Beyond Violet

Three numbers per color means color has a shape. Plot every color a surface can have under white light, in a space built so that equal distances look about equally different (CIELAB), and you get this solid. Its skin is made of the most extreme surfaces possible: ones that reflect everything inside one band of wavelengths and nothing outside it. No paint, pigment, feather or flower lands outside it.

Count the cubes inside it that are one just-noticeable step wide. There are about 2.4 million, close to the 2.28 million that Pointer and Attridge estimated in 1998. So one glance at one spot is worth about 21 bits. The solid shape inside is your screen's range, the sRGB gamut. It holds about a third of the volume. The deep cyan-greens, violets and saturated yellows near the skin exist in the world but not on a standard display.

If your eye keeps three numbers from a spectrum, everything else is space you can write in. Sample the visible band in 65 narrow bands and you have 65 numbers. Your cones pin down only 3 combinations of them. The other 62 can change freely, and the color stays exactly the same.

Type below. Each character becomes 6 bits, and each bit pushes one band up or down. Then the pushes are corrected along the 3 directions your cones measure, so no cone response moves. The two swatches stay identical. A spectrometer, which reads all 65 numbers and knows the plain spectrum, reads your text back.

Color isn't stored in light. It's the short list of numbers a particular eye takes from the light, and other animals take different numbers. A dog has two kinds of cone, peaking near 429 and 555 nm, so it tells colors apart along a single blue–yellow axis. Red and green land in the same place.

A honeybee has three cones like you, shifted toward short wavelengths: ultraviolet, blue and green. It can't see red, and it can see ultraviolet.

Many flowers are printed in ink only a bee can read. The petals reflect ultraviolet at the tips and absorb it near the center, drawing a bullseye around the nectar. To you each petal is one even yellow. Drag the divider to compare, and tap anywhere in the meadow to see the spectrum at that spot and what each eye makes of it.

A soap film is a few hundred nanometers thick. Light reflects from its front and its back, and the two reflections interfere. Where the extra trip through the film is a whole number of wavelengths, that wavelength cancels; half a wavelength off, it doubles. Which colors survive depends only on the thickness, so the colors are a thickness map drawn in light.

A change of about 20 nm is enough to shift the color you see. Reading an interference color, your eye is measuring distances smaller than a virus. Butterfly wings, peacock feathers, beetle shells and oil on a puddle use the same physics. It's called structural color.

Watch the top. The film drains under gravity and thins until it turns black. Below about 30 nm the two reflections cancel at every wavelength, because the one from the front surface flips sign and the one from the back doesn't. There is nothing left to see.

Then switch to a sodium lamp. With one wavelength instead of many, the colors disappear and only dark fringes remain: contour lines of equal thickness. White light carries far more information about the film than a single spectral line can.

In a lecture to the Royal Society in 1801, Thomas Young argued that one point on the retina cannot hold a separate detector for every wavelength, so their number must be limited: “for instance,” he wrote, to three. James Clerk Maxwell put the idea to work with light itself. In 1860 he matched each color of the spectrum with a mixture of three spectral lights, for two observers, himself and Katherine Maxwell. Those were the first measured color-matching functions.

Hermann Grassmann had argued in 1853 that color mixture has three dimensions, and the rules now named after him make matching linear. Add the same light to both sides of a match, or scale both sides, and the match holds. So, for matching, a light behaves like a vector with three coordinates, and matching a test light T with three primaries P₁, P₂, P₃ means solving three linear equations for their amounts: T = c₁P₁ + c₂P₂ + c₃P₃. As long as no primary can be matched by a mix of the other two, the solution always exists. Nothing promises that every amount cᵢ is positive.

Try it below. The disk is split. Its left half is the test, a vivid teal. The right half mixes three lights from your own screen: an orange, a lime green and a violet. Look at the center of the disk and turn the knobs until the halves look the same. You will not get there: the closest mixture is paler and grayer. Now press To test side under the orange light, which shines it into the teal half instead, and try again.

The cone types overlap in what they absorb: every light that excites your M cones also excites your L cones. That bends the colors of the spectrum into a curved boundary, while a mixture of three lights always lands inside the triangle they span, so part of the boundary is always out of reach. W. D. Wright and John Guild measured the matches behind today's standard in the late 1920s. Re-expressed for the three spectral primaries the CIE adopted, their data need a negative amount at 52 of 61 wavelengths from 400 to 700 nm (panel B). The curves r̄, ḡ and b̄ give how much of each primary (700, 546.1 and 435.8 nm) matches one unit of power at each wavelength, and r̄ dips below zero from 440 to 545 nm. To match 515 nm light, red has to go on the test side, and the amount written down is negative.

In 1931 the CIE, the International Commission on Illumination, pooled Wright's ten observers and Guild's seven into one standard observer and re-expressed it for primaries at 700, 546.1 and 435.8 nm: two mercury lines and a red near the end of the spectrum. Then it replaced the primaries. Because matching is linear, any three independent combinations of the matching functions predict the same matches, so the CIE was free to pick a 3×3 matrix with properties that suited 1931. No function would ever go negative, which suited hand and mechanical computation. One function, ȳ, would equal the luminosity curve V(λ), so that Y alone carries luminance. Equal-energy white would sit at x = y = 1/3, and z̄ would be zero beyond 650 nm.

Those demands put two of the new primaries, X and Z, on the alychne, the plane of zero luminance. Every real light has some luminance, so no light can sit there. All three new primaries lie outside the spectrum's horseshoe; they are coordinates, not lights. X and Z are further gone than that. In cone signals, X needs a negative M response and Z a negative L response, so no stimulus of any kind, not even light delivered to single chosen cones, could produce them. Y needs only non-negative cone signals: a device that stimulates chosen cones could make it in principle, though no light can. Panel C builds the 1931 system step by step.

The standard observer is also nobody in particular. Color-normal men fall into two groups on the Rayleigh test, which matches yellow with a mixture of red and green (Neitz and Jacobs, 1986). The cause is one amino acid at position 180 of the L-cone pigment: serine in 62% of (mostly European) men, alanine in 38% (Winderickx and colleagues, 1992). Alanine moves the pigment about 2.7 to 3.5 nm toward shorter wavelengths by psychophysical estimates (Sharpe and colleagues, 1998), and in panel D's model a 2.7 nm shift calls for about 1.2 times as much red. About 47% of women should carry both versions (a Hardy–Weinberg estimate, not a measurement), with two L pigments only about 3 nm apart; nothing suggests that gives them an extra dimension of color. Deuteranomaly, the most common red–green deficiency at about 5% of European men (Birch, 2012), puts a pigment close to L in place of M and moves the match far toward green.

