Rainbow

For other uses, see Rainbow (disambiguation).

Semicircular double rainbow. Supernumerary rainbows on the inside of the primary arc. Shadow of the photographer marks the centre of the rainbow circle (antisolar point).

A rainbow is an optical and meteorological phenomenon that causes a spectrum of light to appear in the sky when the Sun shines on to droplets of moisture in the Earth's atmosphere. It takes the form of a multicoloured arc, with red on the outer part of the arc and violet on the inner section.

A rainbow spans a continuous spectrum of colours; the distinct bands are an artifact of human colour vision. The most commonly cited and remembered sequence, in English, is Newton's sevenfold red, orange, yellow, green, blue, indigo and violet (popularly memorized by mnemonics like Roy G. Biv). Rainbows can be caused by other forms of water than rain, including mist, spray, and dew.

Rainbows may also form in mist, such as that of a waterfall

Rainbow with a faint reflected rainbow in the lake

Visibility

Rainbows may also form in the spray created by waves (called spray bows)

Rainbows can be observed whenever there are water drops in the air and sunlight shining from behind at a low altitude angle. The most spectacular rainbow displays happen when half the sky is still dark with raining clouds and the observer is at a spot with clear sky in the direction of the sun. The result is a luminous rainbow that contrasts with the darkened background.

The rainbow effect is also commonly seen near waterfalls or fountains. In addition, the effect can be artificially created by dispersing water droplets into the air during a sunny day. Rarely, a moonbow, lunar rainbow or nighttime rainbow, can be seen on strongly moonlit nights. As human visual perception for colour is poor in low light, moonbows are often perceived to be white. It is difficult to photograph the complete semicircle of a rainbow in one frame, as this would require an angle of view of 84°. For a 35 mm camera, a lens with a focal length of 19 mm or less wide-angle lens would be required. Now that powerful software for stitching several images into a panorama is available, images of the entire arc and even secondary arcs can be created fairly easily from a series of overlapping frames. From an aeroplane, one has the opportunity to see the whole circle of the rainbow, with the plane's shadow in the centre. This phenomenon can be confused with the glory, but a glory is usually much smaller, covering only 5–20°.

At good visibility conditions (for example, a dark cloud behind the rainbow), the second arc can be seen, with inverse order of colours. At the background of the blue sky, the second arc is barely visible.

Scientific explanation

The light is first refracted entering the surface of the raindrop, reflected off the back of the drop, and again refracted as it leaves the drop. The overall effect is that the incoming light is reflected back over a wide range of angles, with the most intense light at an angle of 40–42°. The angle is independent of the size of the drop, but does depend on its refractive index. Seawater has a higher refractive index than rain water, so the radius of a "rainbow" in sea spray is smaller than a true rainbow. This is visible to the naked eye by a misalignment of these bows. The amount by which light is refracted depends upon its wavelength, and hence its colour. Blue light (shorter wavelength) is refracted at a greater angle than red light, but due to the reflection of light rays from the back of the droplet, the blue light emerges from the droplet at a smaller angle to the original incident white light ray than the red light. You may then think it is strange that the pattern of colours in a rainbow has red on the outside of the arc and blue on the inside. However, when we examine this issue more closely, we realise that if the red light from one droplet is seen by an observer, then the blue light from that droplet will not be seen because it is on a different path from the red light: a path which is not incident with the observer's eyes. The blue light seen in this rainbow will therefore come from a different droplet, which must be below that whose red light can be observed.

Contrary to popular belief, the light at the back of the raindrop does not undergo total internal reflection, and some light does emerge from the back. However, light coming out the back of the raindrop does not create a rainbow between the observer and the Sun because spectra emitted from the back of the raindrop do not have a maximum of intensity, as the other visible rainbows do, and thus the colours blend together rather than forming a rainbow.

Light rays enter a raindrop from one direction (typically a straight line from the Sun), reflect off the back of the raindrop, and fan out as they leave the raindrop. The light leaving the rainbow is spread over a wide angle, with a maximum intensity at 40.89–42°.

White light separates into different colours on entering the raindrop because red light is refracted by a lesser angle than blue light. On leaving the raindrop, the red rays have turned through a smaller angle than the blue rays, producing a rainbow.

