caustics – Bean Thinking

Milky Way, stars, astrophotography — The Milky Way as viewed from Nebraska. Image © Howard Edin (http://www.howardedin.com)

A recent opinion piece about last week’s announcement of the detection of gravitational waves at LIGO drew my attention to a quote from Einstein:

“The most beautiful experience we can have is the mysterious. It is the fundamental emotion that stands at the cradle of true art and true science. Whoever does not know it and can no longer wonder, no longer marvel, is as good as dead, and his eyes are dimmed.“

Einstein was not the only scientist to have expressed such sentiments. Many scientists have considered a sense of wonder to be integral to their practice of science. For many this has involved gazing at the heavens on a clear night and contemplating the vastness, and the beauty, of the universe. Contemplating the twinkling stars suggests the universe outside our Solar System. Watching as the stars twinkle gives us clues as to our own planet’s atmosphere. Of course, it is not just scientists who have expressed such thoughts. Immanuel Kant wrote:

“Two things fill the mind with ever-increasing wonder and awe, the more often and the more intensely the mind of thought is drawn to them: the starry heavens above me and the moral law within me.“*

light patterns on the bottom of a tea cup — Dancing threads of light at the bottom of the tea cup.

The other evening I prepared a lovely, delicate, loose leaf jasmine tea in a teapot. I then, perhaps carelessly, perhaps fortuitously, poured the hot tea into a cold tea cup. Immediately threads of light danced across the bottom of the cup. The kitchen lights above the tea cup were refracted through hot and not-quite-so-hot regions of the tea before being reflected from the bottom of the cup. The refractive index of water changes as a function of the water’s temperature and so the light gets bent by varying amounts depending on the temperature of the tea that it travels through. Effectively the hotter and cooler regions of the tea act as a collection of many different lenses to the light travelling through the tea. These lenses produce the dancing threads of light at the bottom of the cup. The contact between the hot tea and the cold cup amplified the convection currents in the tea cup and so made these threads of light particularly visible, and particularly active, that evening. It is a very similar effect that causes the twinkling of the stars. Rather than hot tea, the light from the distant stars is refracted by the turbulent atmosphere, travelling through moving pockets of relatively warm air and relative cool air. The star light dances just a little, with the turbulence of the atmosphere, this way and that on its way to our eyes.

Marcus Aurelius wrote:

“Dwell on the beauty of life. Watch the stars, and see yourself running with them.“†

Marcus Aurelius of course didn’t have tea. Watch the dancing lights in the tea cup and see yourself sitting with it, resting a while and then watching while dwelling on the beauty in your cup.

*Immanuel Kant, Critique of Practical Reason

†Marcus Aurelius, Meditations

Caustic Coffee

A post that applies equally to tea, just swap the word “tea” for “coffee” throughout!

A cusp caustic in an empty mug of coffee — Have you seen this line?

Look deep into your coffee. Do you see the secrets of the cosmos being revealed? Well, neither do I usually but there is something in your coffee that could be said to have ‘cosmic implications’ and I’m sure it’s something that you’ve seen hundreds of times.

Now, admittedly it is easier to see this effect if you put milk in your coffee. Imagine drinking your (milky) coffee with a strong light source (the Sun, a lightbulb) behind you. You see that curved line of light that meets in a cusp near the centre of the cup? You can see various photos of it on this page. Yes, it is indeed the reflection of the light from the curved mug surface but it is far from just that. It is what prompted a professor at Duke University to say “It’s amazing how what we can see in a coffee cup extends into a mathematical theorem with effects in the cosmos.” To understand why, perhaps it is worth reflecting a bit more on our coffee.

The shape of the curve is called a ‘cusp’ and the bright edge is known as a ‘caustic’. It is fairly easy to play with the angle of the cup and the light so that you can see the first cusp curve but you can go further and create caustics that are the result of multiple reflections. Such multiple reflections can give heart shaped curves or “cardioids” so, in a certain sense adding milk to your coffee is good for (seeing) the heart.

caustic in a cup of tea or coffee — A cusp reflection is just visible in a cup of (soya) milk tea

Caustics were first investigated by Huygens and Tschirnhaus in the late 17th century. Mathematically, the cusp curve is termed an epicycloid, you can draw one by tracing the shape made by a point on the circumference of a circle rotating around a second circle, as this graphic from Wolfram mathematics demonstrates. There is a lot of maths in milky coffee. But just how is it that these curves reveal the “Cosmos in a cup of coffee“? It turns out that once you start to see caustics you start to see them everywhere. Caustics are not just going to be formed on the inside of your coffee mug, they can be formed by light waves getting bent by ripples on the surface of a stream or even by gravity, in a phenomenon known as “gravitational lensing”. Gravitational lensing is when a massive object, such as a black hole or a galaxy, bends the light travelling past it so that it acts analogously to a lens in optics (but a very big one). It is this last type of caustic that prompted the headline quoted above. In a series of papers published in the Journal of Mathematical Physics, Arlie Petters of Duke University and coworkers calculated how light from distant objects was focussed through gravitational lensing and the effects of caustics. Their predictions (and in particular any exceptions to their predictions) could lead to a new way to search for the elusive dark matter, which is thought to contribute to much of the Universe’s mass. They are now waiting for the Large Synoptic Survey Telescope (LSST) to start mapping the sky in order to test their theories.

multiple caustics from multiple LEDs — Multiple light sources are being reflected in this cup.

Before concluding this discussion of cosmic coffee, it is worth taking another look at the mathematician Tschirnhaus. As well as maths, he was known for his philosophy and his chemistry. In fact, it seems that he was responsible for the invention of European porcelain. As noted elsewhere, it has been argued that it was the ability of Europeans to start making their own porcelain that explained the rapid rise in consumption of tea and coffee during the eighteenth century in Europe. Interestingly, one of the tools that allowed Tschirnhaus to succeed in manufacturing porcelain in Dresden where others elsewhere failed was his use of “burning mirrors” to focus the heat and to achieve higher furnace temperatures than were otherwise available. He was using those caustics that he and others had so thoroughly studied mathematically in order to produce the type of cup in which we most often encounter the easiest caustics. A lovely little ‘elliptical’ story on which to end this Daily Grind.

In order to see the caustics in your coffee, it is necessary that the coffee reflects the light incident on it. Meaning, you need to add milk to your coffee. I knew there had to be a good reason to add milk to coffee at some point. Please do share your photos of caustics in your coffee either here or on Facebook or Twitter.