icon

Plus Advent Calendar Door #7: Stokes's phenomenon

Share this page

When you look closely at a rainbow, then if you are lucky you will see one or more faint arcs below the main bow that are predominantly green, pink and purple. These are called supernumerary fringes and back in 1838 they attracted the attention of the then Astronomer Royal George Biddell Airy. Little did he know that his attempt to describe those fringes mathematically would found an entire field of mathematics that is still active today.

Rainbow

A rainbow with supernumerary fringes. Photo: Johannes Bahrdt, CC BY-SA 4.0.

The problem which Airy's question eventually uncovered concerns quantities that are so small — exponentially small — that you might be tempted to ignore them. But this you do at your peril. As George Gabriel Stokes showed when he was working on Airy's question, these quantities can grow exponentially large over time and space. This doesn't just happen with supernumerary fringes of rainbows. Understanding these potentially explosive quantities is essential in all sorts of areas beyond maths: from building jet engines to theoretical physics.

This year the Isaac Newton Institute in Cambridge ran a virtual research programme on the topic, bringing together some of the best minds in the field. We were lucky to speak to one of the organisers, Chris Howls, who provided some fascinating insight into Stokes's phenomenon — it's an asymptotic adventure well worth reading about. Find out more here.

Return to the Plus advent calendar 2021.


This article is part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

INI logo

Read more about...
  • Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.

  • As COP28, the 2023 United Nations Climate Change Conference, kicks off we look at how maths can help understand the climate crisis.

  • How do you create dramatic film out of mathematics? We find out with writer and director Timothy Lanzone.

  • Mathematics plays a central role in understanding how infectious diseases spread. This collection of articles looks at some basic concepts in epidemiology to help you understand this fascinating and important field, and set you up for further study.

  • Find out why the formula we use to work out conditional probabilities is true!

  • We talk about a play that explores the fascinating mathematical collaboration between the mathematicians GH Hardy and Srinivasa Ramanujan.