The Bow-ery

The third of the “-ery” sheets, after The Hatchery’s twelve cosmic eggs and The Quillery’s eight Esmx. Same rule as both: every drawing carries one checkable thing, with its source under it. The companion catalogue is A Field Guide to Rainbows (No. 20), which sorts bows by what physically changed. This sheet asks the smaller question that a catalogue cannot answer in prose — how big is it, and where in the sky do I turn to look?

The one thing to know before the plates

Three units to the degree, and every plate is centred at the same point. The dot at the bottom of each drawing is the centre of that bow — the antisolar point, in the direction of your own shadow, for everything made of water; the sun itself for the two plates made of ice and the one that bounced three times. Radii are true against each other, so a bow that looks half the size of another is half the size of it.

Every plate also carries the 42° primary as a faint dashed ghost, whether or not the primary is what the plate is about. That is the reference line: it is the bow nearly everyone has seen, and it is the only way a drawing can tell you that a 22° halo is not a small rainbow but a different thing at half the radius, on the other side of the sky.

What the scale cannot show: brightness, which varies by more than an order of magnitude across these ten and is drawn only suggestively; and the sky itself, which is left out so the geometry is not competing with weather.

What this is, and what it is not

These are drawings, not photographs, and not data. The radii are computed from published angles and are trustworthy to the width of the stroke; everything else — the thickness of a band, the intensity of a colour, the exact spread of a fogbow — is drawn to be legible rather than measured. Where a figure is doing real work it is printed as a number on the plate, and the source line carries it.

Eight of the ten reproduce their counterpart glyph from Field Guide No. 20. That guide draws each bow at 104×96 with two labels in six-and-a-half-point type, because the glyph has to survive being a diagram in a card. The plates here are 400×232 — roughly eight and a half times the area — and they exist because a diagram is drawn to explain itself and a drawing is not. Putting both on the plate is the honest version of that claim: you can see what the small one had to give up.

Two plates carry no glyph, and the reason is worth stating. The field guide has no entry for a plain rainbow, because it sorts bows by what changed and in an ordinary rainbow nothing has. So the most familiar object on this page is the one the catalogue could not card — and it leads the sheet.

The ten

The Primary

No. 1 · the one the catalogue could not card

The bow nearly everyone has seen, and the only plate here with no counterpart in Field Guide No. 20. That guide sorts bows by what changed, and in this one nothing has. It is the reference, not an entry — which is why its dashed ghost appears on all nine of the others.

The bands, though, are ours rather than the sky’s. Newton’s optical lectures of 1670–72 name five: red, yellow, green, blue, violet. Opticks, in 1704, prints seven — orange and indigo added at exactly the points where the half-steps fall in the Western diatonic scale, the seven-note division of an octave. Newton offered the correspondence between the harmony of sound and the harmony of colour as an analogy and hedged it carefully; the seven survived the hedge and became what children are taught.

The light has no bands in it at all. It is a continuum, and every boundary drawn across it — including the six drawn here — is a decision somebody made about where one name should stop. The sky supplies the gradient. The categories are ours.

Sources: Newton’s five-colour spectrum in the Optical Lectures (1670–72) against the seven of Opticks (1704), and the diatonic-scale explanation for the addition of orange and indigo: Topper, D., “Newton on the number of colours in the spectrum,” Studies in History and Philosophy of Science (1990), which is the scholarly treatment of the question rather than the popular retelling. Newton framed the sound–colour correspondence as an analogy, not a result, and we keep that. Related here: Not a Line (No. 21), which argues the adjacent thing — that a spectrum is not a severity scale — from the emission-line side rather than this one.

No counterpart glyph. Field Guide No. 20 has no entry for a plain rainbow, because nothing about it changed.

The Secondary

No. 2 · the second bounce, drawn at true separation

One internal reflection leaves the drop at about 42.7°. Two leave it at about 50°, and the extra bounce flips the colour order, so the two reds face each other. The outer bow is 43% as bright as the inner — an extra reflection is another chance to lose light, and a longer path loses more.

