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!

Some Thoughts on Ballet Class

So this post came about rather unexpectedly, and stemmed from a video I posted on my Tumblr site (where I post random pictures, videos, quotes or thoughts that generally aren’t important enough to warrant a full blog post). I’ll talk about the video in a moment, but it got me thinking about some of the thoughts that go through my head during a Ballet class.

Let’s take this Saturday. On Saturdays I head into New York and take classes at the Joffrey Ballet School: generally taking Basic Ballet, then Advanced Beginner Ballet and finally, after an hour break, Beginner Ballet. This triplet is affectionately known to those who take it as the “trifecta” and never fails to leave me both satisfied and absolutely shattered.

I absolutely love Ballet class, as I’m sure you know if you read this blog or my tweets. But part of the problem with Adult Ballet classes in general is that there isn’t really a standard definition of the levels. In fact, the actual difficulty of the class often depends on who is teaching or taking it: the Adv Beg and Beg Ballet classes vary week-to-week as to which stretches me more.

As such, sometimes I can feel a bit like Dawn French in this hilarious clip from The Vicar of Dibley with the Royal Ballet’s then-principal Darcey Bussell

It is easy to feel like everyone else in the class is way better than you. It may be someone cranking out a few fouettes like in the video, but it might be something a lot more basic. I’m nowhere near as flexible as other people in class, and when we’re all on the floor with legs in second stretching forward, my attempt feels completely pathetic compared to everyone else who pretty much have their chests on the floor.

But I’ve come to a realization. Everyone in an adult Ballet class is there for themselves. No-one else. And that is exactly how it should be.

With that in mind, it’s clear that it is not a competition; we are all taking class just for ourselves so it doesn’t matter what everyone else is doing. Everyone has things they struggle with in class, and everyone works at their own level. If everyone struggled at the same point then class would be pretty boring! The important thing is to try and concentrate on what you struggle with, work out what your level is, what your achievable goals are, and then try to reach them!

As for my stretching, I went up to my teacher after class and asked if there is anything I can do to help. It turns out I don’t sit far enough forward when in second so I’m going to work on sitting forward with my legs straight in front before moving on to legs apart. Sure, I might not be as flexible as others in class, but the important thing is I’m working on it!

One of the amazing things I find with class is that, without fail, there is always at least one point in class where something goes awesomely right, so I try to concentrate on that if I’m feeling a little demoralized. In the first class on Saturday I nailed a great (for me at least) double pirouette from fourth – the best I’ve ever done! The teacher suggested anyone feeling comfortable with a single just try a double and it just happened. My leg was turned out, my head was up, my spot was quick and I finished in a nice passe before dropping my foot down. The teacher didn’t see it (the class was a good 30 or 40 students) but it didn’t matter. Because Ballet class is there for me, and no-one else. Something my teacher did see was my arabesque in the third class. I’ve been working hard with her on my arabesque because I have a tendency to tilt my shoulders and upper back and to drop my toes, but on Saturday I held a half-decent balance and was rewarded with a “well done”, which meant the world to me. Sure, my leg was nowhere near as high as others, but it was an achievement for me.

It was also great to see a guy in his first ever Ballet class on Saturday! Mark was about my age and took Basic Ballet and did ridiculously well for his first class! When we got to the center work he was having a little trouble with a tombee, pas de bouree, glissade, assemble combination, and I could remember how tricky I found this, so I stayed at the back of the class with him and took him through it slowly. When he got it, he beamed from ear to ear, and I had a smile to match! Like I said earlier, we all have our levels we work at whether it’s our 1st, 101st or 1001st class.

So at the end of the day, even if it feels like you’re way behind everyone else, we should smile and enjoy class, even when it’s tough. Perhaps especially when it’s tough.
Which brings me back to the video, because it makes me smile.
A lot.

Until next time, keep dancing!

Review – A Black & White evening with the English National Ballet

This last week has been rather busy for me. Whereas most students spend Spring Break getting drunk somewhere nice and warm, I was instead spending mine travelling around the UK visiting universities, family and friends. This was due to my impending deadline of choosing between the University of Bath or Warwick University to pursue my PhD at. Incidentally, I have chosen to go to Bath, so if anyone knows of a good Adult Ballet class in Bath or Bristol let me know!

