beauty in a coffee cup – Bean Thinking

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.

Levitating water

V60 from Leyas — Time to look more closely at the surface of your black coffee.

Have you ever sat watching the steam that forms above a hot Americano? Beneath the swirling steam clouds you can occasionally see patterns of a white mist that seem to hover just above the dark brew. Bean Thinking is about taking time to notice what occurs in a coffee cup and yet I admit, I had seen these mists and thought that it was something that was just associated with the evaporation of the water and that “someone”, “somewhere” had probably explained it. So it was entirely right that I was recently taken to task (collectively with others who have observed this phenomenon and taken the same attitude) for this assumption by the authors of this paper who wrote “The phenomenon that we studied here can be observed everyday and should have been noticed by many scientists, yet very few people appear to have imagined such fascinating phenomena happening in a teacup.”

ineedcoffee.com, espresso grind — The water particles in the white mist are a similar size to the smallest particles in an espresso grind. Photo courtesy of ineedcoffee.com, (CC Attribution, No Derivs). The coin shown is a US nickel of diameter 21.21 mm

The authors of the study show that the white mists (these “fascinating phenomena”) are, in fact, layers of water drops that have a typical diameter of around 10 μm (which is roughly the size of the smallest particles in an espresso grind). Although the white mists exist above tea and even hot water as well as coffee, they are probably easiest to see against the black surface of the Americano.

More surprising than the fairly uniform distribution of water droplet size though is the fact that the authors of this study showed that the droplets were levitating above the coffee. Each water droplet was somehow literally hovering above the surface of the coffee at a height of between 10 – 100 μm (which is, coincidentally, roughly the particle size distribution in an espresso grind).

white mists, slow science — You can (just about) see the white mists over the surface of this cup of tea (which is a still from the video below)

One of the questions that the authors of the paper have not yet managed to answer is what is causing this levitation? Could it be the pressure of the hot coffee evaporating that keeps these particles held aloft? This would explain the observation that the mists form patterns similar to those caused by (heat) convection currents. Alternatively perhaps the droplets are charged and are kept away from the coffee by electrostatic repulsion, an explanation that is suggested by the behaviour of the droplets when near a statically charged object (eg. hair comb, balloon, try it). Perhaps the levitation is caused by the droplets spinning and inducing an air cushion under them? Why not design some experiments and try to find out. It would be great if we can drink hot black coffee in the name of science. Let me know the results of your observations in the comments section below. In the meanwhile, here is a video of the white mists in tea, enjoy your coffee:

You can read the study at: Umeki et al., Scientific Reports, 5, 8046, (2015)