The gap between them is Alexander’s dark band, and it is drawn here as what it is: not a shadow and not a missing band, but a direction into which neither bow returns any light at all. Alexander of Aphrodisias noticed the sky was darker there around 200 CE and wrote it down, roughly fourteen centuries before anyone could say why.

The reason this plate exists rather than a photograph: in a photograph the separation is whatever the lens made it. Here it is 7.3° of true angular distance, measurable against every other plate on the sheet.

Sources: Angles, relative brightness and the no-light-returned explanation of the dark band all follow Stull, R., Practical Meteorology, and are the same figures printed in Both Arcs, One Drop (No. 70) and Field Guide No. 20 — deliberately, because a sheet and its zine disagreeing about a number is worse than either being wrong alone. Alexander of Aphrodisias, c. 200 CE, for the band’s first record and its name. The argument this plate does not make — that “secondary” is a rank the physics does not contain — is No. 70’s.

As Field Guide No. 20 draws it: same two arcs, no scale, at 104×96.

The Supernumeraries

No. 3 · the arcs that broke the ray model

The faint pink and green fringes that sometimes crowd the inner edge of a primary. They are not weak repeats of the rainbow. They are the one feature of a rainbow that geometric optics cannot produce at all — not “predicts poorly”, but cannot, in principle, which made them a standing refutation of the complete ray account sitting in plain sight above every rain shower on Earth.

Thomas Young explained them in 1804, by interference: two rays can leave a drop at the same angle having travelled different distances inside it, and on recombining they reinforce or cancel by wavelength. Because that requires light to have a phase, the arcs became evidence for the wave theory at the moment that argument was live.

The half worth keeping is what happened next. Airy dismissed Young in 1838 as “the imperfect theory of interference”, on the real ground that it put the arcs at the wrong angles. Young’s numbers were off because he had not known a ray crossing a focal line picks up a 90° phase shift. Put it in and his account is surprisingly accurate. A correct mechanism, rejected for a missing term, and overshadowed for well over a century.

Sources: Young’s 1804 interference explanation, Airy’s 1838 dismissal of it as “the imperfect theory of interference”, and the missing 90° focal-line phase shift: Lock, J. A. & Adams, C. L., “Supernumerary arcs of rainbows: Young’s theory of interference,” Applied Optics 56(19), G104 (2017), read at abstract and summary level. Fringe spacing is drawn schematically — it depends on drop size and is not printed as a figure.

As Field Guide No. 20 draws it, with the ray model crossed out.

The Fogbow

No. 4 · the same bow, drawn with a wider pen

Rain falls at around a millimetre. Fog and cloud droplets rarely exceed a hundred micrometres, and a fogbow wants them at roughly 0.05 mm and below. At that size the light diffracts around the droplet, and diffraction widens every colour band. Widen them far enough and they stop sitting side by side and start sitting on top of each other; overlapping colours across the visible range add to white.

Nothing is missing. Every colour is present and in order. The white is the sum, arrived at by addition — which is the opposite of what “colourless rainbow” implies.

The plate also shows why sorting these by appearance was never going to work. A fogbow looks nothing like a rainbow and differs from one in a single parameter. And it runs on a slope, not a switch: as droplets grow the bow narrows and colour returns at the edges, so waterfall spray at 100–500 µm gives bows that are neither.

Sources: Droplet-size thresholds, the diffraction mechanism and the intermediate spray forms follow Les Cowley’s Atmospheric Optics (atoptics.co.uk), an expert secondary source and named as one. Radius is drawn as a representative value inside the primary; the real figure is a range, about 30–45°, because it varies with droplet size — which is why no single number is printed on this plate.

As Field Guide No. 20 draws it: colours beneath, white on top.

