Somewhere over the rainbow

Some mathematical questions you might have wondered every time you look at one of those rainbows


Every time I see a rainbow, one of those magical shows that nature has prepared for us, I can only feel amazed by such a special and colourful figure shining up in the sky. What shape is it? From how far away is it still visible? What does a rainbow look like from the side?

Firstly, rainbows appear when there is a refraction of light, caused usually by rain. Their shape is actually a circle, with its centre aligned with the light source and the observer, on the opposite side of the light source. That means that the next time that you are looking for a rainbow, look for it on the section of sky that is opposite to the sun. Those wonderful things have the shape of a full circle, but most of the time we observe rainbows during a sunny but rainy afternoon, in which case the ground covers a large part of it, so it is difficult to realise that it is in fact a circle. Here is a fantastic video of a full circle rainbow observed from a tower.

Since the centre of the circumference is on the straight line between the observer and the sun, then the rainbow actually moves if you move and it follows you, provided that you remain within the zone in which the light is refracted by the rain. This means that you cannot get closer to the rainbow, nor can you see it from the side, so sadly you will never be able to reach that pot of gold!. For one particular storm and during one particular sunset, there is only a small region on Earth in which people can observe those wonderful colours in the sky, and sometimes you are lucky enough to be standing in that region. Rainbows also don’t need to be on the horizon; it’s just the way we usually see them. You might be able to see a rainbow below you, if it just so happens that you are on a hang glider and that the atmospheric conditions are adequate, as in the next video.

Sometimes there is a chance that a second rainbow will appear, which has to be a concentric circle, meaning that it has  the same centre point as the first rainbow. Also, due to light rays following different paths when the length of the raindrops vary, concentric pieces of additional rainbows, called supernumerary arcs, might also be formed, usually appearing close to the first rainbow.

The order of the colours is always the same, with the blue stripe inside the circle and the red stripe on the  outside, with the order being reversed on the second rainbow, if it appears. In fact, due to the refraction phenomena, the inside part of the rainbow looks much more bright than the outside.


A double rainbow, with a brilliant central part and a darker outer part.

Under the right conditions, it is possible to observe a similar optical phenomena when the centre of the “rainbow” is located precisely on the light source. This effect is referred to as a halo, and it usually happens on a day with a high level of humidity, and can surround either the sun, the moon or even a street light. There is even an old saying: ring around the moon means rain soon, because of the high levels of humidity. When halos occur, the order of the colours is reversed, so that the red stripe is on the inside and the blue stripe is on the outside, and the sky is brighter outside the circle, so a halo is basically like a rainbow, but reversed, or “a rainbow observed from the other side”.

Beautiful solar halo observed in Mexico City on the 28th of May, 2015.

Beautiful solar halo observed in Mexico City on the 28th May, 2015.

There are many other phenomena related to the refraction of light, like light pillars or phantom suns, which are just as interesting and difficult to spot. They have intrigued mathematicians and physicists for centuries, from Aristotle to René Descartes, and I can only imagine how difficult it was for them to conduct some research about rainbows or halos, since they are not that frequent, they appear unexpectedly, they last, sometimes, only for a few seconds, and you cannot get close to them! No wonder a rainbow is also frequent in different religions and mythologies.

René Descartes' sketch of how rainbows are formed

René Descartes’ sketch of how a rainbow is formed

Whether you find a rainbow fascinating from an aesthetic point of view, a physical point of view, or simply because of its maths, this optical phenomena certainly has something for everyone, so don’t ever miss the opportunity to admire the extreme beauty that is sometimes hanging in the skies, waiting to be observed.

If you would like to read more about the mathematics of a rainbow, here are my recommendations for you:

Maths behind the rainbow – Plus Magazine

The Mathematics of Rainbows – AMS

And finally, this video which explains Aristotle’s Theory.

Rafael Prieto Curiel is doing a PhD in mathematics and crime. He is interested in mathematical modelling of any social issues, such as road accidents, migration, crime, fear and gossip.

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