An audience of about 60 keen young upper school mathematicians gathered for the inaugural Maths Torch Lecture by Dr Mats Vermeeren, a Belgian post-doctorate lecturer from Loughborough University.
The story began with a shortest distance puzzle about a thirsty, injured cow that ended up in a canal!
Mats engaged with the audience from the start, asking them to interact by solving simple problems and our Oakhamians rose to the challenge. The cow puzzle is analogous to the journey of light through the dry and wet parts of the atmosphere, leading to a derivation of a formula for Snell’s Law.
Dr Mats also used a numerical optimisation on GeoGebra and videos of real life experiments to illustrate the phenomenon of refraction. Extending the ideas of reflection and refraction to three dimensions, Mats explained how different wavelengths are separated, by being reflected at different angles of roughly 42 degrees maximum.
Some maths later a ‘not so beautiful formula’ appeared…
And when the graph of the formula is plotted a maximum is seen around 42 degrees.
Dr Mats showed that a large range of input values achieve angles close to 42, leading to the higher intensity that is observed in the reflection ‘cones’. He used six sample wavelengths to illustrate this, whilst making the point that in fact rainbows are made up of infinitely many colours, being a continuous spectrum.
Dr Mats then moved on from the ‘variational problem’ of the rainbow, where minimising the time for the shortest path results in maximising the light in different parts of space, to talk about the wider, and very common, application of similar principles to other problems in the physical world.
He then took questions from the audience including the question of double rainbows which can be explained by double internal reflections, shown by the blue path, explaining the reversal of colours and lower intensity overall.