Cone counts barely matter for matching. Adaptive-optics imaging found L:M ratios from 1.1:1 to 16.5:1 among color-normal men (Hofer and colleagues, 2005). Drag the ratio: the Rayleigh match stays put, because it depends only on the cone spectra, while the model's luminosity peak slides from 554 to 568 nm. Even the average is off: scaled to the same peak, the cone-based CIE 2015 function is 7.5 times higher than the 1931 ȳ at 410 nm. Yet every color on this page passes through those 1931 functions, coordinates built on primaries no photon can be. Color is a compression showed three numbers standing in for a spectrum, and Two ways to go blind counts what they drop as a 58-dimensional kernel. Negative light is the other face of that compression: three numbers can name every color, but no three lights can make them all. Even six perfect lasers, placed as well as possible, leave 4.4% of the area of real colors in the xy diagram out of reach (2.3% in the u′v′ diagram).

Some hues look pure. There is a yellow with no red and no green in it, and a blue that is neither greenish nor reddish. Ewald Hering built a theory on that observation. He proposed that color is coded as three opposed pairs: red against green, yellow against blue, and black against white. A color can then be reddish or greenish, never both. The unique hues sit where a pair balances: pure yellow is where red and green cancel exactly.

In 1955 Dorothea Jameson and Leo Hurvich measured those balance points on themselves. They took each spectral light, added a canceller of fixed wavelength, and raised the canceller until the opposing hue vanished. The zero crossings are the unique hues. In hue-cancellation and naming studies since, unique blue falls near 470–480 nm, unique green near 500–520 nm and unique yellow near 575–580 nm. Unique red is usually not in the spectrum at all: long-wave red needs a little short-wave light added before it stops looking yellowish.

Physiologists later found color-opponent cells. In the macaque's lateral geniculate nucleus (LGN), the relay between retina and cortex, their preferences cluster along two axes: long- against middle-wavelength cones (L−M), and short-wavelength cones against the sum of the other two, S−(L+M) (Derrington, Krauskopf and Lennie, 1984). Textbooks often call these the red–green and blue–yellow channels. The wheel below holds every hue a standard (sRGB) screen can show at one luminance, with those two axes drawn through it.

Use your own eyes first. Pick a hue and turn it until the patch on the gray looks pure, looking only at the patch while you judge. Rest on plain gray between settings. Each setting becomes a spoke on the wheel, and the readouts compare it with the wiring. Where most people put each hue appears only after you have set yours.

As light crosses expanding space, its wavelength stretches by the same factor the universe grows, written 1 + z. Nothing is lost on the way. Each photon arrives with less energy, and the stream arrives more slowly. A signal sent at one pulse per second from redshift z arrives at one pulse every 1 + z seconds.

Lyman-α is hydrogen's brightest ultraviolet line, at 121.6 nm. From a galaxy at z = 3.5 it arrives at 547 nm, as green light. From z = 10 it lands at 1.34 µm in the infrared, and the galaxy has vanished from visible light entirely. The Webb telescope finds the most distant galaxies by the cliff in their spectrum where intergalactic hydrogen absorbs everything shorter than Lyman-α. That ultraviolet edge arrives stretched into the infrared.

Blueshift runs the translation the other way. Move toward a light source and its waves bunch up: every wavelength shortens by the Doppler factor, D. At everyday speeds D is 1. Near the speed of light, the stars ahead shift toward blue and crowd into a shrinking circle straight ahead, an effect called aberration. Behind you, the sky reddens and goes dark.

The cosmic microwave background, the 2.7 K afterglow of the Big Bang, fills every direction, and ahead of you it blueshifts too. Around γ ≈ 150 it passes 800 K and starts to glow dull red, like iron in a forge. Around γ ≈ 550 it reaches 3,000 K, the temperature it had when it was released. At that speed the sky ahead looks the way the whole universe looked 380,000 years after the Big Bang.

Nothing outruns light in a vacuum. In water, light's wavefronts move at c/n, about 75% of c, because the light sets the water's electrons oscillating and their fields slow the combined wave. A fast charged particle, like an electron thrown out by beta decay in a reactor, can move faster than that.

As it passes, the particle briefly polarizes the molecules along its path, and each one relaxes by emitting a faint pulse of light. Normally those pulses cancel. Faster than c/n, they pile up along a cone, the way a supersonic jet's sound piles into a sonic boom.

One equation sets the cone: cos θ = 1 / (n β), where β is the particle's speed as a fraction of c. Below the threshold β = 1/n there is no cone and no light. In water that threshold is an electron with 0.26 MeV of kinetic energy. At nearly the speed of light the angle approaches 41°, the size of the rings the Super-Kamiokande detector records.

Named for Pavel Cherenkov, who measured it in 1934 in Sergei Vavilov's lab. Ilya Frank and Igor Tamm explained it in 1937, and the three shared the 1958 Nobel Prize in Physics.

A glance at one spot is a measurement. The spectrum coming off the surface is the message, your cones are the receiver, and noise limits how finely each cone can be read: photons arrive at random, and neurons jitter. Claude Shannon's formula gives the most a receiver like that can learn. Split the signal into independent directions, and add up ½ log₂(1 + signal/noise) for each one.

This figure runs that calculation live. The world is a statistical model of surface colors, with a control for how smooth their spectra are. The eye is any set of cones you like. Drag a cone, add one, or press Let information theory design it and watch the cones slide to wherever they pull out the most bits.

Two results fall out without being put in. The first is diminishing returns: when spectra are smooth, the third cone adds far less than the first, and a twelfth adds almost nothing. That is why a mantis shrimp's twelve receptors buy it much less than you'd guess: in this model they yield about six useful channels, and the other six add under a tenth of a bit between them. The second is the opponent channels. Ask which combinations of cone signals are uncorrelated, which is what an efficient code would send down the optic nerve, and you get brightness, blue against yellow, and red against green. Those are the channels neurons in your visual pathway actually use.

