mathematics

Strumming along on a coffee

coffee at Watch House

What links a coffee to a guitar amplifier?

What links a coffee to music by the likes of Eric Clapton and Jimi Hendrix?

As we sit back and enjoy the aroma from our coffee, we may rue the fact that our precious brew is evaporating away. We know from experience that hot coffee evaporates faster than cold coffee and we may dimly remember the physics that explains why this is. But have you ever stopped to consider that it is this bit of your coffee that forms a link between your drink and those famous guitarists?

The link concerns the mechanism behind the evaporation. To evaporate out of the coffee, a water molecule needs to overcome a certain energy barrier, let’s call it W, in order to escape. Given that W is constant, the more energy a water molecule has, the greater its likelihood of escape. So we could say that the probability of a water molecule escaping the coffee goes as exp{-W/kT} which means, the higher the temperature, T, the smaller the ratio W/kT and hence the greater the probability (because the exponential is raised to a negative power and hence is a dividing factor). The k is a constant known as the Boltzmann constant.

thermometer in a nun mug

Hot coffee evaporates more. Something that Halley had noticed in his experiments at the Royal Society

Now think about how the amplifiers used by many musicians work. It seems that many guitarists favour valve amplifiers owing to the type of sound they produce. Certainly Clapton and Hendrix were well known for their use of valve amps. A valve amp works by a process of thermionic emission in which electrons are ‘evaporated’ from a hot metal wire before being accelerated to a positively charged plate. This bit is the ‘valve’. In order to escape the metal wire, the electrons have to overcome a certain energy barrier, let’s call it Ω. Just as with W and the coffee, this barrier is a property of the metal that the electron evaporates from. The more energy an electron has (the higher its temperature), the greater the likelihood of it escaping the metal filament and fulfilling its role in the valve amplifier. Hence the mathematics describing thermionic emission is the same as the mathematics describing the evaporation in your coffee cup¹ and the probability of thermionic emission goes as exp{-Ω/kT}.

Now the size of the barrier is of course different in the two cases (Ω is much larger than W) which is why you have to plug in your amplifier to the electricity supply rather than just let it sit on the table top. But this is a difference of size rather than of kind. It is another of those connections between your coffee cup and the world that can be stranger than you may at first think.

If you think of a connection between your coffee and an interesting bit of physics, why not share it in the comments section below.

¹This discussion originally appeared in (and was adapted from) the Feynmann Lectures on Physics, Vol. 1

Back of the envelope calculations with coffee

coffee at Watch House

Coffee is generally a great help for reading, but to properly see the clouds in your coffee, it may help if you prepared yourself a brew now.

To read this post it will help if you have a cup of lovely, hot, freshly prepared coffee or tea with you.

Got it? Ok, let’s begin.

A few weeks ago, there was a talk given by Prof. Paul Williams of the University of Reading about the Mathematics of turbulence and climate change. An entertaining talk about the importance of, and the effort of comprehension required to, use mathematics in order to understand climate change. There were several thought provoking comments through the talk that demanded further reflection. But one, almost throw-away comment has been bugging me since. Although I’ve forgotten the exact words, they went along the lines of

Of course mostly we think about the impact of climate change on the weather, after all, we live in the bottom few metres of the atmosphere and so that is what mostly affects us. What I would like to talk about is the effect of climate change on airplane turbulence…

The bottom few metres of the atmosphere? It’s true. The bit we’re most experienced with is just a tiny portion of it. It’s about perspective. To us, it seems the atmosphere is very big, we pump all sorts of exhaust fumes into it and they disappear. We have expressions such as “the sky is the limit” that suggests that the atmosphere is a huge volume of gas. We all know it is not really limitless, but day to day, on our human scale, it seems enormous.

Now the mathematics that Prof Williams uses to calculate the effect of changing temperature and carbon dioxide levels on the jet stream (and consequently the turbulence felt by planes) is way beyond the sort of back of the envelope calculation that we can do with a cup of tea (or coffee). Understandably, to even start to comprehend these mathematical models requires years of training in maths and physics. However, assuming that we are not ourselves atmospheric physicists, there are things that we can do to help us to see our atmosphere in a more realistic way. And this is where your coffee comes in.

Earth from space, South America, coffee

Clouds swirling above our common home. But if the atmosphere is represented by the white mists on the surface of a cup of coffee, what size coffee are we drinking?
The Blue Marble, Credit, NASA: Image created by Reto Stockli with the help of Alan Nelson, under the leadership of Fritz Hasler

Take a close look at that coffee. Assuming it is not cold brew, hopefully your coffee or tea is still fairly warm. Watch the surface of the coffee. You may start to see movement such as convection in the mug, perhaps you can see a film of oil on the surface. But do you see something else? In very hot tea or coffee, you should be able to see what appear as white mists hovering over the surface of the cup*. It is easy to miss them, but as you watch, cracks suddenly appear in the mists and then there is a re-organisation of them which allows you to start to see them dancing over the surface of your drink*.