The 22° Halo

No. 5 · the plate this sheet was built for

Put the two on one scale and the argument stops needing to be made in words. A 22° halo is not a small rainbow. It is a different phenomenon, at about half the radius, formed in hexagonal plates of ice rather than spheres of water — and, decisively, it is centred on the sun. To see the dashed arc on this plate you would have to turn around.

That single fact disposes of most of the genre. Lists of “rainbow kinds” routinely file sun dogs, the 22° halo and the circumhorizontal arc alongside real bows because they are coloured arcs in the sky, and coloured arc in the sky is a description of appearance. The mechanism, the medium and the direction you face are all different.

Sun dogs are the same ice at the same 22°, concentrated where the plates are horizontal — which puts them level with the sun rather than anywhere on the ring, and is why they come in pairs.

Sources: Hexagonal plate and column crystals, the 22° minimum-deviation angle, and sun dogs as the horizontal-plate case: standard halo optics via Atmospheric Optics. Drawn with the sun on the horizon so the geometry is legible; at any real solar elevation the ring rides with the sun and the dogs stay level with it. Field Guide No. 20 files all of this under a single withheld card, The Ones That Are Not Rainbows, with no reframe table.

As Field Guide No. 20 draws it: the crystal, and a bow crossed out.

The Circumhorizontal Arc

No. 6 · where “rare” turns out to mean “not from here”

Sold everywhere as a fire rainbow. It is ice — sunlight entering the vertical side face of a hexagonal plate crystal in cirrus and leaving through the horizontal bottom face — and it hangs 46° below the sun, which is why this is the one plate whose centre had to move. The scale is unchanged.

It needs the sun higher than 58°. That is a hard geometric condition, and it has a consequence the genre never prints: the arc is impossible north of 55°N or south of 55°S. Not rare there. Impossible.

Where it is possible, availability varies enormously. The sun clears 58° for roughly 140 hours a year in London and roughly 670 in Los Angeles. Same crystals, same cirrus, same physics — a once-in-a-lifetime sight in one city and a regular summer one in the other. Every rarity ranking in this genre is partly a ranking of where the reader lives, and this is the entry where you can put a number on it.

Sources: The 58° solar-elevation requirement, the impossibility above 55° latitude, and the London/Los Angeles hour figures come from the halo literature via Atmospheric Optics and Wikipedia’s circumhorizontal arc entry, which agree. No primary computation was reached, and these are the numbers on this sheet we would most like to improve — they are also the ones carrying the most argumentative weight, which is exactly when a figure should be flagged rather than trusted.

No counterpart glyph. Field Guide No. 20 files this inside its single ice-halo card rather than giving it one of its own.

The Glory

No. 7 · the open question on this sheet

Coloured rings a few degrees across, tight around the shadow of your own head on cloud or fog — from a ridge, or an aircraft window. The scale is the point of drawing it here: set against the dashed primary, a glory is tiny, and photographs never tell you that because they are taken with whatever lens was to hand.

It is also the one entry on this sheet whose mechanism is openly unfinished. Mie theory computes the glory in full colour and offers no explanation of it. Debye-series analysis says the light has undergone one internal reflection. In 1947 van de Hulst proposed surface waves — light grazing the droplet and shedding radiation as it travels round — and that remains the consensus. The standard statement of that consensus is that the precise mechanism remains rather obscure.

So this is the plate that keeps the sheet honest. Nine of these ten are settled to a fraction of a degree. One of them you can see out of a plane window most cloudy flights, and it is not finished. A specimen sheet that only drew solved things would be teaching that the sky is solved.

Sources: Mie theory computing but not explaining the glory, the Debye-series single internal reflection, van de Hulst’s 1947 surface-wave proposal, and the “remains rather obscure” assessment: Laven, P., “How are glories formed?”, Applied Optics 44(27), 5675 (2005), with the surrounding Atmospheric Optics material. Ring radius is drawn at a representative few degrees; it varies with droplet size and no single figure is printed.