After 14 different train journeys in 6 days I ended up in London on Thursday 17th March with a plan to just head to my hotel and sleep before my 5am pre-flight wake-up. Well, that was until I saw that English National Ballet were performing their Black & White mixed bill at the London Colleseum that night!

After buying a ticket in the Upper Circle I met some friends that afternoon before returning for the 7:30 performance, which incidentally was being filmed – I’m guessing for Agony & Ecstasy, the documentary following ENB recently shown on the BBC. I haven’t been able to watch Agony & Ecstasy while in the States, but I’m hoping it’ll either be repeated when I’m back in the UK, or released on DVD later in the year.

Now it sounds a trivial point, but I wanted to mention how impressed I was with the programme for the evening. Sure, it wasn’t free like Playbills are, but I didn’t mind paying for such a high-quality photos of the pieces and intriguing interviews with the choreographers. I also found it amusing to compare the ranks to those at NYCB or ABT. At the American companies there are three ‘ranks’: Corps, Soloists and Principals. At ENB there are quite a few more: Artists, First Artists, Junior Soloists, Soloists, First Soloists, Principals and Senior Principals!

The bill opened with Eagling’s Resolution, accompanied by a collection of poems by Friedrich Ruckert with music by Gustav Mahler and all sung expertly by Elizabeth Sikora. I enjoyed this piece but feel it almost overshadowed itself by the stunning final poem – a trio which was so striking that it dominates my memory of the piece.

Resolution by Wayne Eagling (Photo credit: ENB)

I was glad to have read the programme entry for this piece prior to the performance as Eagling talked of creating it for a Muscular Dystrophy charity event. This specific form of Muscular Dystrophy particularly affects young men, and Eagling discussed how inspired he had been by the bravery and courage of the young men fighting this dreadful illness. The final dance in this piece was for three men and showed one’s fight against the burden of the others and seemed to express his fear and fight, his helplessness and hope. There was a certain rawness to the dance but it was also danced with much emotion and was one of my favourite segments of the evening.

Next up was the Black Swan Pas de Deux – a reasonably last minute addition to the bill and not perhaps a necessary one. Although danced expertly, I couldn’t help but wonder if this had been added solely due to the Black Swan movie – although if it brings audiences in, I won’t complain! On that note though, MAC makeup had “created a stunning look” for this piece which, from the upper circle, looked almost identical to the movie posters of Natalie Portman, and was a bit ‘too much’ in my eyes (or rather, Odile’s eyes).

Takahasi and Gruzdyev were very accomplished as Odile and Siegfried. Initially I felt Takahashi’s movements were a little too precise and deliberate in the opening section, but she seemed to loosen up a little in the Odile variation, which was fantastic. Siegfried’s variation was great, although just like at NYCB we had the issue of black tights on a black background making those lovely arcing jetés virtually invisible. The coda finished with some great fouettes and a la seconde turns (everyone seems to forget Siegfried keeps dancing after Odile’s 32 fouettes, so here’s a shout-out for all the Siegfried’s out there!). As much as I enjoyed this piece, I think I would have preferred to see it in the ENB’s full-length Swan Lake (being performed next week) as I think part of the thrill of this Pas is the change we see in Odette/Odile.

 

 

 

 

 

Men Y Men by Wayne Eagling (Photo credit: ENB)

 

 

 

 

Men Y Men was a piece Eagling choreographed during the 2009 season as an ‘antidote’ to Giselle. Performing Giselle meant a demanding season for the females in the company, but left the men without all that much to do. Eagling therefore took some Rachmaninov piano preludes (which were later orchestrated) and created a showcase for the men of the company. His aim was to show off the hallmarks of a danseur: turns, jumps and lifts; all while creating an interesting piece.

Well it was fantastic! Like Patrick Corbin’s piece with American Repertory Ballet, this managed to be artistic whilst proving just how athletic and awesome these dancers are. The main section involved a ‘never-ending’ cascade of men walking across the stage, two or three at a time (there were only 10 dancers in total, meaning lots of running backstage I guess!) while various guys would ‘break the norm’ through a pique arabesque leading into a solo.