These mists are the result of the levitation of many thousands of droplets of water just above the surface of the coffee. I have written about them elsewhere. No one knows quite how they levitate above the surface, but what is known is that they are at a distance of up to 100 μm (0.1mm) from the surface of the coffee.

Let’s construct a scale model of our coffee as the Earth and its atmosphere. These mists can then do a fairly good job of representing the atmosphere with its drifting clouds. So, assuming that the mists are the atmosphere and the coffee is the Earth (on the same scale), what size of coffee would you have to have? Would you be drinking:

a) an espresso

b) a long black

c) a venti

d) a ristretto

Think you know the answer? Let’s work it out with a “back of the envelope” calculation. The easy bit is deciding the radius of the Earth, it’s just under 6400 km, our first problem comes with the estimate of the thickness of the atmosphere. There are several layers in the atmosphere. The one that we are most familiar with, the one closest to us is the troposphere. This extends for the first 16 km above the surface of the Earth (though this varies with latitude, it is only 8 km at the poles). Most of our weather happens in this region and it is also the layer of the atmosphere that planes fly in. Above the troposphere is the stratosphere which extends until about 50 km. Beyond that, things get very rarified indeed though the boundary between our atmosphere and “space” does not happen for several hundred km (indeed, the orbit of the International Space Station is in this bit of our extended atmosphere).

Coffee Corona

Look carefully around the central (reflected) white light. Can you see a rainbow like spreading of the colours? Another manifestation of the white mists on the coffee surface.

As we are mostly concerned with the weather (and airplane flight etc) though, it seems sensible to define the atmosphere height to be the top of the troposphere. After all, most of us will tend to think that the Space Station is in, well, space. This definition is further justified by the fact that about 75% of the mass of the atmosphere is found within this region (the atmosphere gets thinner as you go higher).

What size coffee would we be drinking if the white mists (0.1 mm above the coffee surface) represent the 16 km of the Earth’s atmosphere? We’ll call the coffee height, hc. Our first step is quite easy, we can just use the ratios of the heights to calculate the coffee size:

(height of troposphere)/(radius of Earth) = (white mist height)/(height of coffee)

A bit of rearrangement:

height of coffee = (white mist height)*(radius of Earth)/(height of troposphere)

hc = (0.1) * (6400)/16

hc = 40 mm (4cm)

So for the mists to represent the atmosphere in your coffee, you would need to be drinking a 4cm tall coffee which is probably a smallish long black. I would leave it to you to calculate the coffee size for the atmosphere defined as outer space (beyond the orbit of the International Space Station). But perhaps this perspective gives us another way of looking at our atmosphere. Vast indeed, but fragile too.

*As I was writing this, I had a warm, very drinkable, cup of coffee but it wasn’t steaming and so showed no white mists over the surface. The mists are best seen in freshly made, very hot drinks.

An odd one out at Shot Espresso, Parsons Green

tubes playing with perspective Shot Espresso Parsons Green

The view from the ‘conservatory’ in Shot Espresso, Parsons Green

A couple of weeks ago we were wandering around the Parsons Green area in search of a coffee. Near the station, was a small shop front with a familiar name. Not quite a chain, but the logo of Shot Espresso is well known to me from its relatively new outlet in Victoria. It turns out that the Parsons Green branch is one of four outlets for Shot Espresso which started just around the corner in Fulham.

The staff were very friendly and took our order before we found seats at the back of the café. Although there was plenty of seating near the counter it was all taken, clearly this is a popular haunt on a Saturday afternoon. This did mean however that we found a cosy table in a small but very bright area, almost like a mini-conservatory. It seems we often have a long black and a soy hot chocolate and today was no exception. The hot chocolate was apparently perfectly well done my long black was fruity and drinkable, offering a perfect flavour backdrop against which to appreciate the area of the café.

Then a tricky decision. Ordinarily, I am not a fan of reviewing chains (though there is a question, does four branches equal a chain or not?). I’m a great fan of what an independent coffee shop can bring to an area, a place where the owners can be found behind the counter and you can really get to know a friendly space. However Shot Espresso is not that large a chain and the branch at Parsons Green had the feel of a local. The staff when we were there certainly took an interest in the running of the shop and, another factor in my decision to review, there were so many things to notice here.

infinity, shot espresso

Infinite tables? The logo on the table next to us at Shot Espresso Parsons Green.