A rainbow does not actually exist at a particular location in the sky. Its apparent position depends on the observer's location and the position of the Sun. All raindrops refract and reflect the sunlight in the same way, but only the light from some raindrops reaches the observer's eye. This light is what constitutes the rainbow for that observer. The position of a rainbow in the sky is always in the opposite direction of the Sun with respect to the observer, and the interior is always slightly brighter than the exterior. The bow is centred on the shadow of the observer's head, or more exactly at the antisolar point (which is below the horizon during the daytime), appearing at an angle of 40–42° to the line between the observer's head and its shadow. As a result, if the Sun is higher than 42°, then the rainbow is below the horizon and usually cannot be seen as there are not usually sufficient raindrops between the horizon (that is: eye height) and the ground, to contribute. Exceptions occur when the observer is high above the ground, for example in an aeroplane (see above), on top of a mountain, or above a waterfall.

Variations

"Double rainbow" redirects here. For other uses, see Double Rainbow (disambiguation).

Some light reflects twice inside the raindrop before exiting to the viewer. When the incident light is very bright, this can be seen as a secondary rainbow, brightest at 50–53°.

A double rainbow features reversed colours in the outer (secondary) bow, with the dark Alexander's band between the bows.

Frequently, a dim secondary rainbow is seen outside the primary bow. Secondary rainbows are caused by a double reflection of sunlight inside the raindrops, and appear at an angle of 50–53°. As a result of the second reflection, the colours of a secondary rainbow are inverted compared to the primary bow, with blue on the outside and red on the inside. The secondary rainbow is fainter than the primary because more light escapes from two reflections compared to one and because the rainbow itself is spread over a greater area of the sky. The dark area of unlit sky lying between the primary and secondary bows is called Alexander's band, after Alexander of Aphrodisias who first described it.

A third, or tertiary, rainbow can be seen on rare occasions, and a few observers have reported seeing quadruple rainbows in which a dim outermost arc had a rippling and pulsating appearance. These rainbows would appear on the same side of the sky as the Sun, making them hard to spot. One type of tertiary rainbow carries with it the appearance of a secondary rainbow immediately outside the primary bow. The closely spaced outer bow has been observed to form dynamically at the same time that the outermost (tertiary) rainbow disappears. During this change, the two remaining rainbows have been observed to merge into a band of white light with a blue inner and red outer band. This particular form of doubled rainbow is not like the classic double rainbow due to both spacing of the two bows and that the two bows share identical normal colour positioning before merging. With both bows, the inner colour is blue and the outer colour is red.

Higher-order rainbows were described by Felix Billet (1808–1882) who depicted angular positions up to the 19th-order rainbow, a pattern he called "rose". In the laboratory, it is possible to observe higher-order rainbows by using extremely bright and well collimated light produced by lasers. A sixth-order rainbow was first observed by K. Sassan in 1979 using a HeNe laser beam and a pendant water drop. Up to the 200th-order rainbow was reported by Ng et al. in 1998 using a similar method but an argon ion laser beam.

Supernumerary rainbow

A contrast-enhanced photograph of a supernumerary rainbow, with additional green and purple arcs inside the primary bow.

A supernumerary rainbow—also known as a stacker rainbow—is an infrequent phenomenon, consisting of several faint rainbows on the inner side of the primary rainbow, and very rarely also outside the secondary rainbow. Supernumerary rainbows are slightly detached and have pastel colour bands that do not fit the usual pattern.

It is not possible to explain their existence using classical geometric optics. The alternating faint rainbows are caused by interference between rays of light following slightly different paths with slightly varying lengths within the raindrops. Some rays are in phase, reinforcing each other through constructive interference, creating a bright band; others are out of phase by up to half a wavelength, canceling each other out through destructive interference, and creating a gap. Given the different angles of refraction for rays of different colours, the patterns of interference are slightly different for rays of different colours, so each bright band is differentiated in colour, creating a miniature rainbow. Supernumerary rainbows are clearest when raindrops are small and of similar size. The very existence of supernumerary rainbows was historically a first indication of the wave nature of light, and the first explanation was provided by Thomas Young in 1804.

Reflected rainbow, reflection rainbow

Reflection rainbow and normal rainbow, at sunset

When a rainbow appears above a body of water, two complementary mirror bows may be seen below and above the horizon, originating from different light paths. Their names are slightly different. A reflected rainbow will appear as a mirror image in the water surface below the horizon, if the surface is quiet (see photo above). The sunlight is first deflected by the raindrops, and then reflected off the body of water, before reaching the observer. The reflected rainbow is frequently visible, at least partially, even in small puddles.