Side by side, people can tell apart about 2.28 million surface colors by one standard estimate, roughly 21 bits (Pointer and Attridge, 1998). From memory, with nothing to compare against, they identify only about 11 to 15 spectral hues without error under their viewing conditions (fewer in dim light), 3.5 to 3.9 bits (Halsey and Chapanis, 1951). The first count spans lightness and saturation as well as hue, so the fair comparison is along the spectrum alone: side by side, the eye separates about 100 to 200 hue steps, 6.6 to 7.6 bits, and memory keeps about half of those bits. Words sit on the memory side. A color word is a lossy code: many colors go in, one word comes out, and the listener has to guess which color you meant.

Every vocabulary has a cost and a payoff, both measured in bits. Complexity, I(M;W), counts the bits a word W carries about M, the color the speaker has in mind: more words and finer distinctions cost more to learn and to use. Accuracy, I(W;U), counts the bits a listener recovers from the word about the color itself, U. A word can never tell the listener more than it cost the speaker, so accuracy never exceeds complexity.

In 2018 Noga Zaslavsky, Charles Kemp, Terry Regier and Naftali Tishby compared the color vocabularies of the 110 languages of the World Color Survey, plus English, with the best trade-off the information bottleneck allows. They lie near that bound. Each language is best fit by a single number β, the bits of complexity it will spend for one more bit of accuracy, and that one number captures much of the variation between languages.

This figure runs their method live on a generated chart of 330 color chips. Below β = 1 a second word can never pay for itself, and in this model a single word for every color stays stable up to β = 1.076. The figure plays the growth past that point once as it comes into view. Press Play to run it again, drag β, or drag sideways across the chart to turn the names around the hue circle.

Light comes in modes. A mode is one distinct way for light to exist: one pattern in space and direction, one polarization, and one short slot of time and frequency. Each mode can hold no photons, one, or many. For light in thermal equilibrium, such as the glow of a hot, black surface, quantum physics sets the average from a single ratio, hν/kT: the energy of one photon (h is Planck's constant, ν the light's frequency) over the thermal energy (k is Boltzmann's constant, T the temperature). The average number of photons per mode is n = 1/(e^hν/kT − 1), where e ≈ 2.718: the Bose–Einstein occupation.

The numbers are small. At the surface of the Sun, 5,772 K, green light at 550 nm has hν/kT = 4.53, so n = 0.0109: about 99 modes in 100 are empty. Crossing empty space does not change n. At Earth's distance, a mode that points straight at the Sun is as full as one at its surface, which is why no lens or mirror can heat anything above the temperature of the Sun's surface. A white page changes that. It gathers light from the Sun's small disk and scatters it in every direction, so n falls by a factor of about 46,000, to 2.35 × 10⁻⁷, while the light keeps its color. That is one photon for every 4.3 million modes. Ordinary light is almost empty.

Color temperature is a statement about this emptiness. Because n depends on wavelength, a hot source fills its short-wavelength modes relatively more than a cool one. At 5,772 K, n is 0.0039 at 450 nm and 0.022 at 650 nm. Multiply that tilt by the number of modes, which climbs steeply toward short wavelengths, and you get the Sun's spectrum, and so its color. Press Track the peak and slide the temperature to see something stranger. At the wavelength where its own spectrum peaks, every glow holds the same 0.0070 photons per mode, from the 2.7 K afterglow of the Big Bang to a 30,000 K star. Switch the curve to λ × T, wavelength times temperature, and the reason shows: every glow falls on one curve. That curve fixes a second constant. Averaged over its whole spectrum, each photon of a glow carries 5.20 bits of entropy, at any temperature.

Emptiness is what lets a photon say a lot. The most information one mode can carry, when it holds n photons on average, is g(n) = (n+1) log₂(n+1) − n log₂ n bits. This is the classical (Holevo) capacity of light, proved exact even for a lossy channel in 2004. Per photon that is g(n)/n, which grows without limit as n shrinks, close to log₂(e/n): 7.97 bits per photon at the Sun's surface and 23.5 on the sunlit page. The same g(n) is also the entropy of a thermal mode, so the disorder of a glow and the capacity of light are one formula.

A measurement can be blind in one direction: change what is being measured that way, and the data do not change. Statistics makes this exact. Fisher information, F, measures how sharply the likelihood of the data changes as a parameter θ changes, and its inverse bounds the variance of any unbiased estimate of θ. With N independent photons, each carrying Fisher information F, no such estimate can have a standard deviation below 1/√(N·F), the Cramér–Rao bound. Where F is zero the bound is infinite, and no cleverness afterward can recover θ.

Color is blind by construction. A cone reports how many photons it absorbs and nothing about their wavelengths (the principle of univariance, named by Naka and Rushton in 1966), so its average count is a weighted sum over the spectrum. Sample the band from 400 to 700 nm every 5 nm and a light is 61 numbers, which three cones turn into three. A linear map from 61 numbers to 3 ignores a 58-dimensional family of changes, called its kernel. Add any of them to a light and every cone catches the same number of photons on average. The two lights are metamers, and in that direction F is exactly zero.

Blur is the other way. Two stars closer together than a telescope's blur merge into one smudge. A camera still records where each photon lands, and each photon still carries some information about their separation, d. But for two equally bright sources whose light does not interfere, the camera's Fisher information about d falls toward zero as d shrinks, in proportion to d² at small separations. This is Rayleigh's curse. Here zero is approached, never reached.

In both cases the light still holds what the measurement drops. A fourth kind of cone reads one more direction of color's 58-dimensional kernel, and a spectrometer reads all of it. For the stars, Mankei Tsang, Ranjith Nair and Xiao-Ming Lu showed in 2016 that sorting photons by spatial mode instead of by position (into Hermite–Gauss modes, a scheme called spatial-mode demultiplexing, or SPADE) keeps the Fisher information at its quantum maximum at every separation: 1/(4σ²) per photon, for a Gaussian blur whose intensity has standard deviation σ. Both cures change what is measured: more spectral channels for color, a different set of spatial modes for the stars. Below, drag the ring to choose a color and push its light along a hidden direction; in the Rayleigh half, pull the two sources together.

A color is a frequency. Green light of 550 nm is an electric field that swings back and forth 545 trillion times a second, so one cycle lasts 1.83 femtoseconds (fs; a femtosecond is 10⁻¹⁵ s). Your cones cannot follow anything that fast. Each kind absorbs some frequencies more readily than others, and the balance among the three is the color you see. A color is a clock, and the eye reads its rate.

