule the Babylonians had used by example.Built on: Eratosthenes used Euclid’s rule for angles across parallel lines to turn a shadow into the size of the Earth.Built on: Descartes turned Euclid’s shapes into equations.Led to: Riemann asked what geometry looks like if Euclid’s flat-space assumptions are dropped.Built on: Ptolemy’s model of the heavens ran on Hipparchus’s chord tables.Built on: Madhava wrote the sine, the function astronomers had tabulated since Hipparchus, as an endless sum.Analogy only: Circles riding on circles can trace almost any repeating path, just as stacked waves can build almost any shape.Led to: Copernicus kept the circles but moved the Sun to the centre.Led to: Jupiter’s moons showed that not everything circles the Earth, strong support for Copernicus.Led to: Kepler started as a Copernican and replaced its circles with ellipses.Built on: Al-Khwarizmi’s arithmetic book carried Indian numerals, zero included, to the Islamic world and later to Europe.Built on: Descartes drew algebra’s equations as curves.Same technique: Madhava’s infinite series anticipate the power series at the heart of calculus, about 250 years earlier.Led to: Euler identified e as the natural base of logarithms and joined it to the circle.Built on: Calculus works on curves written as equations, which Descartes supplied.Built on: Galileo’s falling bodies and his idea of inertia became Newton’s laws of motion.Led to: Newton showed that an inverse-square pull produces exactly Kepler’s ellipses and his third law.Built on: Getting from a force to an orbit needs the ideas of calculus, though Principia’s proofs are written as geometry.Built on: Euler’s formula comes from writing e, sine and cosine as calculus series.Built on: Maxwell’s equations are calculus applied to fields filling space.Built on: Fourier’s waves are most neatly written with Euler’s formula.Led to: Newton’s gravity let Halley predict a comet’s return.Led to: Lagrange solved a special case of Newton’s three-body problem.Built on: Gauss’s orbit for Ceres is a Newtonian orbit fitted to a few observations.Led to: Newton’s law, trusted completely, said something unseen was pulling Uranus.Same technique: Both turned scattered positions in the sky into a prediction of where to point a telescope.Led to: After Neptune, Le Verrier used the same reasoning on Mercury and predicted a planet, Vulcan, that does not exist.Led to: General relativity explained Mercury’s extra 43 arcseconds per century with no new planet.Built on: Einstein described gravity using Riemann’s geometry of curved spaces.Led to: If light is a wave, what is waving? Michelson and Morley tested for the ether and found nothing.Built on: Einstein’s 1905 paper begins with Maxwell’s theory: light’s speed is the same for everyone.Led to: Relativity made the missing ether drift natural. Historians still debate how much the experiment itself guided Einstein.Led to: E = mc² followed from relativity within months.Led to: Minkowski recast Einstein’s theory as four-dimensional geometry.Built on: General relativity lets Minkowski’s flat spacetime curve.Led to: Schwarzschild found the first exact solution, within weeks.Led to: General relativity predicted how much the Sun bends starlight; the 1919 eclipse tested it.Led to: Friedmann and Lemaître found that Einstein’s equations allow a universe that grows.Led to: Einstein predicted gravitational waves in 1916; it took a century to detect them.Built on: Leavitt’s pulsing stars gave Hubble his distances.Same technique: Both judge distance by how bright a star of known power appears.Led to: If the universe expanded, it was once hot and dense, and that heat should still glow.Led to: Measuring how the expansion changes over time revealed it is speeding up.Built on: 51 Pegasi b obeys Kepler’s laws; its year and its star’s wobble give its orbit and mass.Led to: LIGO’s signal came from two black holes merging into one.Led to: The dark centre in the image is the shadow of a black hole, about the size theory predicts.Analogy only: Gravitational waves travel at the speed of light, like Maxwell’s light waves.Same technique: Each pair of telescopes measures one Fourier ingredient of the image.Led to: Newton wrote cooling as a rate of change: one of the first differential equations used on real measurements.Same technique: Both follow exponential decay: the rate of change is proportional to how much is left.Built on: Taylor’s recipe uses derivatives of every order.Same technique: Madhava’s series for sine and cosine are exactly what Taylor’s recipe gives for those functions.Built on: Writing eˣ, sin x and cos x as series makes Euler’s formula drop out.Built on: Laplace’s s is a complex number: its imaginary part is the pitch and its real part the fade, Euler’s spiral.Built on: The transform is an integral that turns derivatives into multiplication by s.Led to: In pole language: walkers in step dragged the bridge’s pole across the line into growth.Same technique: A ringing black hole is described by complex frequencies, like poles: pitch and fade in one number.Built on: Green’s theorem is the fundamental theorem of calculus, stretched from a line to a flat region.Led to: Stokes’ theorem is Green’s theorem on curved surfaces in three dimensions.Same technique: Both trade a total over a boundary for a total over the inside.Led to: Gauss proved his case while studying the gravity of solid bodies, Newton’s problem.Built on: Maxwell sat the exam with Stokes’ question; his equations switch between edge and inside forms using Stokes’ theorem.Built on: Gauss’s law for electric charge, one of Maxwell’s four equations, is the divergence theorem.Built on: Only the mass inside an orbit sets its speed: Gauss’s theorem, treating the galaxy as roughly round.Built on: Rubin’s argument is Newton’s gravity applied to stars around a galaxy.Built on: Eigenvectors are directions in Descartes’ coordinate plane that a matrix leaves pointing the same way.Same technique: Whether a probe at a Lagrange point drifts or holds is decided by eigenvalues.Built on: Reading an age from a ratio means undoing an exponential: a logarithm.Led to: Dark matter and dark energy together make up most of the universe; we see only about 5%.Babylonian trianglesc. 1800 BCEuclidc. 300 BCEratosthenesc. 240 BCHipparchusc. 150 BCPtolemyc. AD 150Brahmagupta: zeroAD 628al-Khwarizmic. AD 820Madhavac. 1400Copernicus1543Kepler1609–1619Galileo1610Napier: logarithms1614Descartes: graphs1637Calculus1665–1684Newton’s gravity1687Law of cooling1701Taylor series1715Euler1748Halley’s comet1705–1758Lagrange points1772Gauss finds Ceres1801Laplace poles1812Fourier1822Divergence theorem1813–1826Green’s theorem1828Eigenvalues1829Finding Neptune1846Riemann1854Stokes’ theorem1850–1854Mercury’s puzzle1859Maxwell1865Michelson–Morley1887Special relativity1905E = mc²1905Minkowski1908Leavitt1912General relativityNovember 1915Schwarzschild1916Eclipse of 191929 May 1919Expanding universe1922–1929Age of the Earth1956Cosmic microwave background1965Rubin: dark matter1970First exoplanet1995Dark energy1998Millennium Bridge10 June 2000Gravitational waves14 September 2015Black hole imageApril 2019

Guided journeys

How are these connected?

Pick any two ideas and see the shortest chain of links between them.

Your sky

Every idea you open lights up. Links between two of yours form constellations. It stays in this browser only.

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