Where sunlight reflects off a body of water before reaching the raindrops (see diagram), it may produce a reflection rainbow (see photo at the right), if the water body is large, quiet over its entire surface, and close to the rain curtain. The reflection rainbow appears above the horizon. It intersects the normal rainbow at the horizon, and its arc reaches higher in the sky, with its centre as high above the horizon as the normal rainbow's centre is below it. Due to the combination of requirements, a reflection rainbow is rarely visible.

Six (or even eight) bows may be distinguished if the reflection of the reflection bow, and the secondary bow with its reflections happen to appear simultaneously.

Circumhorizontal arc

The circumhorizontal arc is sometimes referred to by the misnomer "fire rainbow". As it originates in ice crystals, it is not a rainbow but a halo.

Rainbows on Titan

It has been suggested that rainbows might exist on Saturn's moon Titan, as it has a wet surface and humid clouds. The radius of a Titan rainbow would be about 49° instead of 42°, because the fluid in that cold environment is methane instead of water. A visitor might need infrared goggles to see the rainbow, as Titan's atmosphere is more transparent for those wavelengths.

Scientific history

The Greek philosopher Aristotle (384–322 BC) was first to devote serious attention to the rainbow. After Aristotle's death, much rainbow theory consisted of reaction to his work, although not all of this was uncritical.

The Arab physicist and polymath, Ibn al-Haytham (Alhazen; 965–1039), attempted to provide a scientific explanation for the rainbow phenomenon. In his Maqala fi al-Hala wa Qaws Quzah (On the Rainbow and Halo), he "explained the formation of rainbow as an image, which forms at a concave mirror. If the rays of light coming from a farther light source reflect to any point on axis of the concave mirror, they form concentric circles in that point. When it is supposed that the sun as a farther light source, the eye of viewer as a point on the axis of mirror and a cloud as a reflecting surface, then it can be observed the concentric circles are forming on the axis." He was not able to verify this because his theory that "light from the sun is reflected by a cloud before reaching the eye" did not allow for a possible experimental verification. This explanation was later repeated by Averroes, and, though incorrect, provided the groundwork for the correct explanations later given by Kamāl al-Dīn al-Fārisī (1267–ca. 1319/1320) and Theodoric of Freiberg (c.1250–1310). Ibn al-Haytham supported the Aristotelian views that the rainbow is caused by reflection alone and that its colours are not real like object colours.

Ibn al-Haytham's contemporary, the Persian philosopher and polymath Ibn Sīnā (Avicenna; 980–1037), provided an alternative explanation, writing "that the bow is not formed in the dark cloud but rather in the very thin mist lying between the cloud and the sun or observer. The cloud, he thought, serves simply as the background of this thin substance, much as a quicksilver lining is placed upon the rear surface of the glass in a mirror. Ibn Sīnā would change the place not only of the bow, but also of the colour formation, holding the iridescence to be merely a subjective sensation in the eye." This explanation, however, was also incorrect. Ibn Sīnā's account accepts many of Aristotle's arguments on the rainbow.

In Song Dynasty China (960–1279), a polymathic scholar-official named Shen Kuo (1031–1095) hypothesized—as a certain Sun Sikong (1015–1076) did before him—that rainbows were formed by a phenomenon of sunlight encountering droplets of rain in the air. Paul Dong writes that Shen's explanation of the rainbow as a phenomenon of atmospheric refraction "is basically in accord with modern scientific principles."

The Persian astronomer, Qutb al-Din al-Shirazi (1236–1311), gave a fairly accurate explanation for the rainbow phenomenon. This was elaborated on by his student, Kamāl al-Dīn al-Fārisī (1260–1320), who gave a more mathematically satisfactory explanation of the rainbow. He "proposed a model where the ray of light from the sun was refracted twice by a water droplet, one or more reflections occurring between the two refractions." He verified this through extensive experimentation using a transparent sphere filled with water and a camera obscura. As he noted in his Kitab Tanqih al-Manazir (The Revision of the Optics), al-Farisi used a large clear vessel of glass in the shape of a sphere, which was filled with water, in order to have an experimental large-scale model of a rain drop. He then placed this model within a camera obscura that has a controlled aperture for the introduction of light. He projected light unto the sphere and ultimately deduced through several trials and detailed observations of reflections and refractions of light that the colours of the rainbow are phenomena of the decomposition of light. His research had resonances with the studies of his contemporary Theodoric of Freiberg (without any contacts between them; even though they both relied on Aristotle's and Ibn al-Haytham's legacy), and later with the experiments of Descartes and Newton in dioptrics (for instance, Newton conducted a similar experiment at Trinity College, though using a prism rather than a sphere).