A clock has to run for a while before its rate means anything, and Fourier's mathematics makes that exact. A flash that lasts a time Δt is built from a spread of frequencies Δν, and measured as standard deviations, the two obey Δν·Δt ≥ 1/(4π) ≈ 0.080. In 1946 Dennis Gabor built a theory of communication on this limit. Multiply by Planck's constant h, which turns a frequency into a photon energy (E = hν), and it becomes Heisenberg's uncertainty relation for energy and time, ΔE·Δt ≥ ħ/2, with ħ = h/(2π): one inequality in two sets of units. A flash with a Gaussian (bell-curve) shape and no chirp, every frequency in step, meets it exactly. Gabor called such a signal a logon.

So a short enough flash of green cannot be green. The eye never isolates one frequency. It adds up the whole spread, weighted by three kinds of cone, and keeps one color: the compression from Part one, which the CIE 1931 standard observer models. From here on, durations and widths are full widths at half maximum (FWHM), 2.35 times the standard deviations, and in those units an unchirped Gaussian's product is 0.441. A 100 fs flash of 550 nm light spreads over 4.4 THz, about 4.5 nm of wavelength, and looks like pure spectral green. In this model its excitation purity, which is 1 for a single wavelength and 0 for the equal-energy white E, is 1.00. At 10 fs the spread is 45 nm and the purity 0.97. At 3 fs the purity is 0.47. A flash one cycle long, 1.83 fs, spreads over 255 nm, most of the visible band, and looks greenish white: purity 0.09. The hue drifts on the way. The dominant wavelength moves to 559 nm at 5 fs, a yellower green, then swings back to 526 nm at one cycle.

Drag the flash shorter below and watch its color drain. Past one cycle of green there is no green flash left to make. The figure raises the carrier frequency so that a cycle still fits, and the light slides into the ultraviolet, leaving less and less of its energy in the visible band until the swatch goes dark. Listen plays each flash 40 octaves down, the transposition used across this site, and slowed by the same factor of 2⁴⁰, so Δν·Δt is unchanged: a 100 fs flash becomes a 110 ms tone at 496 Hz, and a 3 fs flash a 3.3 ms click with no clear pitch.

Earth absorbs about 1.2 × 10¹⁷ watts of sunlight, and it radiates almost exactly as much back into space. The difference is the heat the planet stores, which satellites and ocean measurements track. From 2005 to 2019 it stayed under half a percent of the flow (Loeb et al. 2021). Life doesn't run on the Sun's energy by keeping it. The energy passes straight through.

What changes on the way through is the number of photons. Sunlight leaves the Sun's surface at 5772 K. Earth returns the same power as infrared at about 255 K, the temperature at which it radiates to space. The average photon of blackbody light, the glow of a body at one temperature, carries 2.70 k_BT of energy, where k_B is Boltzmann's constant and T the temperature. So the same energy at a temperature 22.7 times lower takes 22.7 times as many photons: for every photon of sunlight it absorbs, Earth sends about 23 back out. Roger Penrose, who made this point in Cycles of Time (2010), put it at roughly 20.

Photon count is entropy count. A photon of blackbody light carries on average 3.60 k_B of entropy, about 5.2 bits, whatever its temperature. Earth takes its energy in as few photons and gives it back as many, so it sends out far more entropy than it receives. Treating both streams as blackbody light, as the figure does, that comes to about 6.4 × 10³⁷ bits every second, net. Published budgets, which account differently for sunlight thinning out on its way from the Sun and for the real infrared, range from about 4.8 to 6.9 × 10³⁷. That export is the budget for every ordered thing on the surface: weather, life, thought. Anything that builds order must push entropy out to pay for it, and in the end all of it leaves as infrared. Ludwig Boltzmann made the underlying point in 1886: the struggle for existence, he argued, is not a struggle for energy, which is plentiful, but for entropy, made available as energy passes from the hot Sun to the cold Earth.

What makes sunlight valuable is its color: how hot it is, and so how few photons carry each joule. The figure keeps the books for a range of stars and planets. Press Follow one photon to watch the energy of a single photon of sunlight leave again as a ring of infrared ones. Each followed photon is drawn in its own color, and most are invisible: 68% of the Sun's photons are infrared, and a photon of the average energy, 1.34 eV, lies at 923 nm. White is the color of the whole stream, never of one photon.

In 1974 Stephen Hawking showed that a black hole is not quite black. Quantum fields near its horizon make it glow at a temperature set by its mass alone (for a hole that neither spins nor carries charge), T = ħc³/(8πGMk_B). Here ħ is Planck's constant divided by 2π, c the speed of light, G Newton's constant of gravitation, M the mass and k_B Boltzmann's constant. The lighter the black hole, the hotter it glows. One with the Sun's mass sits at 62 billionths of a kelvin, 44 million times colder than the cosmic microwave background, so today it absorbs more than it emits. No Hawking glow has ever been detected.

The figure below opens on a black hole of 2.13 × 10¹⁹ kilograms, roughly the mass of the asteroid 16 Psyche, with a horizon 63 nm across, about the size of a small virus. It glows at 5772 K, the temperature of the Sun's surface. A blackbody that hot is white, like sunlight. The black hole is blue, a deeper blue than a blackbody reaches at any temperature.

The reason is size. A blackbody at 5772 K peaks at 502 nm, 15.9 times the horizon's radius, and that ratio is the same at every mass: a black hole is always smaller than the light it makes. At 1.5 horizon radii, curved spacetime forms a barrier for light waves. Short waves pass over it, and long waves mostly reflect and fall back in. The fraction that gets through is called the graybody factor; Don Page computed it for photons, neutrinos and gravitons in 1976. In this model, at 5772 K it lets out 82% of the violet light at 400 nm that simple ray optics would predict, and 8% of the red at 700 nm, which moves the peak of the hole's light to 375 nm, in the ultraviolet. What is left has CIE chromaticity coordinates (x, y) = (0.229, 0.228), against a blackbody's (0.327, 0.336).

Temperature is also an exchange rate between energy and information. At temperature T, a joule of heat carries 1/(k_BT ln 2) bits of entropy. In Bekenstein and Hawking's formula, a horizon stores bits as area, one for every 7.24 × 10⁻⁷⁰ m², and every joule the Sun-hot black hole radiates takes 1.81 × 10¹⁹ of them away. The average Hawking photon carries 5.71 k_BT of energy, so it takes 8.24 bits off the horizon, whatever the mass. The light carries away more entropy than the horizon loses, about 12.4 bits per photon, so the second law of thermodynamics holds.

