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?
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
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!