In Europe, Ibn al-Haytham's Book of Optics was translated into Latin and studied by Robert Grosseteste. His work on light was continued by Roger Bacon, who wrote in his Opus Majus of 1268 about experiments with light shining through crystals and water droplets showing the colours of the rainbow. In addition, Bacon was the first to calculate the angular size of the rainbow. He stated that the rainbow summit can not appear higher than 42° above the horizon. Theodoric of Freiberg is known to have given an accurate theoretical explanation of both the primary and secondary rainbows in 1307. He explained the primary rainbow, noting that "when sunlight falls on individual drops of moisture, the rays undergo two refractions (upon ingress and egress) and one reflection (at the back of the drop) before transmission into the eye of the observer". He explained the secondary rainbow through a similar analysis involving two refractions and two reflections.

René Descartes' sketch of how primary and secondary rainbows are formed

Descartes' 1637 treatise, Discourse on Method, further advanced this explanation. Knowing that the size of raindrops did not appear to affect the observed rainbow, he experimented with passing rays of light through a large glass sphere filled with water. By measuring the angles that the rays emerged, he concluded that the primary bow was caused by a single internal reflection inside the raindrop and that a secondary bow could be caused by two internal reflections. He supported this conclusion with a derivation of the law of refraction (subsequently to, but independently of, Snell) and correctly calculated the angles for both bows. His explanation of the colours, however, was based on a mechanical version of the traditional theory that colours were produced by a modification of white light.

Isaac Newton demonstrated that white light was composed of the light of all the colours of the rainbow, which a glass prism could separate into the full spectrum of colours, rejecting the theory that the colours were produced by a modification of white light. He also showed that red light gets refracted less than blue light, which led to the first scientific explanation of the major features of the rainbow. Newton's corpuscular theory of light was unable to explain supernumerary rainbows, and a satisfactory explanation was not found until Thomas Young realised that light behaves as a wave under certain conditions, and can interfere with itself.

Young's work was refined in the 1820s by George Biddell Airy, who explained the dependence of the strength of the colours of the rainbow on the size of the water droplets. Modern physical descriptions of the rainbow are based on Mie scattering, work published by Gustav Mie in 1908. Advances in computational methods and optical theory continue to lead to a fuller understanding of rainbows. For example, Nussenzveig provides a modern overview.

Culture

Mythology

The end of a rainbow

Main article: Rainbows in mythology

The rainbow has a place in legend owing to its beauty and the historical difficulty in explaining the phenomenon.

In Greek mythology, the rainbow was considered to be a path made by a messenger (Iris) between Earth and Heaven. In Chinese mythology, the rainbow was a slit in the sky sealed by goddess Nüwa using stones of five different colours.

In Hindu mythology, the rainbow is called Indradhanush, meaning "the bow (Sanskrit and Hindi: dhanush is bow) of Indra, the god of lightning, thunder and rain". Another Indian mythology says the rainbow is the bow of Rama, the incarnation of Vishnu. It is called Ramdhonu in Bengali, dhonu (dhanush) meaning bow. Likewise, in mythology of Arabian Peninsula, the rainbow, called Qaus Quzaħ in Arabic, is the war bow of the god Quzaħ.

In Norse Mythology, a rainbow called the Bifröst Bridge connects the realms of Ásgard and Midgard, homes of the gods and humans, respectively. The Irish leprechaun's secret hiding place for his pot of gold is usually said to be at the end of the rainbow. This place is impossible to reach, because the rainbow is an optical effect which depends on the location of the viewer. When walking towards the end of a rainbow, it will appear to "move" further away (two people who simultaneously observe a rainbow at different locations will disagree about where a rainbow is).