A guitar string can ring only at certain pitches: the ones where a whole number of half-waves fits between its fixed ends. An atom is held to the same rule. Its electron is a wave (Louis de Broglie's proposal of 1924, later confirmed by electron diffraction), and the electric pull of the nucleus confines it, so it can settle only into certain standing patterns. Each pattern has one exact energy.

Light comes from the difference between two of them. When the electron drops from one standing wave to a lower one, the atom sheds the energy gap as a single photon, whose frequency is the gap divided by Planck's constant. Put the electron partly in both patterns, as light tuned to the gap does, and its charge sloshes back and forth at exactly the difference of their frequencies, the way two slightly mistuned strings beat. In this semiclassical picture the sloshing charge is the antenna, and the full quantum theory gives the same frequency. An atom's colors are its beat notes.

For hydrogen the notes fit on one line: 1/λ = R(1/n₁² − 1/n₂²), where the whole numbers n₁ and n₂ label the two standing waves. The Rydberg constant R sets the scale, and apart from a 0.05% correction for the proton's recoil it is built, in energy units, from just two numbers of nature. One is the fine-structure constant, α ≈ 1/137.036, which sets how strongly charge couples to light. The other is the electron's rest energy, mₑc² = 0.511 MeV, which in the Standard Model comes from the electron's coupling to the Higgs field and that field's value in empty space. Together they set the energy scale of atoms, ½α²mₑc² = 13.606 eV, and every color hydrogen makes is a fraction of it. The figure lets you choose the two levels, change the nucleus, and turn both constants.

It is tempting to call an atom trapped light. It isn't: the standing wave belongs to the electron's own field, and the electromagnetic field only holds it in place. One part of the intuition is literal, though. A hydrogen atom in its ground state weighs 13.598 eV/c² less than a free proton and electron, 2.42 × 10⁻³⁵ kg, and that missing mass left as light when the atom formed.

In the last chapter the box was an atom: the nucleus's pull confines the electron's wave, so the atom can ring only at certain notes. A crystal can be a box too, and this one people can size. Trap an electron in a small enough crystal and it, too, can hold only certain notes. Shrink the crystal and every note rises.

Quantum dots are that box. They are crystals of a semiconductor, here cadmium selenide (CdSe), a few nanometers across: about 130 to 9,600 atoms in the sizes below. The chemistry fixes the band gap, the least energy that can free an electron to move through the crystal, leaving behind a hole (a missing electron that moves like a positive charge). In a large lump of CdSe the gap is 1.74 eV, the energy of a 713 nm photon. The lump absorbs every color bluer than that, nearly the whole visible band, so it looks almost black.

An electron and its hole attract. In bulk CdSe they can bind into a loose pair, an exciton, about 11 nm across. Make the crystal smaller than that and its walls squeeze the pair. Squeezed waves cost energy, and that energy adds to the gap. The same material then glows blue, green, orange or red depending only on its size. That is why dots are called artificial atoms: they have discrete levels like an atom, set by a size people choose. Alexei Ekimov, Louis Brus and Moungi Bawendi shared the 2023 Nobel Prize in Chemistry for discovering and making them.

In 1984 Brus also wrote down the textbook formula for the effect: an electron and a hole in a sphere with perfect walls, plus their electric attraction. It works for large dots and fails for small ones. At 3.04 nm, where real dots start absorbing in the green at 550 nm, it predicts 388 nm, at the violet edge of vision. Drag along the shelf and watch the measured colors and the formula's part ways.

Why the formula fails. Among its simplifications, two stop being true at this scale. Its walls are infinitely high, but a real dot's walls are finite, so the electron's wave leaks out a little and the box acts larger than it is. And it gives the electron one fixed effective mass, while an electron squeezed to high energy in CdSe responds as if it were heavier. Both lower the note. A 2 nm dot is also less than five lattice cells across, hardly the smooth, uniform box the formula imagines. Halve a dot from 4 nm to 2 nm and the formula says the energy added by the box should grow 4.3 times; real dots show 2.6 times. The rule survives and the numbers don't, so dots are sized from curves fitted to measurements, like the one this figure uses.

The size ladder in the figure holds one more surprise. The wavelength of the light a dot gives off is about 80 to 240 times the dot's diameter. The dot is a box for the electron, not for the light: in the perfect-wall picture the electron's wavelength inside equals the dot's diameter. Under Listen, press Dots or Formula to hear either glow at its frequency divided by 2⁴⁰. In this model, shrinking the dot raises the note from about 408 Hz for the largest dots here to about 595 Hz for the smallest. For the smallest dot the formula's note is 1,214 Hz, about an octave too high.

It is tempting to picture matter as light caught in a resonance: a wave running in circles, held by its own ringing. The last two chapters came close. Electrons in atoms and in quantum dots are waves, and a box lets each one hold only the notes that fit. This chapter tries the literal version. Put light itself in a box and ask what it becomes. The answer is exact: it moves like a particle with mass.

In empty space a photon has no mass. Its energy E and momentum p obey E = pc, where c is the speed of light, and nothing is left over. Now put it between two parallel mirrors a distance D apart, in a material of refractive index n, where light travels at c̃ = c/n. Only standing waves survive, so a whole number q of half-waves must fit across the gap. That locks the wave's motion along the axis: its wavenumber in that direction (2π divided by the wavelength) is fixed at k_z = qπ/D. Along the mirrors the photon is still free, with wavenumber k_⊥ and momentum p = ħk_⊥, where ħ is Planck's constant divided by 2π.

The photon's energy is then E = ħc̃√(k_z² + k_⊥²). Einstein's rule for a particle of mass m is E² = (mc²)² + (pc)². They are the same equation, with c̃ in place of c and m = ħk_z/c̃. The locked motion has become rest energy, and the free motion has become momentum. A photon bouncing straight between the mirrors is a particle at rest. Tilt it and it drifts sideways, slower than light, the way a massive particle moves. Its rest energy mc̃² is the energy of the longest wavelength the box allows at that q, the cutoff λ_c = 2nD/q. The trapped photon's mass is a color.

In 2010 Jan Klaers, Julian Schmitt, Frank Vewinger and Martin Weitz at the University of Bonn built this box: two curved mirrors about 1.5 µm apart, with a dye dissolved in methanol between them. The cutoff sat at 585 nm, yellow-orange light, which gives each photon an effective mass of 6.7 × 10⁻³⁶ kg, about 7.3 millionths of an electron's mass. Then they did something ordinary light never does. At room temperature, they turned the photons into a Bose–Einstein condensate.

