beauty in mathematics

Some perspective at Over Under, Earls Court

Over Under Coffee Earls Court

Follow the arrow! Over Under Coffee in Earls Court.

Whenever I’m heading somewhere that I haven’t been to for a while, I check the London’s Best Coffee app to see if any new cafés have popped up in the area since my last visit. So when I was in Earls Court recently, I was very happy to be alerted to a new café on the map with a review by Beanthereat.

Over Under Coffee is at 181A Earls Court Road but is tucked around the corner from the main road and so thank goodness for the helpful arrow (and the map which told me I should be on top of it). Once found, we ordered coffee and banana bread and took a table to sit down. The friendly staff behind the counter were quite confident of the ingredients in the banana bread as it is made locally by a lady in Fulham (whose name I have sadly forgotten). The coffee and banana bread formed a great combination for a mid-morning snack. Coffee is roasted by Assembly roasters over in Brixton and came with lovely interference patterns in the bubbles on the surface together with dancing white mists, which never fail to fascinate me.

On the table next to ours was a small Kilner jar for sugar and two succulent plants. The Kilner reminded me of the use of air valves in coffee packaging (which are non-recyclable plastic) and the interesting experiment by Roasting House coffee roasters to investigate whether they are actually needed for freshly roasted coffee (which you can read about here). However it was a picture above the table that prompted the thought-train for today’s Daily Grind. A charcoal sketch, the picture featured a tree in the foreground with a fence behind it. From a very early age we are taught how to represent 3D objects on a 2D sheet of paper, the idea of perspective seems ingrained on our minds. But how intuitive is it really?

perspective in coffee

A picture at Over Under. Note the smaller reflections of the (more distant) light fittings.

Although the ancient Greek artists could convey an idea of depth in their art, the development of a mathematical understanding of perspective only came with Filippo Brunelleschi (1377-1446), although a written account of the mathematics of perspective did not arrive until Leon Battista Alberti (1404-1472). Alberti’s method for drawing in perspective used not just a vanishing point, but an additional diagonal vanishing point in order to construct a sense of depth and an accurate depiction of perspective (a description of Alberti’s method is here). The development of the understanding of perspective during the Renaissance meant that for some paintings, the ‘viewing depth’ can actually be calculated, while other artworks managed to create optical illusions whereby objects would jump out at the viewer as if they are in 3D. Works such as Andrea Pozzo’s ceiling in the chiesa di Sant’Ignazio in which a flat ceiling appears magnificently domed. Or, closer to home,  Samuel van Hoogstraten’s work in the National Gallery in London in which the viewer looks through a peep hole to see the interior of a house complete with a dog that appears to be sitting up inside the painting. Such paintings required a knowledge of the mathematical rules behind the depiction of perspective. Isn’t it surprising that the understanding of these rules is so recent?

Over Under Earls Court

Coffee with bubbles showing interference patterns at Over Under Coffee

Another art work with an interesting use of perspective that will bring us, in some way, back to Over Under Coffee is Raphael’s fresco “School of Athens“. The two figures of Plato and Aristotle stand at the centre of a diverse group of philosophers including Socrates, Zoroaster, Euclid, Diogenes the Cynic and, possibly, an image of Hypatia of Alexandria. Although the use of perspective for the architecture draws your eye towards the centre of the picture, two spheres (held by Zoroaster and Ptolemy) on the right hand side of the picture are drawn as circles rather than ellipses. Spheres viewed from an angle should be represented as ellipses if drawn correctly according to the rules of perspective. Did Raphael make an error in perspective (that may work better for our eyes?) or is the degree to which these two spheres are distorted within the limits of the fresco brush and so not visible in the picture? An episode of Radio 4’s In Our Time discusses this picture at length including a deep conversation about the significance of Plato pointing upwards towards the heavens and Aristotle indicating towards the Earth. Plato’s wisdom and Aristotle’s knowledge, above and below, much like the weave logo that brings us back to Over Under Coffee.

Over Under Coffee can be found at 181A Earls Court Road, SW5 9RB.



Beautiful coffee

beauty in a coffee, coffee beauty

Interference patterns on bubbles in a coffee cup.

In the UK Science Museum’s library there is a book, written in 1910, by Jean Perrin called “Brownian Movement and Molecular reality”. To some extent, there is nothing surprising about the book. It describes a phenomenon that occurs in your coffee cup and the author’s own attempts to understand it. Nonetheless, this little book is quite remarkable. It is perhaps hard, from our perspective in 2016, to imagine that at the time of Perrin’s work, the idea of the existence of molecules in water was still controversial. It was even debated whether it was legitimate to hypothesise the existence of molecules (which were, almost by definition, un-detectable). However, none of that is really relevant to the question confronting today’s Daily Grind. Today, the question is how can this book help us to find beauty in a coffee cup?