I’ve already mentioned the light in the conservatory, there were also the light fittings in the main part of the bar. Wooden outlines of cubes around a light bulb that played with your image of perspective. On the tables next to us, the symbol of the manufacturers was similar to the symbol of infinity, why? But then, an oddity that prompted a mathematical curiosity. On each table was a miniature watering can holding sugar. It’s almost a game:

You have a mug of coffee, a cup of hot chocolate, a doughnut and this watering can on your table. Which is the odd one out and why?

If you answered the watering can, you would have been correct. Topologically the mug of coffee, cup of hot chocolate and the doughnut are the same whereas the watering can is quite different. What does that even mean? It means that in terms of shape, a doughnut can be morphed into a coffee mug which can clearly be morphed into a tea cup as they each have one hole through them. The watering can however has multiple holes, not just to hold it and to let the water out but also, in this ornament design, at the join of the body to the spout (look carefully). This means that there is no way that you can transform a watering can into a doughnut, they are different categories of shape.

table at Parsons Green Shot Espresso

A coffee cup and a miniature watering can. But which has more in common with a doughnut?

This field of mathematical study (which is known as topology) has, in recent years, taken on enormous significance to physics in terms of understanding some odd effects including the way that some materials conduct electricity (or not). Indeed, it has become so important that it was the subject of the 2016 Nobel Prize (you can read the citation here). And yet, even for someone who works in solid state physics and should have a mathematical background, trying to get my head around this subject is extremely difficult.

Which got me thinking about something similar. When teaching, it is sometimes apparent how much mathematics appears as if it is another language. And in parallel with language, it requires a fluency to appreciate its beauty. And further, even with a fluency, to appreciate some use of the language requires more than just fluency but immersion, a concentration, an attention to the words. Perhaps an analogy is needed. Although fluent in English, I do not usually immerse myself in reading it. Consequently, I find the poetry of John Betjeman amusing and ‘readable’, but the poetry of Gerard Manley Hopkins very difficult. With patience, and advice from others, occasionally I can gain a flash of insight into a poem of Hopkins and realise the brilliance of the language but more often I struggle. What I would never imagine doing is saying “I can’t read English, I was never good at it in school”.

And it is here that it seems to me the parallel with mathematics ends. For while we can have fun with algebra, understand some of the beauty in calculus and perhaps struggle with topology, we nonetheless seem happy in our society to say “I can’t do maths, I was useless at it in school”. We accept boasts about mathematical illiteracy when we would blush to say similar things about our native language (whether it is English or another language).

Why is that?

watering can, Shot Espresso, Parsons Green

A closer look at the watering can. The number of holes in the join to the spout would make this useless as a plant watering device.

Surely there are few who are genuinely mathematically illiterate, at least, not to the extent that it is ‘boasted’ about within society. Indeed, you find many who are happy to admit that they don’t “do” maths, actually just mean that they would prefer to use their phone to calculate something. Just as with a spoken language, the language of mathematics requires practise. For it is practise that allows us to appreciate the fun of mathematics just as it is practise that allows us to read poetry. Why do we deny ourselves the fun of a language because it is fashionable to admit illiteracy in it?

If you would like to push yourself with some mathematical poetry, you can read about topology, coffee and doughnuts here or in more detail here and more information on the 2016 Nobel Prize can be found here. In the meantime, if you see something mathematically beautiful in a café, please do share it, either here in the comments, on twitter or on Facebook.

Enjoy your coffee, tea or doughnuts.

Shot espresso can be found at 28 Parsons Green Lane, SW6 4HS

 

 

 

 

In their Elements at Bean Reserve, Bangsar, KL

coffee in Bangsar at Bean Reserve

Bean Reserve, Bangsar, Kuala Lumpur. Note the logo on the window.

The first thing that struck me as I entered Bean Reserve in KL was the geometry. Somewhat hidden along a street behind Jalan Maarof, Bean Reserve offers a quiet space amidst the bustle of Bangsar. The 2D representation of a 3D object that is Bean Reserve’s logo is somehow mirrored in the choice of the tables and chairs that are contained in the cuboid space of this café. Triangular tables are arranged to form larger, quadrilateral tables. Circular stools nestle underneath square tables. Light streams into the café from a large window on one side of the room. The other side features a sliding door that was occasionally opened, revealing the desks of The Co, a co-working space that shares the building of Bean Reserve.

Although we only tried the drinks (an exceptionally fruity long black and a very cocoa-y iced chocolate), there looked to be an interesting selection of edibles on offer, with a bottle of chilli sauce stored behind the counter. Soy milk was available if you prefer non-dairy lattes and there were a good range of drinks on offer from nitro-cold brew to iced chocolate, just what can be needed in the heat of KL! Coffee is roasted by Bean Reserve themselves (who are both a café and a roastery), thereby providing the residents of (and visitors to) Bangsar with a seasonally varying range of great, freshly roasted coffee.

geometry at Bean Reserve

Triangular tables and circular stools.