Another ancient portrayal of the rainbow is given in the Epic of Gilgamesh: the rainbow is the "jewelled necklace of the Great Mother Ishtar" that she lifts into the sky as a promise that she "will never forget these days of the great flood" that destroyed her children. (The Epic of Gilgamesh, Tablet Eleven)

Then Ishtar arrived. She lifted up the necklace of great jewels that her father, Anu, had created to please her and said, "Heavenly gods, as surely as this jewelled necklace hangs upon my neck, I will never forget these days of the great flood. Let all of the gods except Enlil come to the offering. Enlil may not come, for without reason he brought forth the flood that destroyed my people."

In the Dreamtime of Australian Aboriginal mythology, the rainbow snake is the deity governing water.

In Amazonian cultures, rainbows have long been associated with malign spirits that cause harm, such as miscarriages and (especially) skin problems. In the Amuesha language of central Peru, certain diseases are called ayona’achartan, meaning "the rainbow hurt my skin". A tradition of closing one's mouth at the sight of a rainbow in order to avoid disease appears to pre-date the Incan empire.

In New Age and Hindu philosophy, the seven colours of the rainbow represent the seven chakras, from the first chakra (red) to the seventh chakra (violet).

Christianity and Judaism

According to Christianity and Judaism, after Noah's flood the rainbow gained meaning as the sign of God's promise that terrestrial life would never again be destroyed by flood (Genesis 9:13–17):

I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth. And it shall come to pass, when I bring a cloud over the earth, that the bow shall be seen in the cloud: And I will remember my covenant, which is between me and you and every living creature of all flesh; and the waters shall no more become a flood to destroy all flesh. And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth. And God said unto Noah, This is the token of the covenant, which I have established between me and all flesh that is upon the earth.

Art

Rainbows are generally described as very colourful and peaceful. The rainbow occurs often in paintings. Frequently these have a symbolic or programmatic significance (for example, Albrecht Dürer's Melancholia I). In particular, the rainbow appears regularly in religious art (for example, Joseph Anton Koch's Noah's Thanksoffering). Romantic landscape painters such as Turner and Constable were more concerned with recording fleeting effects of light (for example, Constable's Salisbury Cathedral from the Meadows). Other notable examples appear in work by Hans Memling, Caspar David Friedrich, and Peter Paul Rubens.

The Blind Girl, oil painting (1856) by John Everett Millais. The rainbow – one of the beauties of nature that the blind girl cannot experience – is used to underline the pathos of her condition.

Noah's Thanksoffering (c. 1803) by Joseph Anton Koch. Noah builds an altar to the Lord after being delivered from the Flood; God sends the rainbow as a sign of his covenant (Genesis 8–9).

Literature

The rainbow inspires metaphor and simile. Virginia Woolf in To the Lighthouse highlights the transience of life and Man's mortality through Mrs Ramsey's thought,

"it was all as ephemeral as a rainbow"

Wordsworth's 1802 poem "My Heart Leaps Up When I Behold The Rainbow" begins:

My heart leaps up when I behold

A rainbow in the sky:

So was it when my life began;

So is it now I am a man;

So be it when I shall grow old,

Or let me die!...

The Newtonian deconstruction of the rainbow is said to have provoked John Keats to lament in his 1820 poem "Lamia":

Do not all charms fly

At the mere touch of cold philosophy?

There was an awful rainbow once in heaven:

We know her woof, her texture; she is given

In the dull catalogue of common things.

Philosophy will clip an Angel's wings,

Conquer all mysteries by rule and line,

Empty the haunted air, and gnomed mine –

Unweave a rainbow

In contrast to this is Richard Dawkins; talking about his book Unweaving the Rainbow: Science, Delusion and the Appetite for Wonder:

"My title is from Keats, who believed that Newton had destroyed all the poetry of the rainbow by reducing it to the prismatic colours. Keats could hardly have been more wrong, and my aim is to guide all who are tempted by a similar view, towards the opposite conclusion. Science is, or ought to be, the inspiration for great poetry."

Music

• In the song "Over the Rainbow" from The Wizard of Oz, lead character Dorothy Gale fantasises about a place over the rainbow, where the world is in peace and harmony.
• In "Rainbow Connection", a song known for being sung by Kermit the Frog, the idea of a rainbow is seen as something to wish on, as it is popularly seen as a vision, or symbol of hope.
• In the "End of the Rainbow" by September, the singer sings about the rainbow, and how she will be at the end of the rainbow and her ex could see her there when he reaches the end of the rainbow.
• End of the rainbow is an award winning stage play with music (or musical drama) by Peter Quilter.
• The group Rainbow and the song "Rainbow Demon" by Uriah Heep.
• "I Can Sing a Rainbow" is a popular children's classic song written by Arthur Hamilton, despite the name of the song, not all the colours mentioned are actually colours of the rainbow
• Ronnie James Dio used rainbows as a thematic element in many of his songs, particularly as singer and lyrics-writer for Ritchie Blackmore's band Rainbow. Most notable among these are the songs Catch the Rainbow, Rainbow Eyes and the Dio song Rainbow in the Dark.