What does a one hundred year old book have to do with finding beauty in a coffee cup? Perrin received the Nobel Prize in 1926 for his work establishing the molecular origins of Brownian motion and, associated with it, his determination of the value of Avogadro’s constant. It is perhaps why he wrote the book. (The experiment that he used to do this is described in a previous Daily Grind article that can be found here.) It is in his description though, both of the theory and the experiments involving Brownian motion that this little book is relevant for today. One word repeatedly crops up in Perrin’s description of Brownian motion. It comes up when he describes the theory. It comes up when he describes other people’s experiments. It comes up when he describes bits of the maths of the theory. The word? Beautiful*.

Michael Polanyi

Michael Polanyi,
by Elliott & Fry, vintage print, (1930s),
Thanks to National Portrait Gallery for use of this image.

Throughout history, many scientists have recognised, and worked for, the beauty that they see in the science around them. In a 2007 TED talk, Murray Gell-Mann said

“What is striking and remarkable is in fundamental physics a beautiful or elegant theory is more likely to be right than a theory that is inelegant.”

So it is interesting that, although we may agree that scientific theories can be “beautiful” or “elegant”, we do not seem to have a way of quantifying what precisely beauty is. It is similar for those things that are beautiful that we find in every day life. The beauty of a sunset, or the way the light catches the ripples on the surface of a lake, these are things that we recognise as beautiful without being able to articulate what it is about them that makes them so. Instead we recognise beauty as something that strikes us when we encounter it. Elaine Scarry has talked about this as a “de-centering” that we experience when we come across beauty. Scarry writes that, when we encounter the beautiful:

“It is not that we cease to stand at the center of the world, for we never stood there. It is that we cease to stand even at the center of our own world”.¹

It is therefore quite concerning that she goes on to suggest that conversations about beauty (of paintings, poems etc) have been banished from study in the humanities “…we speak about their beauty only in whispers.”¹ This does not seem to have happened yet in science where it is still common to hear about a beautiful equation or an elegant experiment. But is there a creeping ‘ideological utilitarianism” in the scientific community? According to Michael Polanyi ²

“Ideological utilitarianism censures Archimedes today for speaking lightly of his own practical inventions and his passion for intellectual beauty, which he expressed by desiring his grave to be marked by his most beautiful geometrical theorem, is dismissed as an aberration.”²

While we may recoil from this sentiment, what do we write (or expect to read) in grant applications, scientific papers, popular science or even scientific outreach? How often is the utility of a piece of research emphasised rather than its elegance?

Earth from space, South America, coffee

Does an appreciation of beauty help with a wider understanding of justice and environmental concerns?
The Blue Marble, Credit, NASA: Image created by Reto Stockli with the help of Alan Nelson, under the leadership of Fritz Hasler

Another interesting question to ponder is whether our ability to appreciate (and discuss) beauty has wider ramifications. As many others have argued before her, Scarry suggests that the appreciation of the beauty in the world connects with our sense of justice¹. Recently the Pope too, in his great environmental encyclical, Laudato Si’ wrote³:

“If someone has not learned to stop and admire something beautiful, we should not be surprised if he or she treats everything as an object to be used and abused without scruple.”

Could it be true that part of the motivation that we need to change our ecological habits or stimulate our search for wider social justice is enhanced by our ability to slow down and appreciate the beautiful, wherever and whenever we find it?

So to return to our coffee. Is there something, anything, about our coffee or our tea that gives us such a radical de-centering experience? Can we, like Jean Perrin, appreciate the subtle beauty of the molecular interactions in our cup? Do we appreciate the moment as we prepare our brew? Or are we ideological utilitarians, seeing in our cup just another caffeine fix?


* Technically, the book in the Science Museum Library is a translation of Perrin’s work by Frederick Soddy. It is possible that it is Soddy’s translation rather than Perrin’s work itself that uses the word ‘beautiful’ repeatedly. It would be interesting to read Perrin’s book in its original French.

I would like to take this opportunity to say thank you to the Science Museum Library for being such a valuable resource and to the staff at the library for being so helpful.


“Brownian movement and molecular reality”, Jean Perrin, translated by F. Soddy, Taylor and Francis Publishers (1910)

1 Elaine Scarry, “On Beauty and Being Just”, Duckworth Publishers, 2006

2 Michael Polanyi, “Personal Knowledge, towards a post-critical philosophy” University of Chicago Press, 1958

3 §215 Laudato Si’, Pope Francis, 2015

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