Rainbows are one of the most beautiful things that we see in the sky during the day. They are the circular arcs formed by the Sun’s rays and water droplets in the Earth’s atmosphere. For thousands of years, mathematicians and scientists wondered about its mystery. Is there a mathematical explanation why rainbows appear as they do?

The video below shows a modern analysis of the structure and the mathematics of rainbows. What is surprising is that the modern explanation is very similar to that of Aristotle’s theory who lived about 300 BC. The mathematics involved are parallel lines, circles, and arcs.

Below are some main points shared in the video that maybe useful to you.

- Rainbows are composed of a could of rain illuminated by the Sun’s rays.
- Since the rays of the sun are parallel, they are deflected at the same angle anywhere on the cloud.
- There are parts of the cloud where the deflected rays hit the eyes of the observer. These parts of the cloud appear brighter.
- The set of points in the sky which appear bright form a circular arc — the rainbow.
- The rainbow appears to have different colors because the colors that make up the sunlight has slightly different angle of deflection.

Daniel Pearcy has a more detailed mathematical explanation about rainbows in his blog.

That’s a really cool video, Guillermo. I always understood that you could never find an end to a rainbow, but never explored the physics of that. I didn’t know that the height of the rainbow is determined by the height of the sun. Good find!

Thanks Shaun.

Best explained. Although, mathematically it is quite beyond the level here. I encountered a great essay by Dr. John A. Adam “The mathematical physics of rainbows and glories”, and will try to get a copy and enjoy it.