As Field Guide No. 20 draws it, question mark and all.

The Third and Fourth Order

No. 8 · present for centuries, photographed in 2011

Light can go round inside a drop more than twice, and at three and four bounces the bows turn around. They do not appear outside the secondary; they appear on the other side of the sky, at roughly 40° and 45° from the sun itself. The glare on this plate is not decoration — it is the entire reason they went unphotographed.

Their existence was worked out long before anyone saw one. The obstruction was never theory; it was contrast. Both were first photographed in 2011, the third-order by Michael Großmann with Elmar Schmidt and Alexander Haußmann, and a natural fourth-order by Michael Theusner, in the same issue of Applied Optics. It took knowing the exact angle, a rain shaft in the right place, heavy contrast processing, and persistence.

The common correction follows straight from the drawing: three arcs in a photograph are almost never a “triple rainbow.” They are the primary, the secondary and a reflection bow. The real tertiary is at the centre of this plate’s geometry, behind the photographer.

Sources: Großmann, M., Schmidt, E. & Haußmann, A., “Photographic evidence for the third-order rainbow,” Applied Optics 50(28), F134–F141 (2011), and Theusner, M., “Photographic observation of a natural fourth-order rainbow,” Applied Optics 50(28), F129 (2011); both confirmed at the publisher. These are a small number of confirmed images obtained by specialists under heavy processing, not a survey — the phrase the literature uses is “photographic evidence,” and we keep it.

As Field Guide No. 20 draws it: rings around the sun, no scale.

The Moonbow

No. 9 · the bow was complete; the instrument was not

Moonlight is sunlight that bounced off the Moon, so a moonbow is a rainbow in the most literal sense: same geometry, same 42°, same colour order. The only difference is intensity — and intensity is enough, because of how eyes are built.

In low light vision falls back from the cone cells, which discriminate colour, to the rod cells, which are far more sensitive and report no colour at all. A moonbow generally sits below the threshold at which cones respond, so the arc arrives at the retina with its full spectrum intact and is reported in greyscale. Open a shutter for thirty seconds and every colour appears.

The plate draws both readings of one arc, because that is the honest picture. The bow is not deficient, and neither are the rods — they buy their sensitivity by not sorting wavelengths, which is the job they are for. Two instruments, two accounts, and a third settles it. The colours were present and unreported.

Sources: Scotopic vision falling to the rods below the cone threshold, and the full spectrum appearing in long-exposure photographs of moonbows: well-established visual science and atmospheric optics, given as such rather than from a paper read for this sheet. The colour wedge is drawn as a quarter of the arc for legibility; a real long exposure records the whole bow.

As Field Guide No. 20 draws it: grey bow, colour bow, moon and camera.

The Full Circle

No. 10 · the arc was never an arc

A rainbow is the set of directions 42° from the antisolar point, and that set is a circle. It is always a circle. Nobody sees one because the antisolar point is below the horizon whenever the sun is above it, so the lower half would have to be made by raindrops that are underground.

Get above the drops — an aircraft, a cliff, the top of a waterfall — and the obstruction goes away and the bow closes. Which means the famous “rare full-circle rainbow” is not a kind of rainbow. It is the only kind there is, seen from a position where nothing is standing in front of its lower half.

This plate is the sheet’s own argument about itself. Nine drawings above it stop at the horizon line because that is what the ground does to a cone of directions — and the shape you were taught to call a bow is a fact about standing on a planet.

Sources: Geometry, and it needs no citation beyond the antisolar point being below the horizon whenever the sun is above it. Drawn with the centre exactly on the horizon, which corresponds to the sun exactly on the opposite horizon — the case that gives the largest arc anyone on the ground can see. Field Guide No. 20 makes the same point in prose at The Arc Was Never An Arc.

As Field Guide No. 20 draws it: solid above, dashed below.

