I was really keen to share with Numberphile viewers a discovery that I made which I was really
surprised by. I'm sure everyone out there has grown up on these numbers, the Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13- you know what more can there be to say about these numbers?
They appear all over nature, cut an apple open - a five-pointed star, Fibonacci number; number of
petals on a flower. And you know Fibonacci made them famous by writing about their
occurrence in nature. But it turns out they really shouldn't be called the Fibonacci numbers because
Fibonacci was not the first to discover these numbers. In fact they were first discovered
not by mathematicians or scientists at all but by musicians and poets in India. So how did poets and
musicians discover these numbers? A tabla player playing on a drum was quite interested to make as
many complex rhythms as possible. What are the sort of rhythms you can make out of say a long beat or
a short beat? Let's just take rhythm of length four and I could do four short beats, I could go:
Or I could do two long beats, so a long beat is twice a short beat. So it could just go:
and that fits in our kind of length four. But I can mix those up; so I could do short short long,
short long short, or long short short. Now if you explore that that's all you can do with uh kind
of a length four rhythm with long and short beats - and they turn out to be five of those
if you've counted them. So five rhythms you can make with these long and short beats. So
a tabla player would like to really mix this up and might be interested, okay, with a length
16 rhythm what- how many different rhythms can I make with long and short beats? And as they
started to count the number of different rhythms as the rhythm gets longer they discovered that
it was these Fibonacci numbers that counted them. So if I have a rhythm of length five you'll find
that there are eight different rhythms I can make starting with five short beats or or long
long short; the different combinations are eight. And actually you can really see why the Fibonacci
rule - remember what it is, add the two previous numbers to get the next number in the sequence -
why this helps you to generate the new rhythms. Because suppose I want to generate those rhythms
of length five; well I could take the rhythms of length four and add a short beat to those.
Or I could shake the rhythms of length 3 and add a long beat to those, that will be a rhythm
now of length five. The number of combinations to make rhythms of length five are the ones of
length four with the short beat on or the ones with length three with a long beat on; so that's
why adding the two previous number of rhythms together gives you the next one in the sequence.
So these were discovered by these musicians, poets also interested in uh lines of poetry with long and short beats,
and so these numbers were actually already being talked about in India several hundred
years before Fibonacci came across them. Indeed, there's an Indian poet Hemachandra who talks
about these numbers, so frankly they really should be called the Hemachandra numbers and
not the Fibonacci numbers at all. - (Brady: Did Fibonacci give a nod to the Indians or not?) - I don't think
actually Fibonacci would necessarily been aware of the fact that these numbers were in in rhythms.
I'm sure he would have talked about it had he realized that. Fibonacci actually of course
learned about a lot of the ideas of India and the Arabs because he spent time in Northern Africa;
that's why he wrote Liber Abaci, he discovered these extraordinary ideas that were bubbling
up the Arab world and he brought them to Europe with that first book. He could have been exposed
to them but I'm sure he would have talked about that. There's another interesting way actually
to think about that long and short beats which is kind of a practical challenge you can try at
home actually which is, you know, if you've got a set of stairs in yours- in your house um how many
different ways are there to climb to the top of the stairs if you take one step or two steps at a
time? So you could do each one step individually or you could do it in two steps and maybe you
have to add another one and if you've got an odd number of stairs, but it's the same challenge. The
number of different ways to get from the bottom of the stairs to the top, if you're either taking
one or two steps at a time, is exactly the same challenge of the long and short beats rhythms.
Actually the Fibonacci numbers started from poetry and music and actually in the modern
age they've actually had a kind of resurgence in poetry because there's a new poetic form which
uses the Fibonacci numbers for its structure. And it's something called a fib - so in England
a fib is uh a lie but in this case it's a sort of poetry. And this poet Gregory Pincus
came up with this idea of actually why don't we use the Fibonacci numbers to to kind of give a
structure for a poem? So his idea was each line of the poem should have the number of syllables
of that Fibonacci number, so it should start one syllable, one syllable, two syllables, three
syllables, five, eight and so on. And he felt just in the same way as Fibonacci discovered there was a
sense of growth that connects these numbers that when you read a poem with these Fibonacci rhythms
there's a kind of naturalness to it. And so here was Gregory Pincus' first fib; and you can
probably hear just kind of the nice way the Fibonacci helped build up a sort of uh a poetic structure.
One small precise poetic spiraling mixture math plus poetry tree yields the fib. Rather beautiful.
So I thought actually I would offer my Twitter followers a challenge, because I thought it's
kind of a nice small type form - Twitter, ferfect place uh to to write your own fib. And so I offered
this challenge out to my my followers and I got a wonderful load of fibs back but I've chosen my
favourite for you because I think it's it's rather beautiful. It's by- it was by @benbush:
Tweet, Tweet, Marcus, here's my fib, an unwise ad-lib, wait fib on Twitter I'm confused, how
many of my 140 have I used. It's pretty cool and you might also notice 140 so it was uh pre
uh the move to 280 which actually probably has the same number of syllables- it still will it
might still work. But I think it's rather nice because he's gone 1, 1, 2, 3, 5, 8,
he actually got to 13 in his fib which is pretty impressive. After that it starts to not sound like
poetry and sounds more like uh sort of prose so that's the way I think you've got to stop.
(All right Marcus, can I can I throw the same challenge to the YouTube commenters on this video?)
I think I'd love to see uh uh all the the fibs that are going to appear at
the bottom of this little Numberphile video. Well watched what next? Links to click? Well we
have plenty. See the video description for books by Marcus and other stuff that you should see.
Here we see sort of symmetry, playing the two together to generate music...
His artistry really is being able to write a piece of music then apply a little bit
of mathematics to it to create something that together makes something beautifully
harmonic and complex. - (Professor, not being a musician I have no clue by looking at that...)