IOne drop, one ray
A raindrop is, to a good approximation, a tiny sphere of water. Sunlight that strikes it bends going in (refraction), can bounce off the far inside wall (reflection), and bends again coming out. The total turn the ray makes is its deflection. Steer the ray up and down the drop and watch the deflection change — it is not the same for every ray, and that is the whole story.
Slide slowly from the top of the drop (b → 1, grazing the edge) down toward the centre (b → 0). The deflection swings — but it does not swing freely. It runs down to a lowest value and then turns back up. There is a minimum deflection, and the rays that happen to hit near that special height all leave at nearly the same angle. Uniformly spaced going in, they come out bunched. Bunched rays mean concentrated light. The rainbow is that bunch.
The bow is not painted on the sky at 42°. It is the one angle the drop cannot avoid sending light to.
IIThe curve that makes a rainbow
Plot the deflection against where the ray hit, and the rainbow draws itself. The bottom of the curve is flat — and a flat bottom means a whole range of incoming rays map onto almost the same outgoing angle. That flatness is the pile-up, made visible.
There is a clean closed form for where the bottom sits. For a ray that bounces k times inside, the minimum is at the incidence angle where
cos²i = (n² − 1) / k(k+2)
— with n the refractive index of water, about 1.333. The verifier finds the same angle two independent ways: by this formula, and by numerically hunting the minimum of the traced deflection. They agree to four decimals. For one bounce that puts the bottom at an incidence of 59.4° and a deflection of 137.9° — and a deflection of 137.9° means the light comes back 180° − 137.9° = 42.1° from the point directly opposite the sun. That is the radius of the bow.
Descartes counted this by hand · 1637
In Les Météores, René Descartes did exactly what the Rain of rays button does — only with arithmetic, no calculus and no computer. He traced rays through a glass sphere of water at many heights, tabulated where each emerged, and saw them concentrate near 41–42° for the primary and near 51° for the secondary. He had found the caustic by census. In the instrument above, fire the rain and look at the tally: of the rays binned by exit angle, the fullest sliver near 42° holds about fourteen times as many as a typical one. Descartes got the geometry and the brightness exactly right — and got the colour entirely wrong, blaming a spin imparted to light particles. Colour waited for Newton.
IIIColour, the second bow, and the dark between
Water bends violet light a hair more than red. So there is no single rainbow angle — there is one per colour, and the bow is the thin fan between them. Newton's insight, made literal.
Run the minimum-deflection formula for each colour's own refractive index and the bow spreads into a band:
| colour | n (water) | primary radius | secondary radius |
|---|
The primary bow runs from violet at its inner edge to red on the outside — a fan only about 1.9° wide. The secondary, with its second bounce, reverses the order: red turns inside, violet out. And between the two bows lies a strip of sky that no single- or double-bounced ray can reach at all. Single-reflection light only ever emerges inside 42°; double-reflection light only outside 51°. The gap between — roughly 42° to 50° — receives neither. It is measurably darker than the sky on either side, and it has a name older than any of the geometry: Alexander's dark band, after Alexander of Aphrodisias, who wrote it down around 200 AD.
IVThe edge of the edge — where ray optics ends
Everything above treated light as rays. Rays alone get the 42°, the second bow, the dark band, the colours — a remarkable amount. But they also predict an impossibility, and they cannot explain the faint pink-and-green fringes that sometimes ripple just inside the primary. This is the honest seam.
Ray optics says the brightness at the caustic is infinite — a perfectly sharp line of light. Real bows have a sharp-ish but finite edge, and just inside the strongest ones, in fine mist or spray, you can see supernumerary arcs: a few extra pale bands, too closely spaced and too pastel to be a second spectrum. Pure rays have no way to make them.
The mechanism is the one fact the ray picture hands you for free and then can't use. Inside the bow, every angle is fed by exactly two rays — one from each side of the minimum (the instrument in §II shows the curve crossing any horizontal line twice). At the rainbow angle the two merge; just outside, there are none. Those two rays travel different distances through the drop, so they arrive out of step — and waves that arrive out of step interfere, reinforcing at some angles and cancelling at others. Thomas Young saw this in 1804; the verifier confirms it, computing the two rays' optical-path difference straight from the traced polylines and watching it grow from zero at the rainbow angle.
One thing falls straight out of that picture and is worth keeping because it is visible: the fringe spacing scales as (λ⁄a)2/3 — wider for shorter drops. Shrink the drop eightfold and the verifier finds the first fringe move out about fourfold (82/3 = 4, confirmed to 3.98). That is exactly why supernumeraries show in fine mist and spray, where drops are small, and almost never in a downpour of fat drops.
What this page does not compute — named, not hidden
The two-ray model gives the mechanism and the scaling, but not the exact place of each fringe or the true shape of the edge. That needs the full wave theory — George Biddell Airy's 1838 "rainbow integral," whose first maximum sets the real bow and whose later maxima are the supernumeraries. That integral is cited here, not run. The honest claim of this layer is the part it checks end to end: the angles, the caustic, the colour order, the dark band, and the existence-and-scaling of the interference. Where the model stops, the page says so.
A rainbow is a place, not a thing. It is the set of directions in which a sky full of drops happens to crowd its light — and it moves with you because the geometry is anchored to your own eye.
A wink, kept honest: the primary radius rounds to the integer 42, the same 42 Douglas Adams made the Answer to Everything in 1979. It is a coincidence of our base-ten, 360°-to-a-circle convention and means nothing about the sky — but the verifier checks it rounds to 42 anyway, because it would be a shame not to.
✓Show the check
Every angle on this page is recomputed from scratch by one offline verifier — none is quoted from memory — by tracing rays through a sphere of water and reading the geometry off the traces.
The verifier builds a vector ray-tracer (refract → reflect → refract), checks it against the closed-form deflection to better than a millionth of a degree, then derives the rainbow angle as the extremum two independent ways, reproduces Descartes' ray-census pile-up, places every colour edge of both bows, proves Alexander's dark band is empty of rays, shows the higher-order bows sit around the sun (why the 3rd and 4th are nearly unseeable — first convincingly photographed only in 2011), and confirms the two-ray interference and its (λ⁄a)2/3 scaling. The numbers it writes to artifact.json are the numbers printed above.
Inputs that are measured, not derived: the refractive index of water at three wavelengths (red 1.331, yellow 1.333, violet 1.344), standard textbook values near 20 °C. There is no single n — it depends on wavelength, which is the whole reason a rainbow has colour, and drifts slightly with temperature and purity. Everything else is traced from those three numbers.