Why this light can condense. Ordinary light cannot. Cool a hot oven and its light fades instead of gathering into one mode, because the walls absorb and emit photons freely and their number follows the temperature. (Physicists say blackbody light has zero chemical potential.) The Bonn cutoff, 2.12 eV, is 82 times the thermal energy k_BT at room temperature, where k_B is Boltzmann's constant, so in this model heat alone would put about 10⁻³¹ photons into the modes near the cutoff. The pump laser keeps the dye excited, and the dye's light replaces the photons that leak through the mirrors, so the count holds on average. Dye molecules (rhodamine 6G) absorb and re-emit each photon many times, bringing the gas to the dye's temperature, and the curved mirrors add a shallow bowl. Massive particles, a fixed number of them, a bowl and a temperature: that is the recipe for Bose–Einstein condensation. Past about 77,000 photons, the critical number N_c, every extra photon drops into the single lowest mode, in this model a spot about 13 µm across that glows at the cutoff color. Press Turn up the pump and watch the cloud stop growing while the spot takes the rest. Semiconductor cavities have since done the same (Schofield et al. and Pieczarka et al., both 2024).

None of this gives the photon a true rest mass. Free photons are massless to within 1 × 10⁻¹⁸ eV/c², the Particle Data Group's bound, which is 3.7 × 10¹⁸ times smaller than the Bonn photon's mass. The box gives an effective mass: a gap at the bottom of the dispersion curve, the lowest energy (and so the reddest color) that light can have there. Every mass light acquires is a gap of this kind. In the ionosphere's densest layer the gap is the plasma frequency, about 3 to 15 MHz. Radio below it is reflected, and radio above it travels as if each photon had a mass, 7 × 10⁻⁴⁴ kg where the plasma frequency is 10 MHz. Inside a superconductor, magnetic fields die out within the London depth, about 40 nm in niobium, which gives light there 8.8 × 10⁻³⁶ kg, close to the Bonn photon. In 1963 Philip Anderson argued that the same mechanism could give mass to the particles that carry forces. Its relativistic version, worked out by several groups in 1964, is the Higgs mechanism, which gives the W and Z bosons their mass.

A proton weighs 938.272 MeV. (Particle physicists quote a mass as its rest energy, mc².) Inside it are three quarks, two up and one down, and each gets its mass from the Higgs field: 2.16 MeV for an up quark and 4.70 MeV for a down. Add them and you get 9.0 MeV, about 1% of the proton. That comparison is rough bookkeeping (the careful version comes below the figure), but its lesson survives: almost none of a proton's mass is the mass of its parts. It is energy, E = mc² read from right to left.

Most of that energy is tied up in the gluon field, the field that holds quarks together, and in the quarks' own motion inside it. Gluons are close cousins of photons: massless, with spin 1. The difference is charge. A photon carries no electric charge, so light passes through light almost untouched. A gluon carries the same kind of charge it responds to, called color, so gluons pull on other gluons. Between two quarks the field lines attract one another and squeeze into a narrow tube, instead of spreading out the way electric field lines do.

A tube stores the same energy in every femtometer of its length (10⁻¹⁵ m; a proton's charge radius is 0.84 fm), so the force needed to stretch it stays nearly the same however far you pull. Lattice QCD, which solves quantum chromodynamics (the theory of quarks and gluons) numerically on a grid of points in space and time, puts that tension at about 0.9 GeV per femtometer. In everyday units that is about 15 tonnes-force: the weight of 15 tonnes, hanging on one quark.

Pull farther and the tube still does not let go. By about 1.2 fm it holds roughly 1 GeV, and making a new quark and antiquark, one on each broken end, becomes cheaper than stretching any further. The tube snaps, and one meson becomes two. No one has ever seen a lone quark. Drag the quarks below to feel the difference, and pull the tube sideways and let go to pluck it.

An atom's colors come from its electrons. They sit on energy levels a few electron-volts apart, and the photons they trade when they jump between levels are mostly infrared, visible or ultraviolet light. A nucleus has energy levels too, but its protons and neutrons are bound far more tightly, so its steps are thousands to millions of electron-volts. Those photons are gamma rays, far past anything a tabletop laser can make.

Thorium-229 is the exception. Its nucleus has an excited state, called an isomer, only 8.3557 eV above its ground state. The photon that links the two has a wavelength of 148.382 nm, in the vacuum ultraviolet, 1.36 octaves past the violet edge of vision. It is the only known nuclear transition low enough in energy for a tabletop laser. In a paper published in April 2024, a team from PTB and TU Wien reported tuning one across it, with thorium held in a calcium fluoride crystal, and watching the nuclei glow for minutes afterward (Tiedau et al.).

It was not the first nucleus a laser had excited: an X-ray laser had reached a 12.4 keV level of scandium-45 the year before. It was the first in the optical and ultraviolet range, where clocks work. That September a JILA team reported measuring the line with a vacuum-ultraviolet frequency comb tied to a strontium atomic clock: 2,020,407,384,335 kHz, give or take 2 kHz (Zhang et al.). A nuclear energy level became a color known to 12 digits.

Why chase it? The 8.36 eV is what is left when two much larger changes nearly cancel. The electric repulsion among the protons totals about 1 GeV. Between the two states it is estimated to change by tens of keV, and the energy of the strong force that binds the nucleus changes by almost exactly the opposite amount. The remainder is thousands of times smaller. So if the fine-structure constant α, which sets the strength of electromagnetism and with it the scale of every atomic color, ever changed, the thorium frequency would change, as a fraction, by about 5,900 times α's fractional change (give or take 2,300). That is roughly 100,000 times the response of a strontium clock. No change in α has ever been seen.

Empty space is not empty. In quantum theory the vacuum is the lowest-energy state of every field, and even there the electromagnetic field still fluctuates: its average is zero, but its measured values are not. Each of its modes (one way for a light wave of a given wavelength, direction and polarization to fit into space) holds a small jitter that nothing can remove, called its zero-point fluctuation. In 2015 a group at the University of Konstanz sampled it directly with laser pulses a few femtoseconds long, and saw it as a small excess of noise in the probe signal.

