Pythagoras and Pliés – Combinations of Combinations

I’m in class, we’ve got through our pliés (with lots of delightful cracking of joints) and we move onto our first tendu combination. As I’ve mentioned before, I’m a mathematician. So I count things. In fact, my main area of research is in combinatorics which can be broadly thought of as counting combinations of interesting objects (hence the name).

At the barre it occurred to me that most teachers use very similar tendu combinations – so it got me thinking during the Barre/Center break: how many do they have to choose from?

Normally we do tendus at the Barre... (© 2011 Oliver Endahl)

Now I’m going to make some assumptions here (mathematicians love assumptions – they make life easier!): let’s assume the teacher is going to set a nice tendu combination from first with a single tendu every beat, and the combination is going to last 32 beats. No crazy polyrhythms or inside leg work here! How many different combinations can the teacher do?

In first position, we can tendu to one of three directions: front, side or back. So at each step we have three choices. If we just do two tendus we have 3×3 choices: FF, FS, FB, SF, SS, SB, BF, BS, BB.

In fact for any number of steps, we simply have to multiply that number of threes together. So the total number of combinations is a staggering:

3^32 = 1853020188851841 combinations

That is 1 quadrillion, 853 trillion, 20 billion, 188 million, 851 thousand and 841! To realise just how big that is, this is approximately 93 times the number of red blood cells in the average human. Wow. No shortage of combinations there!

But most teachers use a symmetric combination – after the first 16 counts you repeat the combination but reversed. How does this change our count? Now the only steps that “matter” are the first 16 steps. So in fact, the number of combinations is now

3^16 = 43046721 combinations

So that’s simplified things a bit – if we could convince every person in Spain to do a tendu combination, we could cover them all.

In fact, let’s simplify things a bit further. Tendu combinations always tend to start with a tendu front – so we don’t actually have an choice about the first step, and our calculation becomes

3^15 = 14348907 combinations

£14.5 million for this?!? I can think of better ways to spend that much money...

So if you had a pound for every combination you could afford this lovely roman statue of ‘Artemis and the Stag’.”

And now let’s assume our teacher is feeling nice and gives us a nice juicy plié as the final beat to switch our legs or soutenu to switch sides. Well that’s another move gone and we end up with

3^14 = 4782969 combinations

Which at least is down to 7 figures! In fact, its the number of times Firefox 4 was downloaded in the first 24hrs – so if we could convince each downloader to do a combination we would be sorted!

If we assume the teacher is going to be seriously nice and give us a plié every fourth beat (because we all haven’t been particularly diligent in our stretching) then the combinations reduce down to depending on only 12 steps, one of which (the first) has already been decided.
So there are now only:

3^11 = 177147 combinations

Doesn’t seem like too many right? Well assuming you teach 5 classes every single day of the year, it’ll still take you around 100 years before you cover them all!

But we’ve only covered tendus from first – what about from 5th? Well this is actually a bit simpler, because whenever we are in fifth we have only two choices: with working foot front we can move front or side, and if the working foot is back we can only tendu back or side. So in fact our numbers become:
4294967296, 65536, 32768, 16384 and 2048 combinations
All more manageable, but still pretty huge numbers.

So in fact there is no shortage of combinations for the teacher to choose from. Which made me realise that there is a reason teachers tend to follow the same combination. Because in fact, the tendu combination is about the TENDU, not the combination. It’s about the quality of the movement, not the confusing pattern we trace out. Sure it can be interesting to do a crazy combination every so often, but really we should be thinking of getting the tendu perfect first.

And all of this went through my head during the break between Barre and Center. Told you mathematicians like to count…

Until next time, keep dancing!

Pythagoras and Pliés – Mathematical Beauty

For my first post in P&P, I’m going to talk about beauty in ballet. Now I know I promised that if I ever ran out of ideas I would just post pictures of gorgeous ballerinas, but I’m not talking about that kind of beauty.

Nope. Not this kind of beauty (unfortunately!)

Instead, I’m talking about mathematical beauty.

Yup. Beauty of a mathematical nature. Did you even know such a thing existed?

To digress, let me assure you that mathematics is full of beauty, it just sometimes takes some digging to find. At high-level maths, we can talk of a result’s beauty for a number of reasons: it may be a beautifully concise statement, it may be a result that has beautiful applications, it may have a beautiful proof or simply for a certain “je ne sais quoi” the result has.

For example, Fermat’s Last Theorem is considered ‘beautiful': this was a deceptively simple statement (that you can’t solve the equation x^n + y^n = z^n for n>2 unless one of x,y or z was 0) that ended up taking over 400 years before someone found a proof. That being said, it doesn’t have a ‘nice’ proof, or ‘nice’ applications.