What we left out, and the colour that is not here

The colour these are drawn in is in none of them. This site’s palette runs on a pink, #f472b6, and it appears on nine of the ten plates above. Magenta is not in the spectrum. There is no wavelength that produces it: it is what the visual system builds when the long-wavelength and short-wavelength cones fire together and the middle one does not, and the brain closes a ring that the physics leaves open. So a sheet of ten bows, drawn as carefully to scale as we could manage, is drawn partly in a colour that no rainbow contains — and there is no version of these drawings that fixes it, because the pink is doing a job the spectrum has no way to do. We are not going to pretend that is a flaw we overlooked. It is the most interesting thing on the page.

We left out the sea spray bow, the twinned bow and the Titan bow, and the reason is this sheet’s own rule. All three are in Field Guide No. 20 and all three are good. None of them can be drawn to scale in a way that tells you anything: a sea spray bow differs from a rainbow by 0.8°, which at three units to the degree is 2.4 units — about the width of the stroke it would be drawn in. A twinned bow’s split comes from drops flattened by one or two per cent. A Titan bow is at 49°, which would look like a slightly larger rainbow and teach nobody anything, because the interesting part is the liquid and not the angle. A specimen sheet organised around scale has to decline the entries where scale is not the story, and saying so is better than quietly dropping them.

The reflection bow was cut for a worse reason, and it is worth admitting. It belongs here by every test — it is a scale-and-position phenomenon, it is beautiful, and it is the single commonest cause of a photograph captioned “triple rainbow.” It came out because drawing it honestly needs a second centre on the plate (the reflected sun has its own antisolar point above the horizon), and two centres on a sheet whose entire premise is one shared centre would have undermined the thing the sheet is for. That is a limitation of the format we chose, not a fact about the bow. It is carded properly at Field Guide No. 20.

Two plates break the shared centre, and both say so on their own faces rather than here. The Circumhorizontal Arc moves the sun to the top of the frame, because the arc sits 46° below the sun and would otherwise be off the bottom of the plate; the scale is unchanged. The 22° Halo and The Third and Fourth Order are centred on the sun rather than the antisolar point, which is not a break in the convention but the whole content of those two plates — they are on the other side of the sky, and the drawing is how you find that out.

What the scale is worth, stated plainly. The radii here are computed from published angles at three units to the degree, and they are as good as those angles. Everything else is drawn for legibility: band widths are exaggerated (the primary’s by about 2.8×, and the plate prints that), brightness is suggestive rather than measured, and the sky is left out entirely. Where a figure is load-bearing it is printed on the plate and sourced under it. The two figures we would most like to improve are the circumhorizontal arc’s London and Los Angeles hours, which reached no primary computation and carry more of the argument than any other numbers on the page.

On paper, this sheet loses its colour, and that is the right trade. The shared print stylesheet turns every SVG stroke to a single dark grey, because our five accents measure between 1.7:1 and 2.9:1 against white and not one of them clears the 3:1 a graphic needs to be legible. So a printed Bow-ery is line art. The geometry — which is the whole point — survives intact: a 22° halo still prints at half a rainbow’s radius, and Alexander’s dark band still prints 7.3° wide. What is lost is band order, which is why every plate whose argument is a colour order also prints that order as a label. A page about colour that cannot print its colour is worth writing down rather than hiding.

Printed in grey. Every stroke on this sheet is a colour the shared print stylesheet flattens for legibility — the radii and the labels are unaffected, and the colour version is at starstuff.earth/bow-ery.

Why a third “-ery”, and why it is an Easter egg. The Hatchery came from the small egg at the foot of the front page; The Quillery from the mascot beside it. This one comes from a catalogue that had to draw sixteen bows at 104×96 and could not show you how big any of them was. An egg is not a lower standard here — same sourcing, same fact-check ledger, same eight gates, same changelog entry as anything else on this site. Only the door moves.