## Fingerprints in the sky explained

### Elliptic integrals help scientists explain polarization patterns in skylight

Today, a group of physicists published the most compact and elegant explanation of one of nature's simplest phenomena: the way light behaves in the sky above us. This research appears today (Tuesday, 9th November) in the New Journal of Physics, published jointly by the Institute of Physics and Deutsche Physikalische Gesellschaft (German Physical Society).

Michael Berry and Mark Dennis from the University of Bristol, in collaboration with Raymond Lee of the US Naval Academy, have successfully predicted the patterns of polarisation of skylight, explained in broad outline by Lord Rayleigh in 1871, using elliptic integrals – a type of mathematics with deep geometrical roots, often described as "beautiful".

The blue sky seen through polaroid sunglasses gets darker and brighter as the glasses are rotated. This reveals something almost invisible to our unaided eyes: daylight is polarized light. This means that the light waves vibrate differently in different directions. The effect is strongest at right angles to the sun, and weaker elsewhere. It creates patterns in the sky that look similar to the ridges in human fingerprints and are used by many species of birds and flying insects as an aid to navigation.

A striking feature of the pattern is a pair of points near the sun where the light is not polarized at all (this point is a singularity and the pattern breaks down here). Although they have been studied for nearly two centuries, no one attempted to construct a model using the most obvious feature - the singularities - until now.

Sir Michael Berry said: "We wondered: what if you start with the singularities and write the simplest description of polarisation that puts the singularities in the right places? We found that this gives a remarkably good fit to the observational data, and predicts the pattern across the whole sky."

"This is beautiful mathematics in the sky. Using elliptic integrals, we've been able to replace pages and pages of formulae with one very simple solution that predicts the pattern extremely well"

"After almost 200 years there's now a way of understanding this natural phenomenon which is very different from previous models, but utterly natural. It's a modern theme of physics to study things by looking at their singularities – to think about them geometrically."

In order to test their theory, co-author Raymond Lee took four different polarized photographs of each of two clear-sky cases at the United States Naval Academy in Annapolis, Maryland, using a Nikon digital camera with a specially converted fisheye lens. When they compared these detailed observations to the pattern predicted by their model, they found that the fit was very good, indicating that the arrangement of the singularities could be vital in shaping the overall "fingerprint in the sky".

Many scientists and mathematicians believe that simple, concise explanations of natural phenomena are better or closer to some underlying truth than more complex ones. Professor Marcus du Sautoy, from the Mathematical Institute at the University of Oxford, said: "Having a sense of beauty and aesthetics is an important part of being a scientist. Nature seems to be a believer in Occam's Razor: given a choice between something messy or a beautiful solution, Nature invariably goes for beauty. This is why those scientists with an eye for aesthetics are often better equipped for discovering the way Nature works. We might find a complicated ugly solution but that is probably a sign that we haven't yet found the best explanation. The fact that there is so much beauty at the heart of Nature is what gives scientists a constant sense of wonder and excitement about their subject."