The beauty I’m going to talk about is all to do with a certain number: ?, the golden ratio. The name alone should give you a hint that something special is going on here, you don’t call something golden without due reason!

So what is ?? Well to start, let’s have a look at the Fibonacci Numbers – forever immortalised by The Da Vinci Code.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

To start, these numbers might seem a bit of an ugly sequence, but there is a beautiful rule to constructing them. To work out a term in the sequence, all you have to do is add together the previous two numbers. Easy-peasy lemon-squeezy!

Now take any Fibonacci number and divide it by the number immediately before it in the sequence: for example 89/55. If you keep doing this you will find that the further along the sequence you go, you get closer and closer approximations to a number. That number is ?!

? is a bit like pi in that you can’t write down it as an exact fraction, the nearest we can do is state ? to a certain number of decimal places: 1.61803399… But it is also like pi in having some pretty cool properties. For example, say you wanted to know what 1/? was – all you have to do is subtract one from ? (to get 0.61803399…) and you’re done. Simple.

So why is ? considered so ‘beautiful’? Well artists and architects generally agree that having two lengths such that one is ? times the other is the most beautiful ratio you can get. This is because the ratio of the larger to the smaller, is the same as the ratio of the whole to the larger. Pretty neat, right?

The Golden Ratio property - a and b have ratio phi (thanks Wikipedia!)

But what has this got to do with Ballet, you ask? Well to start, let’s look at the human body. Take your height in inches, in my case 73. Now take the distance from the floor to your belly button, in my case 45 inches. Divide your height by your belly button height – what do you get? In my case: 1.62, pretty close to ?, right?

In fact, most of our body is roughly in proportion ? to each other: the length elbow to fingertips compared to wrist to fingertips; length of our heads compared to the width; even the dimensions of our front two teeth! In fact, in clinical studies, it has been shown that the more somebodies features ‘obey’ the golden ratio, the more beautiful they are perceived to be by society. I’d be really interested to see a study of principal dancers from different companies and see how much they obey ? – I’d be willing to bet pretty closely!

Now what about more obvious uses of the golden ratio? Well, let’s head back to the Fibonacci Numbers. These things crop up everywhere: from the number of petals on a flower and leaves on a stem (ever wonder why four-leaf clovers are so rare? Fibonacci numbers!), to the mating patterns of bees.

In fact, composers have often worked with these numbers, thinking that they were a key to great compositions. Debussy was a little obsessed with these numbers and composed pieces according to them: La Mer was composed like this, and also Cathedrale Engloutie. Here’s our first major Ballet link, Jiri Kylian choreographed one of his first pieces to Cathedrale Engloutie for Nederlands Dans Theater. Luke Jennings recalls seeing it “at Sadlers Wells in the late 70s. Rather wonderful.” It was recently performed by the Limón Dance Company in a December 2010 mixed bill entitled Masters & The Next Generation.

What about other works choreographed to the tune of the Fibonacci Numbers? Did you know that Mr B made a piece with these numbers? Metastaseis was a work he created in 1968 to a composition by Xenakis (along with a companion piece Pithoprakta). Xenakis composed Metastaseis to strict rules relating to the Fibonacci numbers and his 61-piece orchestra (unfortunately not a Fibonacci number) played 61 distinct parts. There is a fascinating video showing a graphical representation of the piece:

Although I can’t find much information about the piece itself, Arthur Mitchell said of the piece that it showed Balanchine’s emphasis on rhythm, rather than steps – which makes sense given the precise rhythm structure of the music. According to the Balanchine Trust, it was choreographed on 22 women and 6 men and “In the ballet METASTASEIS (Greek, meaning ‘[action] after stillness’) the dancers, in white, form a mass in the shape of a giant wheel that moves and changes, ending as it began.” Now if that doesn’t sound mathematical, I don’t know what does!

Oh, and how could I leave out Twyla Tharps The Golden Section? This piece, for 13 performers (no coincidence that this is a Fibonacci Number) was premiered in 1983 and most recently performed by Miami City Ballet last year. It culminates a group of pieces entitled The Catherine Wheel and, much like Balanchine’s Metastaseis, the dancers are monochromatic – this time in gold. I’ve included below a clip from Miami City Ballet’s performance (although due to copyright infringement by the company on YouTube the audio has been disabled). I’m sure you’ll agree it certainly has a ‘mathsy’ feel to it!