This was following by a three by three square of the men whence they did a sequence of rapid chaînes – the front and back rows doing two right then two left and so on, whilst the middle row went in opposite directions. It was great that the piece featured a full range of dancers from Principal down to Artist and yet they were all very tight and precise.

Eagling has said that in the future he hopes to choreograph a companion piece for the women in the company, and then a piece for the whole company to combine the three into a mixed bill – if they are anything like Men Y Men then I can’t wait to see them!

Vue de l’Autre was a new work (that had only been premiered the night before) by Van le Ngoc, a principal in the company. Choreographed to music by Ludovico Einaudi I was intrigued to see some fresh, young choreography. The last new work I had seen was Plainspoken by Millepied at the New York City Ballet fall gala, which although impressive had not really been to my taste (I found it a little too abstract).

Opening with the dancers in running poses backed by nothing but silence I thought we were going to experience something not unlike Wayne McGregor’s Entity – sharp and abstract. Instead, a romantic work followed highlighting relationships and love.

The first duet was set among the stars and rather pretty. At one point the man held the girl up in the air as if trying to give her the night sky, or putting her up there as his guiding light. A trio followed involving the giving and receiving of a single red rose. I’m not entirely sure that the rose was appropriate here: although there was some inventive choreography going on, I felt the rose verged on being a gimmick. Indeed my only real issue with the piece was at times it verged on cheesily romantic. Also I wasn’t too sure about the costumes – in light blue unitards with metallic details the men looked more suited for figure skating than Ballet!

To contrast the rose section, the next duet involved the ballerina draped in a long flowing chiffon scarf and created some stunning “freeze frame” moments – I hope the Company’s photographer got some good shots! Next followed a trio reversing the earlier trio – rather than two women and a man, we now had two men and a sole woman.

Following that was a segment where all the ballerinas wore the same long scarves as we had seen earlier and included a rather telling moment with the girls throwing their scarves off for the men to run and pick up for them. A metaphor for many relationships methinks? Finishing we had a duet by the two men from the earlier trio, which was particularly touching, before a final couple rounded off the piece.

Clement Crisp described this piece as being “too fragile” for the public and I can see what he is alluding too. However, I thoroughly enjoyed it and am excited to see Ngoc’s work as he grows and matures in the coming years. Having previously choreographed to Yann Tiersen’s soundtrack from Ameliè, I personally would like to see Ngoc choreograph a full length ballet in the future – a modern love story with similar music to Ameliè or this piece. I think this could be a way to interest younger audiences into Ballet, especially if paired with the distribution of Sky Arts HD (ENB’s season sponsor).

Suite en Blanc by Serge Lifer (Photo credit: ENB)

To finish we were treated to the brilliant Suite en Blanc, choreographed by Serge Lifer to music by Edouard Lado which has been  deservedly getting rave reviews. Not knowing anything about the piece I was first pleased to have an overture! All too often we spend so much time concentrating on the dance that it is easy to forget about the orchestra and it was nice to appreciate them without getting distracted by beautiful dancing!

When the curtain finally lifted we were treated to a stunning scene with a plethora of white tutu-clad ballerinas and danseurs. The scenery was simple yet effective; a pair of staircases at the back of the stage leading to an elevated walkway which kept the stage interesting without detracting from the dancers.

What followed was an exemplary one-act ballet, which avoided a plot and instead simply treated us to stunning dancing. I find it interesting that Lifer was in the Ballet Russes as I found myself being reminded of Balanchine choreography at times.

One thing is sure though, Lifer sure likes his entrechats! It seemed every segment was heavy on entrechat sixes and the like, all expertly executed by the ENB dancers.

Highlights for me included the Pas de Cinq portion: I mean, with one ballerina being partnered by 4 men what is there not to like? Let’s just say that Nancy Osbaldeston certainly had control of the four men (Anton Lukovkin, Van le Ngoc, Laurent Liotardo, Yonah Acosta). The intriguingly named ‘Cigarette’ solo was expertly danced by Anaïs Chalendard but I think one of the biggest rounds of applause of the night went to Vadim Muntagirov dancing the Mazurkas solo. Muntagirov danced with such passion and vigor it was a true pleasure to watch. I also did not envy Fernanda Oliveira for her ‘Flute’ solo, which had a most peculiar and intriguing rhythm to it. Needless to say she danced it superbly.

