Pythagoras – Bean Thinking

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

From Beethoven to Pythagoras via Kin Cafe, Fitzrovia

I had been waiting for an opportunity to try Kin Cafe in Fitzrovia for a while. Having followed them on Twitter, I had been tempted by the large selection of great-looking vegetarian and vegan food choices tweeted almost daily. Although I’m no longer a vegetarian, appetising meat-free meals are always appealing. So it had been on my “to try” list for a long time (preferably for lunch). However, sometimes things don’t work out quite the way you had initially hoped and so it was late afternoon by the time we ended up at Kin, sadly no lunch then. So we settled on an Americano, soya hot chocolate and a slice of Butternut and ginger cake. The coffee (from Clifton Coffee) was very fruity and full of character, highly enjoyable while sitting in the window overlooking the street outside. The cake meanwhile deserves a special mention. Not only was the cake very good, the helpful staff at Kin were very confident in their knowledge that this cake was nut-free and they also ensured that the new member of staff (being trained) used a new cake slice to serve it. Extra ‘points’ for a nut-allergy aware café and definitely a tick in the “cafes with good nut knowledge box”.

As we sat with our drinks, one of Beethoven’s quartets was playing through the loudspeakers. For me, Beethoven being played in the background is a bonus for any café but it did, perhaps, mean that I was less sociable than normal with my frequent companion in these reviews; the quartets are too absorbing. I do hope the hot chocolate made up for it.

Interior of Kin cafe — Tables are supported by struts forming triangles. But this is not the Pythagorean link.

Inside the café, tables along the wall were each stabilised by a diagonal support. A practical arrangement that had the visual effect of forming a triangle with the wall. While this did make me think about force-balancing and Pythagoras, this is not the link to Pythagoras alluded to in the title. No, instead the connection goes back to the Beethoven and the links between music and mathematics. Perhaps we no longer immediately think of music and mathematics as being particularly connected, after all one is an ‘art’ and the other a ‘science’. But music and mathematics have, traditionally, been so inextricably linked that, as Susan Wollenberg wrote in ‘Music and Mathematics’* “… it is their separation that elicits surprise”.

Some of the links between music and mathematics are explored in this TED-Ed talk about the maths to be found in Beethoven’s Moonlight Sonata. This part of the link between music and mathematics comes in the relation between what is known as consonant and dissonant notes. The first part of the Moonlight Sonata is made up of triplets of notes that sound good to our ears when they are played together. As Pythagoras is said to have discovered (see link here, opens as pdf), there is an interestingly simple relation between notes that are consonant with each other. Whether you look at the frequency of the notes or the length of a string required to play them, the ratio of two consonant notes seems to be a simple number ratio.

For example, the A of an oboe has a frequency of 440 Hz*. The A one octave higher is at 880 Hz, a factor of 2. If we took instead a series of notes of frequency f, then we could find a series of consonant notes at f:2f:3f. But now, remembering that octaves are separated by a factor of 2 and that they ‘sound good’ together, this will mean that the ratio of frequencies f:1.5f:2f will also sound good. This set of frequencies just happens to coincide with the C-G-C’ chord that forms the basis of many guitar based pieces of music. As you continue looking at these simple number ratios you can start to build a set of notes that eventually forms a scale.

Blue plaque Foley St — The artist Fuseli once lived diagonally opposite Kin Cafe. J. James notes that Fuseli was part of the artistic revolution that was paralleled by Beethoven and the Romantics in the musical sphere**.

But the links go deeper than this. In the same book “Music and Mathematics”, JV Field wrote “..in Ancient, medieval and Renaissance times, to claim that the order of the universe was ‘musical’ was to claim that it was expressible in terms of mathematics.” Indeed, Kepler looked for these musical harmonies in the maths of the planetary system. Although he found no ‘harmonies’ in the ratio of the periods of the planets then known, he did find musical scales in the ratios of the speeds of the planets (measured when they were closest to the Sun, at the perihelion, and furthest from the Sun, at the aphelion). Other simple number ratios can be found when we look to different regions of the Solar System. The periods of three of the Galilean moons of Jupiter for example have the ratio 1:2:4 (Io:Europa:Ganymede). While we would no longer describe these patterns as reflecting the harmony of the Universe (see here instead for current understanding), perhaps we ought to ponder the next sentence that Field wrote in the chapter on Musical Cosmology:

“We still believe [that the universe is expressible in terms of mathematics] now. Indeed, mathematical cosmology has proved so powerful that it is perhaps difficult to take a sufficiently cold hard look at the metaphysical basis on which it rests. On the other hand, the explicitly musical cosmologies derived more directly from the Ancient tradition seem sufficiently fantastic to invite instant questioning of their underlying metaphysics…”

One to consider next time you happen to wander into Kin Cafe, or another café playing such mathematical composers as Beethoven.

Kin Cafe can be found at 22 Foley St, W1W 6DT

*Music and Mathematics, Edited by J. Fauvel, R. Flood, R. Wilson, Oxford University Press (2003)

** The Music of the Spheres, J. James, Copernicus (Springer-Verlag), (1993)

Lastly, a video of Wilhelm Kempff playing Beethoven’s Moonlight Sonata. I would really recommend playing it twice, the first time to listen only, the second to watch while Kempff plays. His performance is fascinating.