Finally, to modern dance. Recently I took part in the Terpsichorus online dance community (run by DanceAdvantage) where we discussed Wayne McGregor’s piece Entity. If you haven’t seen this work, then I urge you to check it out (it’s available on iTunes and Amazon through TenduTV) and join the discussion! What I found really interesting (as well as the dance) was McGregor’s use of Fibonacci imagery in the second half of the piece. At two separate points were “Golden” images projected on the floor – first a sequence of Fibonacci squares, followed later by the Golden Spiral. These images portray an immediate sense of growth, especially the spiral, and have their basis in nature (golden spirals are seen in Sunflower heads and in peoples ears to name but two instances) which I think is in keeping with McGregor’s piece which conveys, to me at least, the growth of a community, or entity.

A Fibonacci Spiral inscribed in squares with Fibonacci-lengthed sides (up to 34)

Finally, I got to thinking of Fibonacci Numbers, and was thinking of how interesting it would be to have a piece choreographed entirely on the Fibonacci Numbers. By this I mean have a set number of dancers (a Fibonacci Number of course!) and give each a Fibonacci portion of a set phrase. Then you work Fugue-like, getting the dancers to repeat their phrase with variations (en l’air, en terre, in retrograde and so forth) and see how the Fibonacci numbers interlink. In fact, we can show mathematically that every k-th term in the Fibonacci sequence (where k = 2, 3, 4, ..) is a multiple of the k-th Fibonacci Number so there would be lots of nice sections which would work in unison.

So that’s my first dose of Pythagoras and Pliés! I hope you’ve found it interesting, and learnt a little maths and Ballet. I’m going to try and keep the posts pretty regular, but as I’m currently trying to complete my mathematics Masters there may be times when my work gets on top of me! Don’t worry though, I’ll be keeping up my regular posts too – so there will be plenty to read in the coming months :)

Until next time, keep dancing!

New Feature – Pythagoras and Pliés!

Okay, so I’ve decided to add a new “feature” to the blog – and it’s going to link two very distinct, and so far very separate, sides of my personality.

In case you didn’t know, my ‘day job’ is studying mathematics – I’m just about to complete my postgraduate Masters in maths from Rutgers, and am heading back to the UK to complete a PhD in the stuff. My evening/weekend alter-ego is the beginner Ballet dancer you all know and (hopefully) love and after a recent talk with Adult Beginner, Tights and Tiaras, You Dance Funny@seenfromafar and others on Twitter, I realised just how related these two sides of my personality are.

There has always been much discussion regarding the link between maths and music; and indeed in my experience there are always many talented musicians in a university’s mathematics department. For example one of the leading researchers in Lie Groups at Rutgers also plays bassoon in the Princeton Symphonic Orchestra, and one of my college-mates at Oxford trained at the Birmingham Conservatory for piano, flute and voice. Personally, I play classical guitar and sang Bass in the Oxford University Student Chorus.

But what about the link between maths and dancing? I think if a link is going to be found anywhere, it would certainly be in Ballet rather than other ‘performance’ dance styles. There’s certainly a lot of logic to Ballet, and the fact that steps are relatively absolute definitely suggests mathematical links.

Miami City Ballet dancers in Twyla Tharp's "The Golden Section", named after a mathematical constant (Photo: Alexandre Dufaur)

So this feature is going to be all about exploring this link in cool and interesting ways. But don’t be scared! I’m going to assume that you, dear reader, know no maths whatsoever – and keep it nice and accessible for everyone. Most people seem to have a fear of mathematics, scarred by trigonometry and the quadratic equation in high school. Well there is no need to be scared here – all the maths I’ll be talking about will not only be fun, but hopefully nice and easy to understand. And if all else fails, I’ll just post a load of pretty pictures of dancers!

So what will I be talking about? Well I don’t want to spoil the surprise, but some of the topics I’ve got lined up include random (“drunken”) walks, Fibonacci numbers/the Golden Ratio, combinations (in a mathematical and dance sense), choreology… and much more!

In fact, if there is a particular ‘mathematical’ aspect of Ballet you want me to discuss then please feel free to suggest it in the comments – I’ll do my best to answer any of your questions.

Hopefully the first of these posts will be going up later this week, so keep your eyes peeled. In the meantime, I’ve got to get back to the office and prove some Theorems…

Until next time, keep dancing!

P.S. Thanks to @rlcsurf for the awesome name suggestion! Thanks to him and Ms. Adult Beginner, I’ve had my eyes opened to plenty of mathematical/Ballet puns to be had: Pascal en Pointe, Algebra in Attitude, Two plus Two = Soutenu… The list is endless!