Finally we were treated to a reprise of the sections as the dancers reappeared and finished in formation. It was truly a showstopping end to the evening and I hope the dancers felt the audiences appreciation as applause filled the theatre.

All in all this was a perfect end to my trip back home, and a most enjoyable evening (although any evening I get to watch Ballet is enjoyable!). It’s also prompted me to start saving a “David’s trips to London to watch Ballet while at Bath” fund – donations welcome! :P

Until next time, keep dancing!

Review – American Repertory Ballet, Douglas Martin’s Inaugural Season

On Saturday 5th March I attended a performance by American Repertory Ballet to celebrate Douglas Martin’s Inaugural Season – Douglas took over as ARB Company Director last year and the company has gone from strength to strength under his leadership. With a core of around 16 dancers it performs a variety of works from traditional works to fresh new choreography.

So, who is Douglas Martin? Well he trained at San Jose Ballet School before being picked to study at the American Ballet Theatre School. He then was a Principal at the Joffrey Ballet where he danced a range of roles, including Nijinksy’s La Sacre du Printemps and danced on PBS Dance in America. He then moved to American Repertory Ballet where he danced as a lead before taking the Ballet Master position and later Company Director. More importantly, he’s my Ballet teacher! He has taught me from when I first started Ballet, and in fact my other teacher at Princeton Ballet School, Edward Urwin, is a dancer with the company. I’ve tried to be impartial with my review, but I definitely had extra excitement when seeing ARB perform – I couldn’t wait to see what Douglas had done with the Company after seeing the dancers around Princeton Ballet School and at the ARB On Pointe sessions.

Our Town (Ch. Phillip Jerry, Co. Aaron Copland)

First up was Our Town, a piece based on Thornton Wilder’s play of the same name. This was originally choreographed on Douglas Martin and Mary Barton (Douglas’ wife, and ARB’s Ballet Mistress), and they placed the piece on ARB dancers earlier this season. It follows a girl, Emily and the town she lives in. Emily (Barton’s original role) falls in love with George (Martin’s original role) and they have a child together, before Emily’s untimely death. Emily then wanders back through her town, seeing how she had touched the people in her life, before accepting her fate.

Wow, was this piece emotional! You know those first ten minutes of the film “Up”? And you know how heart wrenchingly sad they were? Well this was just like that but in Ballet form. I’m willing to bet at least half the theatre was in tears my the end (I had a few tears welling by the end myself) especially when George walked off carrying his son after Emily’s death.

Brittany Fridenstine and Marc St-Pierre in Philip Jerry's Our Town. (Photo: Valerie Ford Photography)

Brittany Fridenstine danced the role of Emily and gave a glorious performance. She poured so much heart and emotion into it, you felt like you were taking the journey with her. Her partner was Marc St-Pierre dancing the role of George and they made a lovely pairing. They truly conveyed a feeling of young love and when they held hands for the first time they caught perfectly the sense of nerves and anticipation.

My favourite scene was a sequence done entirely to the sound of rain and thunder. The simplicity of this backing heightened the dance sequence although the music by Copland during the rest of the piece was great backing too. The depth of brass in the music gave a sense of suspension, and it included the magnificent Fanfare for the Common Man. All in all, a marvelous piece that I’m looking forward to seeing again in the future. And make sure you bring tissues!

Ephemeral Possessions (World Premiere, Ch. Douglas Martin, Co. Samuel Barber)

Douglas’ premiere piece was a complete triumph. Set to Barber’s Adagio For Strings, it was a piece for 5 couples, with the lead couple Michael Crawford and Michelle de Fremy. Starting as an intimate duet, the piece swelled and ebbed increasing to the full complement of ten dancers at the end of the first movement.

As the piece continued, the dancers worked both as a unit and in their individual pairings – blending seamlessly their movements. Throughout, there was a distinct feeling of grace and beauty. Douglas’ choreography seemed to truly inspire the dancers to a stunning performance, just as I am sure the dancers inspired Douglas during the choreographic process.

There was a fantastic movement between Crawford and de Fremery which took my eye: she turned in passé on pointe and as Crawford span her she pliéd and straightened her standing leg. I don’t know why, and I’m not entirely sure I’m describing the movement correctly, but it just looked awesome – especially with the pair flanked by the rest of the company.

