Rainbow

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Double rainbow and supernumerary rainbows on the inside of the primary arc. The shadow of the photographer's head on the bottom 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. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.

In a "primary rainbow", the arc shows red on the outer part, and violet on the inner side. This rainbow is caused by light being refracted while entering a droplet of water, then reflected inside on the back of the droplet and refracted again when leaving it.

In a double rainbow, a second arc is seen outside the primary arc, and has the order of its colours reversed, red facing toward the other one, in both rainbows. This second rainbow is caused by light reflecting twice inside water droplets.

Overview

The rainbow is not located at a specific distance, but comes from any water droplets viewed from a certain angle relative to the Sun's rays. Thus, a rainbow is not a physical object, and cannot be physically approached. Indeed, it is impossible for an observer to manoeuvre to see any rainbow from water droplets at any angle other than the customary one of 42 degrees from the direction opposite the Sun. Even if an observer sees another observer who seems "under" or "at the end" of a rainbow, the second observer will see a different rainbow further off-yet, at the same angle as seen by the first observer.

A rainbow spans a continuous spectrum of colours. Any distinct bands perceived are an artefact of human colour vision, and no banding of any type is seen in a black-and-white photo of a rainbow, only a smooth gradation of intensity to a maximum, then fading towards the other side. For colours seen by a normal human eye, the most commonly cited and remembered sequence is Newton's sevenfold red, orange, yellow, green, blue, indigo and violet.

Rainbows can be caused by many forms of airborne water. These include not only rain, but also mist, spray, and airborne 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)
Rainbow after sunlight bursts through after an intense shower in Maraetai, New Zealand.

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.[1] 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.

Spectrum

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,[2] with normal colour vision, see the rainbow.

Newton originally (1672) named only five primary colours: red, yellow, green, blue and violet.

Red Yellow Green Blue Violet
                        

Later he included orange and indigo, giving seven colours by analogy to the number of notes in a musical scale.[3]

Red Orange Yellow Green Blue Indigo Violet
                                  

The colours visible in the rainbow are not pure spectral colours. There is spectral smearing due to the fact that for any particular wavelength, there is a distribution of exit angles, rather than a single unvarying angle.[4]


Explanation

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 due to dispersion, causing red light to be refracted less than blue light.

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.[5]

The amount by which light is refracted depends upon its wavelength, and hence its colour. This effect is called dispersion. 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. Due to this angle, blue is seen on the inside of the arc of the primary rainbow, and red on the outside.

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.[6]

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

Multiple rainbows

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

Although most people will not notice it because they are not actively looking for it, a dim secondary rainbow is often present 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.


Tertiary and quaternary rainbows

In addition to the primary and secondary rainbows which can be seen in a direction opposite to the sun, it is also possible (but very rare) to see two faint rainbows in the direction of the sun. These are the tertiary and quaternary rainbows, formed by light that has reflected three or four times within the rain drops, at about 40° from the sun (for tertiary rainbows) and 45° (quaternary). It is difficult to see these types of rainbows with the naked eye because of the sun's glare, but they have been photographed; definitive observations of these phenomena were not published until 2011. [7].[8] [9]

Higher-order Rainbows

Higher-order rainbows were described by Felix Billet (1808–1882) who depicted angular positions up to the 19th-order rainbow, a pattern he called a "rose of rainbows".[10][11] In the laboratory, it is possible to observe higher-order rainbows by using extremely bright and well collimated light produced by lasers. Up to the 200th-order rainbow was reported by Ng et al. in 1998 using a similar method but an argon ion laser beam.[12]

Supernumerary rainbow

A contrast-enhanced photograph of a supernumerary rainbow, with additional green and violet 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, cancelling 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.[13]

Monochrome rainbow

An unenhanced photo of a red (monochrome) rainbow.

Occasionally a shower may happen at sunrise or sunset, where the shorter wavelengths like blue and green have been scattered and essentially removed from the spectrum. Further scattering may occur due to the rain, and the result can be the rare and dramatic monochrome rainbow.

Rainbows under moonlight (Moonbows) are often perceived as white and may be thought of as monochrome. The full spectrum is present but our eyes are not normally sensitive enough to see the colours. So these are also classified (on the basis of how we see them) into seven coloured rainbow, three coloured rainbow and monochrome rainbow. Long exposure photographs will sometimes show the colour in this type of rainbow.

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.[14]

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.[15]

Scientific history

The classical Greek scholar Aristotle (384–322 BC) was first to devote serious attention to the rainbow. According to Raymond L. Lee and Alistair B. Fraser, "Despite its many flaws and its appeal to Pythagorean numerology, Aristotle's qualitative explanation showed an inventiveness and relative consistency that was unmatched for centuries. After Aristotle's death, much rainbow theory consisted of reaction to his work, although not all of this was uncritical."[16]

In the Naturales Quaestiones (ca. 65 AD), the Roman philosopher Seneca the Younger devotes a whole book to rainbows, heaping up a number of observations and hypotheses. He notices that rainbows appear always opposite to the sun, that they appear in water sprayed by a rower or even in the water spat by a launderer on dresses; he even speaks of rainbows produced by small rods (virgulae) of glass, anticipating Newton's experiences with prisms. He takes into account two theories: one, that the rainbow is produced by the sun reflecting in each water-drop, the other, that it is produced by the sun reflected in a cloud shaped like a concave mirror. He favors the latter theory. He observes other phenomena related with rainbows: the mysterious "virgae" (rods) and the parhelia.