The different geometrical features in the café immediately suggested Euclid to my thoughts. Written over 2300 years ago, Euclid’s The Elements was, for many years, the text book on geometry and mathematics. It is said that Abraham Lincoln taught himself the first 6 books of The Elements (there are 13 in total) at the age of 40 as training for his mind¹. Working from 5 postulates and a further 5 common notions, Euclid describes a series of elegant mathematical proofs, such as his proof of the Pythagoras theorem. And so, it may be appropriate that there is one more geometrical connection between the ancient Greeks and Bean Reserve: That sliding door that connects the café to the working space of The Co.

The space, occupied by The Co, behind the sliding door seems to be much larger than the café. But how much larger is it? Double the length? Double the volume? This is similar to the problem that perplexed the Delians. The idea is simple: Find the length of the side of a cube that has a volume exactly double that of a given cube. It is thought that the problem may have been formulated by the Pythagoreans, who, having succeeded in finding a method of doubling the square (see schematic), extended that idea to 3D. Could a simple geometrical method be used to double the cube? (There is of course the alternative legend about the problem having been given to the Delians by the Oracle)

A geometrical method for finding the length of a square with twice the area of a given square… now for 3D

It turns out that this is a tough problem, but one that may again have relevance for our world today. While researching this café-physics review, I came across a book by TL Heath² that had been published in 1921. In his introduction he wrote:

The work was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, ‘It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another’.

 

 

Bean Reserve can be found at 8 Lengkok Abdullah, Bangsar, 59000 Kuala Lumpur, Malaysia

¹History of Mathematics, An Introduction, 3rd Ed. DM Burton, McGraw-Hill, 1997

²A History of Greek Mathematics, Thomas Heath, Oxford at the Clarendon Press, 1921

 

Hanging out at J+A Cafe, Clerkenwell

Exterior of J and A cafe (the bar is on the other side of the passageway)

Exterior of J and A cafe (the bar is on the other side of the passageway)

Tucked down a little alley, in the back streets of Clerkenwell is the J+A Cafe. Not just a cafe, but also a bar, J+A is a satisfying place to find, particularly if you happen to find it serendipitously. As you head down the alley, the café is on your right whereas the bar opens up on your left. The café is simply furnished, with bare brick walls adorned with a few impressionist paintings. There are plenty of seats at which to enjoy good coffee and home-made cake. Their website suggests that J+A specialise in Irish baking and so we dutifully had a slice of Guinness and chocolate cake with our coffees. Importantly, the dreaded “does it contain nuts?” question was met with a knowledgable answer and without the ‘frightened bunny face’ that I often encounter when I ask this question. J+A definitely gets a tick in the ‘cafe’s with good nut knowledge’ box on my website.

Lights were suspended from the ceiling, connected by wiring that was allowed to hang down, a section of electrical wire held at both ends and freely hanging. While I’m sure that this was done for aesthetic reasons (and certainly it works on that level), such hanging wires are in fact far more than merely pleasing to the eye. Such hanging wires were a mathematical puzzle just four centuries ago. Indeed, these simple hanging wires form curves that are so important they get their own name; they are catenary curves, from catena, the Latin for chain.

lights at J and A coffee Clerkenwell

Between each lamp, the electrical cord formed a catenary curve.

Galileo had thought that a wire hanging under its own weight and suspended at its two end points formed a parabola. A fairly simple curve that is easy to describe mathematically. It was natural for Galileo to assume that these catenary curves were really parabolic. He had earlier shown that objects that fell with gravity followed parabolic paths, and after all, the hanging wires did look almost parabolic. It fell to Joachim Jungius to show that the curve was not parabolic and then to Huygens, Bernoulli and Leibniz to derive the equations determining the form of the curve. Although the differences between the parabola and the catenary curves are subtle, they have profound consequences.

When a chain, or a wire, is suspended and allowed to hang under its own weight, it forms a catenary. Flipping this around, quite literally, a catenary arch will be self-supporting. This means that a vault made of a series of catenaries or a dome that is made into the shape of a catenary will be self-supporting with no need for buttresses. This property of the catenary curve was used by Antonio Gaudi in his designs of the Casa Mila in Barcelona and also by Christopher Wren. The famous dome of St Pauls is not a catenary, but it is not one dome either. It is in fact 3 domes stacked together. The outer dome is spherical (which is weak from a structural point of view) while the inner dome is a catenary. The dome between these two was designed, using the mathematics of the day, to support the impressive outer dome (more info here and here). Wren, was not just an architect, he was also a keen mathematician, there is maths, physics and beauty throughout many architectural designs.

Mathematics in the city reflected in the lights of J+A.

J+A is at 1+4 Sutton Lane, London EC1M 5PU