The staging was simple – blue costumes on a blue background, and Barber’s music (the full 10-minute version) was a great accompaniment. At a recent “On Pointe” session with Martin at Princeton he spoke of how, when choreographing, he listens to the music repeatedly until it “tells him the movement”. I think he certainly captured the essence of the composition in his choreography.

I think what pleased me most about the piece though, was that it was definitely Douglas’ work. Taking class with him every week I’ve got a feel for his sense of humour, outlook on life and personality – and the entire piece spoke of this. Great work Doug!

Folia (World Premiere, Ch. Patrick Corbin, Co. Francesco Geminiani)

The third piece was another premiere – this time by Patrick Corbin (who has his own company CorbinDances). Along with Douglas’ piece, this had been the subject of an ARB ‘On Pointe’ session, which talked of the structure of the piece: one set phrase that is repeated en terre, with turns, en l’air, in retrograde and combinations of the above.

This certainly appealed to the mathematical side of my brain, but moreover I was excited to see a piece choreographed just for the men of the company – a showcase of male dancing.

It would have been very easy for Corbin to produce a very straightforward and almost stereotypical piece with lots of acrobatics between the men. What he created, however, was something that managed to convey the strength and physicality of the performances, yet at the same time retained the artfulness. The curtain lifted to the men of the company stood with their backs to the audience, muscles tense and a sense of anticipation in the air.

Starting with the main phrase, we were transported with the music (which was surprisingly passionate) as the phrase built and receded. There was a stand-out solo by Alexander Dutko, starting unaccompanied in a single spotlight, whence we glimpsed the depth of the main phrase.

This led to a solo by Edward Urwin (another of my teachers at ARB) where he performed the whole phrase then it’s retrograde at lightning speed. Marc St-Pierre then joined him on stage for a fascinating Pas de Deux between them.

Again, it would have been easy for Corbin to make a ‘standard’ Pas de Deux, but instead what followed was a different and exciting interplay as both dancers corrected and placed each other – both taking on roles as the precise controller and the malleable mannequin. One particular move I found impressive involved St-Pierre standing on Urwin’s thighs in a second plié before launching over Urwin’s head – great stuff.

As the piece continued we saw how cleverly Corbin’s phrase had been constructed – the phrase fitted so well to it’s retrograde and variations that it became a testament to Corbin’s genius. And then it finished with the single phrase again, bringing us back to our initial moment and leaving us with a sense of continuity.

Glazunov Variations (Ch. Kirk Peterson after Petipa, Co. Alexander Glazunov [Raymonda])

To finish, the company performed a collection of variations from Glazunov’s Raymonda. These took the form of collective partnering, individual variations and single duets.

To begin, our five princes led their tutu clad ballerinas out for the initial variation. What followed was a tour de force in classical technique and form. With instantaneous changes between the different sections, it was a marvel that the dancers survived this piece, never mind with such flair.

Stand out performances include Audra Johnson performing a devilishly flirtatious variation that reminded me of the Siren in The Prodigal Son. She was partnered by Marc St-Pierre (who truly seems born to partner) who performed his own impressive variation. We saw fun variations from the women of the company, and a technical variation from the five men – in sync while performing turns, tours and more.

The pace was kept up unabated to it’s thrilling finale which led to a well-deserved standing ovation for the company, including a substantial cheer for Mr Martin’s arrival on stage.

Overall, this was triumphant night for American Repertory Ballet and Douglas Martin. Personally, I felt it was especially a showcase for the men of the company, both the dancers and Douglas – is showed Douglas’ depth and creativity and the dancers’ unquestionable stamina and skill. Not to detract from the women’s performances (all of whom were excellent) but to even think of  performing in four consecutive pieces tired me, and yet the men gave both an artful and precise delivery throughout the entire evening. Bravo!

ARB will be holding their gala this coming weekend; unfortunately I will be back in England and so am unable to attend. However, they will be performing again in May (at the Mason Gross School of Arts), when I will certainly be attending. I do urge you to make it if possible, if it is anything like the inaugural performance you will be treat to a fantastic night of dance!

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: http://wn.com/Metastasis_(Xenakis)

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!