That jitter is one way to see why an excited atom glows. The vacuum's modes shake it, and sooner or later it drops to a lower energy level and hands a photon to one of them. This is spontaneous emission, the process behind the light of lamps, LEDs, fluorescence and stars. (An equally valid account gives part or all of the credit to the atom's reaction to its own field instead. Either way, the photon needs a mode to go into.) So how fast an atom glows is not a property of the atom alone. It depends on which modes surround it.

Mirrors decide which modes exist. A perfect mirror forces the electric field along its surface to zero, so across a gap of width a between two mirrors, a field that points along the mirrors must form a standing wave with a whole number m of half-waves. A wave of wavelength λ can do that while traveling along the gap only if λ < 2a/m, so no such wave longer than 2a fits at all. For an atom whose dipole (the direction its charge oscillates) lies along the mirrors, a gap narrower than half its wavelength leaves nowhere to put the photon. The color is switched off.

This has been measured. In 1985 Hulet, Hilfer and Kleppner put cesium atoms in highly excited states between conducting plates about 230 µm apart. Their transition had a wavelength of about 0.45 mm, so half a wave only just fit; an electric field then tuned the wavelength slightly longer, until half a wave no longer fit. The atoms' lifetime grew by a factor of at least 20. Two years earlier Goy, Raimond, Gross and Haroche had shown the reverse: sodium atoms in a cavity tuned to their transition emitted about 500 times faster than in free space.

Close the mirrors and, for a dipole along them, the red end goes first. At a gap of 300 nm every color longer than 600 nm is gone, and below 190 nm there is nowhere for even violet to go. Just wider than the first cutoff the opposite happens: the one standing wave that fits is squeezed into a thin gap, so its zero-point field at the atom is stronger, and in this model an atom midway between the mirrors emits three times faster than in free space. Turn the dipole across the gap and nothing switches off, because one mode, whose field points straight across, fits any gap. For ideal mirrors only the ratio of gap to wavelength matters, which is why an experiment with plates just under a quarter of a millimeter apart tests the same rule as this figure.

The missing modes also push. Outside the mirrors every mode is present. Between them most of the long-wavelength modes are missing (only the one whose field points straight across survives), and the imbalance presses the mirrors together. For ideal mirrors this Casimir pressure is P = π²ħc/(240a⁴), where ħ is Planck's constant divided by 2π and c is the speed of light: 1.30 mPa at a gap of 1 µm, 13.0 Pa at 100 nm, and 8.13 kPa at 20 nm, 8% of an atmosphere, though at gaps that small real metals give far less than this ideal value. It has been measured by Lamoreaux in 1997 with a torsion pendulum, by Mohideen and Roy in 1998 with an atomic-force microscope, to about 1% of the force at their closest separation, and between truly parallel plates by Bressi and colleagues in 2002, to about 15%.

The same force has a second, equally valid description. Robert Jaffe showed in 2005 that it follows without mentioning zero-point energy at all, as a van der Waals attraction between fluctuating charges and currents in the two plates, delayed by the time light takes to cross the gap. It vanishes if the fine-structure constant α, the strength with which charge couples to light, goes to zero; the formula above is the limit of perfect mirrors. So the Casimir force shows that quantum fluctuations, whether described as the field between the plates or as the charges within them, have mechanical effects. It does not, by itself, prove that empty space stores energy.

Until it was about 372,000 years old, the universe was opaque. Its hydrogen was ionized, and light scattered off the free electrons so often that photons, electrons and nuclei moved as one fluid. It was a strange fluid, made mostly of light. There were about 1.6 billion photons for every baryon (a proton or neutron), and even at the end the light carried 1.21 times as much energy as the baryons' mass, counted by E = mc². The light gave the fluid its pressure, the spring, and most of its inertia too. The baryons added mass without adding pressure, a weight on the spring.

Anything with a spring and a mass can ring. Gravity pulled the fluid toward places that started out slightly denser, the light's pressure pushed it back out, and every region oscillated. Sound traveled at c/√(3(1 + R)), where c is the speed of light and R = 3ρ_b/4ρ_γ compares the baryons' mass density ρ_b with the light's, ρ_γ. At the end R was 0.62, which puts the speed of sound at 0.453c, about 136,000 km/s.

Then the hydrogen caught its electrons, the light went free, and the sound froze where it stood. A wave that had just reached its first full squeeze was caught at maximum compression. A wave oscillating twice as fast had squeezed and rebounded, and was caught at maximum rarefaction; three times as fast, it had squeezed twice. All their wavelengths are set by one length, the sound horizon: how far sound could travel before the light went free. Waves much wider than that had barely begun to swing, and they make the flat plateau at the left of the figure below. In Planck's fit the sound horizon is 144.43 megaparsecs (Mpc) when stretched to today's scale, about 471 million light-years. It spans 0.5965° on the sky, an angle Planck measured to 0.03%. The harmonics show up as peaks of power at multipoles ℓ (patterns about 180°/ℓ across) near ℓ = 220, 540, 810, 1120 and 1440. They are spaced about 300 apart, 180° divided by that 0.5965°, but the whole ladder is shifted down: the first peak sits near 220 rather than 300, because gravity's driving shifts every wave's phase.

Harmonics in space are harmonics in time. Count the first peak's wave as half an oscillation in 372,000 years and its average frequency is 4.26 × 10⁻¹⁴ Hz, one cycle every 744,000 years. That is an illustration, not a frequency an instrument could have recorded: the oscillation ran fastest early and had slowed to about 1.7 × 10⁻¹⁴ Hz by release. Visible light vibrates about 40 octaves above hearing; this sound vibrated 50 below it. Raise it by 2⁵⁰ and the fundamental is 48 Hz, a low G a third of a semitone flat, with the other peaks as its overtones at 96, 144, 192 Hz and up. Press Play the chord to hear them, each with power in proportion to its peak's height, and watch the sky beside the curve fill in note by note, from the fundamental's degree-wide spots to the finest ripples.