Flags

Main article: Rainbow flag

Rainbow flags tend to be used as a sign of a new era, of hope, or of social change. Rainbow flags have been used in many places over the centuries: in the German Peasants' War in the 16th century, as a symbol of the Cooperative movement; as a symbol of peace, especially in Italy; to represent the Tawantin Suyu, or Inca territory, mainly in Peru and Bolivia; by some Druze communities in the Middle east; by the Jewish Autonomous Oblast; to represent the International Order of Rainbow for Girls since the early 1920s; and as a symbol of gay pride and LGBT social movements since the 1970s.

The seven colours of the rainbow

A rainbow spans a continuous spectrum of colours—there are no "bands." The apparent discreteness is an artefact of the photopigments in the human eye and of the neural processing of our photoreceptor outputs in the brain. Because the peak response of human colour receptors varies from person to person, different individuals will see slightly different colours, and persons with colour blindness will see a smaller set of colours. However the seven colours listed below are thought to be representative of how humans everywhere, with normal colour vision, see the rainbow.

Newton originally (1672) named only five primary colours: red, yellow, green, blue and violet. Later he included orange and indigo, giving seven colours by analogy to the number of notes in a musical scale.

Red:  . Orange:  . Yellow:  . Green:  . Blue:  . Indigo:  . Violet: .

Effects to be distinguished from rainbow

• Circumhorizontal arc
• Circumzenithal arc
• Fog bow
• Glory
• Halo
• Iridescent colours in soap bubbles
• Moonbow
• Sun dog

References

• Greenler, Robert (1980). Rainbows, Halos, and Glories. Cambridge University Press. ISBN 0195218337.
• Lee, Raymond L. and Alastair B. Fraser (2001). The Rainbow Bridge: Rainbows in Art, Myth and Science. New York: Pennsylvania State University Press and SPIE Press. ISBN 0-271-01977-8.
• Lynch, David K.; Livingston, William (2001). Color and Light in Nature (2nd ed.). Cambridge University Press. ISBN 0-521-77504-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Minnaert, Marcel G.J. (1993). Light and Color in the Outdoors. Springer-Verlag. ISBN 0-387-97935-2. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Minnaert, Marcel G.J. (1973). The Nature of Light and Color in the Open Air. Dover Publications. ISBN 0-486-20196-1. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Naylor, John (2002). Out of the Blue: A 24-Hour Skywatcher's Guide. Cambridge University Press. ISBN 0-521-80925-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Boyer, Carl B. (1987). The Rainbow, From Myth to Mathematics. Princeton University Press. ISBN 0-691-08457-2.
• Graham, Lanier F. (editor) The Rainbow Book Berkeley, California: Shambhala Publications and The Fine Arts Museums of San Francisco (1976) (Large format handbook for the Summer 1976 exhibition The Rainbow Art Show which took place primarily at the De Young Museum but also at other museums. The book is divided into seven sections, each coloured a different colour of the rainbow.)
• De Rico, Ul (1978). The Rainbow Goblins. Thames & Hudson. ISBN 0-500-27759-1.

External links

• Images by Crayford Manor House Astronomical Society
• National Center for Atmospheric Research, About Rainbows
• Supernumerary and Multiple Rainbows
• About rainbows
• Incredible Rainbows Worldwide - slideshow by Life magazine
• Interactive simulation of light refraction in a drop (java applet)
• Spectacular rainbow at Elam Bend (McFall, Missouri)
• Walter Lewin's Discussion on colours and rainbow physics
• Straight Dope on double rainbows
• Rare photo of the ‘end’ of the rainbow
• Rainbow seen through infrared filter and through ultraviolet filter
• Rare double rainbow (video)
• Atmospheric Optics website by Les Cowley
  • Description of multiple types of bows, including: "bows that cross, red bows, twinned bows, coloured fringes, dark bands, spokes", etc.