According to Hüseyin Gazi Topdemir, the Persian 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."[17] 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.[18] This explanation was later repeated by Averroes,[17] 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).[19] 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.[20]

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."[21] This explanation, however, was also incorrect.[17] Ibn Sīnā's account accepts many of Aristotle's arguments on the rainbow.[20]

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.[22] Paul Dong writes that Shen's explanation of the rainbow as a phenomenon of atmospheric refraction "is basically in accord with modern scientific principles."[23]

According to Nader El-Bizri, 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." An experiment with a water-filled glass sphere was conducted and al-Farisi showed the additional refractions due to the glass could be ignored in his model.[18] 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).[24][25][26][27][verification needed][clarification needed]

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.[28] 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.[29] 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".[30] 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.[31][32]

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 is refracted less than blue light, which led to the first scientific explanation of the major features of the rainbow.[33] 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.[34]

Culture

Religious belief

The "end" of a rainbow. An observer located at this point would not see this rainbow.

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

In Greco-Roman 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 religion, 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 Rangdhonu 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 Armenian mythology the rainbow is a belt of Tir, a Sun god.

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."

According to Genesis, after Noah's flood God put the rainbow in the sky as the sign of His promise that He would never again destroy the earth with flood (Genesis 9:13–17):[35]

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.

The Church of Jesus Christ of Latter-Day Saints founder and prophet Joseph Smith stated that the second coming of the Christ would not occur in any year in which a rainbow is seen.[36]

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.[37][38]

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).

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 Thank Offering). 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 Thank Offering (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" 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

Films

  • In A Shine of Rainbows, the young protagonist is promised to be taken into a rainbow.
  • In Marianne, a double rainbow was filmed by chance when Sandra is introduced for the first time.
  • In Rainbow, damage to a rainbow threatens the world at large.
  • In the film The Wizard of Oz, lead character Dorothy Gale sings the song "Over the Rainbow" where she fantasises about a place over the rainbow, where the world is in peace and harmony.

Flags

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;[39] 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.[40][41]

See also

Notes

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References

External links

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  2. Berlin, B. and Kay, P. (1969). Basic Color Terms: Their Universality and Evolution. Berkeley: University of California Press. ISBN 1575861623. 
  3. "Umn.edu". Umn.edu. Retrieved 2010-10-16. 
  4. "Atmospheric Optics". 
  5. Cowley, Les. "Sea Water Rainbow". Atmospheric Optics. Retrieved 2007-06-07. 
  6. Cowley, Les. "Zero order glow". Atmospheric Optics. Retrieved 2011-08-08. 
  7. Triple Rainbows Exist, Photo Evidence Shows, Science Daily Oct. 5, 2011
  8. Theusner, Michael (01-Oct-2011). "Photographic observation of a natural fourth-order rainbow". Applied Optics. The Optical Society. 50 (28): F129–F133. Bibcode:2011ApOpt..50F.129T. ISSN 1559-128X. doi:10.1364/AO.50.00F129. Retrieved 06-Oct-2011.  Check date values in: |access-date=, |date= (help)
  9. Großmann, Michael (01-Oct-2011). "Photographic evidence for the third-order rainbow". Applied Optics. The Optical Society. 50 (28): F134–F141. Bibcode:2011ApOpt..50F.134G. ISSN 1559-128X. doi:10.1364/AO.50.00F134. Retrieved 04-Nov-2011.  Unknown parameter |coauthors= ignored (help); Check date values in: |access-date=, |date= (help)
  10. Billet, Felix (1868). "Mémoire sur les Dix-neuf premiers arcs-en-ciel de l'eau". Annales scientifiques de l'École Normale Supérieure. 1 (5): 67–109. Retrieved 2008-11-25. 
  11. Walker, Jearl (1977). "How to create and observe a dozen rainbows in a single drop of water". Scientific American. 237 (July): 138–144 + 154. Retrieved 2011-08-08. 
  12. Ng, P. H.; Tse, M. Y.; Lee, W. K. (1998). "Observation of high-order rainbows formed by a pendant drop". Journal of the Optical Society of America B. 15 (11): 2782. Bibcode:1998JOSAB..15.2782N. doi:10.1364/JOSAB.15.002782. 
  13. Terje O. Nordvik. "Six Rainbows Across Norway". APOD (Astronomy Picture of the Day). Retrieved 2007-06-07. 
  14. Cowley, Les. "Circumhorizontal arc". Atmospheric Optics. Retrieved 2007-04-22. 
  15. Science@NASA. "Rainbows on Titan". Retrieved 2008-11-25. 
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  17. 17.0 17.1 17.2 Topdemir, Hüseyin Gazi (2007). "Kamal Al-Din Al-Farisi’s Explanation of the Rainbow" (PDF). Humanity & Social Sciences Journal. 2 (1): 75–85 [77]. Retrieved 2008-09-16. 
  18. 18.0 18.1 O'Connor, J.J.; Robertson, E.F. (November 1999). "Kamal al-Din Abu'l Hasan Muhammad Al-Farisi". MacTutor History of Mathematics archive, University of St Andrews. Retrieved 2007-06-07. approximation obtained by his model was good enough to allow him to ignore the effects of the glass container 
  19. Topdemir, Hüseyin Gazi (2007). "Kamal Al-Din Al-Farisi's Explanation of the Rainbow" (PDF). Humanity & Social Sciences Journal. 2 (1): 75–85 [83]. Retrieved 2008-09-16. 
  20. 20.0 20.1 Raymond L. Lee, Alistair B. Fraser (2001). The rainbow bridge: rainbows in art, myth, and science. Penn State Press. pp. 141–144 ISBN =0271019778. ISBN 9780271019772. 
  21. Carl Benjamin Boyer (1954). "Robert Grosseteste on the Rainbow". Osiris. 11: 247–258. 
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