Baryons add inertia but no pressure, so they tip every oscillation toward compression, the way a weight hung on a spring lowers its resting point. The compressions, the odd peaks, gain; the rarefactions, the even ones, lose. In this model, at Planck's value the third peak nearly matches the second even though damping has begun to erase it. Halve the baryons and it drops to two thirds of the second; double them and it stands half again as tall. That contrast is one of the main ways the cosmic microwave background weighs ordinary matter. Planck finds Ω_bh² = 0.02237 ± 0.00015, where Ω_b is the baryons' share of the critical density and h is the Hubble constant in units of 100 km/s per megaparsec: one baryon for every 1.63 billion photons. A laboratory agrees. The LUNA experiment, deep under the Gran Sasso mountains in Italy, measured how fast deuterium captures a proton and emits a gamma ray of about 5.5 MeV. Big Bang nucleosynthesis turns that rate and the measured primordial deuterium into 0.02233 ± 0.00036. A reaction measured in a laboratory and run a few minutes after the Big Bang, and sound frozen at 372,000 years, agree: the two values differ by 0.2%, well inside the laboratory route's 1.6% uncertainty.

The sky beside the chord is one draw. The primordial lumps were random, so theory predicts only the average power at each ℓ, and our sky is a single sample of it. At each ℓ it offers 2ℓ + 1 independent numbers, so every power measurement carries an error no telescope can remove, the cosmic variance, on a full sky √(2/(2ℓ+1)): 63% at ℓ = 2, 6.7% at ℓ = 220, 2.0% at ℓ = 2500. That is why one draw's dots scatter widely at the left of the figure and hug the curve at the right. Counted from ℓ = 2 to 2500, the temperature sky holds (2500 + 1)² − 4 = 6,254,997 independent numbers. No better telescope can add to them; only other signals can, such as the light's polarization and its lensing. Cosmic variance is why the sky's large-scale oddities, such as the Cold Spot, stay at about 2 to 3 standard deviations: no new data can settle them, and how significant they look also depends on statistics chosen after looking. Part one's eye kept three numbers from a spectrum; cosmology compresses Planck's maps of 50,331,648 pixels into the six parameters of its standard model.

Strike a bell and it rings at a few pitches set by its shape, each one fading as the vibration drains away into sound and heat. Physicists grade anything that rings by its quality factor, Q = πfτ, where f is the frequency and τ the time the amplitude takes to fall by a factor of e (about 2.72). A resonator with a large Q rings for many cycles and its tone is pure. One with a small Q dies almost at once. Part five has treated matter as ringing: an electron's standing wave in an atom, light held between mirrors, sound in the plasma of the young universe. The last resonator in it is made of nothing but spacetime.

When two black holes merge, the new horizon is born misshapen, and it sheds the lumps as gravitational waves, ripples in spacetime that stretch and squeeze whatever they cross. They come out at a set of tones called quasinormal modes, each a pitch with its own decay. In general relativity every one of them depends on just two numbers, the black hole's mass M and its spin a, which runs from 0 for none to 1 for the fastest possible. (A third, electric charge, is negligible for real black holes.) Mass sets only the scale. Double it and every tone drops an octave and lasts twice as long. Spin sets the shape of the chord. Each tone is labeled by three integers (l, m, n): l and m describe its pattern around the hole, and n counts overtones, each fading faster than the one before.

On 14 September 2015 LIGO made the first direct detection of gravitational waves: GW150914, two black holes of about 36 and 29 solar masses merging into one of about 62, spinning at 0.67. About 3 solar masses of energy left as gravitational waves. For that final black hole the standard fit of the tones (Berti, Cardoso and Will, 2006) puts the fundamental, (2,2,0), at 272 Hz in the black hole's own frame, near the C♯ above middle C, with Q = 3.2. (The expanding universe stretched it to about 250 Hz by the time it reached LIGO.) Its amplitude falls by e every 3.70 ms and is down to a thousandth after 7.0 cycles. An exact calculation, which the figure below runs live, gives 271 Hz and Q = 3.18. The line a strontium clock ticks on has a natural Q of about 4 × 10¹⁷: tuned to the same pitch, it would take about 100 million years to fade that far.

Why is the black hole so lossy? Around one that does not spin, at r = 3GM/c² (1.5 times the horizon's radius; G is Newton's constant of gravitation and c the speed of light), light can travel in a circle. The orbit is unstable, like a ball balanced on a ridge. Light that starts a hair outside it spirals away, and its distance from the orbit multiplies by e^2π ≈ 535 on every lap. In the limit of short waves this becomes exact: the tones' pitch is set by how fast light circles that photon ring, and their decay by how fast light slides off it (Cardoso and colleagues, 2009). For the long waves a real black hole rings with, it stays close.

Could consciousness have something to do with light beyond violet, all the way out in the extreme ultraviolet? That band, EUV for short, runs from 10 to 121 nm, where each photon carries 10.25 to 124 eV (an electronvolt, eV, is the energy one electron gains crossing one volt). The idea deserves a fair test, and the answer comes in two halves. One half holds: something fundamental does sit in the EUV. The other half fails, on physics that can be checked number by number.

Start with the half that holds. The energy scale of atoms is the Rydberg energy, ½α²mₑc² = 13.6 eV, where α ≈ 1/137 is the fine-structure constant and mₑc² the electron's rest energy; the atom chapter builds every color of hydrogen from it. As light, 13.6 eV is a 91.1 nm photon, inside the EUV. So are the energies that pull an electron off hydrogen, carbon, nitrogen and oxygen, and off liquid water (11.33 eV). Water's electrons respond most strongly near 21 eV (59 nm), so a fast charged particle crossing a body most often gives up 20 to 22 eV at a time, an EUV-sized packet, as it would in any dense matter. The fine-structure constant puts the end of chemistry in the EUV: there, every photon that is absorbed ionizes or breaks a molecule. (The damage starts lower, in the ultraviolet, as the figure shows.)

Now the floor. At body temperature, 310 K, the energy of thermal jostling is about kT = 0.0267 eV, where k is Boltzmann's constant and T the temperature. The Rydberg is 509 times larger, and every molecular step a brain takes happens inside that factor. A hydrogen bond holds 2 to 11 kT. Splitting one ATP, the cell's fuel, releases about 20 kT. The 70 mV across a nerve cell's membrane is worth 2.6 kT per charge. The photon that switches retinal in your rods carries 2.49 eV, and silicon, where chips compute, has a band gap of 1.12 eV.

Below, one photon sits at 13.5 nm, the EUV light that prints today's chips. In principle it could pay for 4,960 bits against the noise of body heat, and it carries as much energy as 148 to 177 ATP. In water it gets 139 to 179 nm, its 1/e depth (the distance over which a beam of such photons dims to 37%), and it ionizes what it hits. Drag it from heat toward violet and on into the EUV, and watch where light stops being read and how far it still gets.