So the growth of the plant is what gives rise to the Fibonacci pattern.
So how does that work?
you think of this is very, very tiny shoot just come up out of the ground
this is enormously magnified and it's growing.
blobs appear those are called primordia. Those are what are going to turn into
all the interesting bits and pieces of the plant: the petals the seeds
everything and they pop into existence one by one.
there were none of them and then you see one then another one then another one
then another one and they appear near the middle and they move outwards to get
this sort of pattern. Looks a bit like a pizza.
In fact Leonardo's nickname ought not to be Fibonacci it ought to be Leonardo of
Pizza.
Suppose we slice across this growing shoot.
pattern the same spiral pattern.
pattern work and there were two French scientists Auguste and Louis Bravais.
They were crystallographers.
You measure angles and the Bravais brothers discovered this whole pattern is
governed by one special angle and again I need a volunteer from the audience.
Okay you want to come help me measure the angles and see what's going on here.
Okay, here's your protractor. Got a protractor to measure and let's just
move it around this way.
primordia appear they start in the middle and they move somewhere and the
first one to appear ends up here. Second ones over there a long way away.
Third one's over here.
Now what I want you to do is to measure the angle between the first one
in the middle and the second one.
We'll fit that in on the second one and let's see what angle we get around here
Okay and what you make that? - 140 degrees.
Marcus. - Okay Marcus thanks now what about the angle between number two and three?
Can you just move it around and compare? Just sort of turn it that way. That's it.
And let's see how the angle between number two and three compares
with the angle that we had. Okay we want to go to the middle here and we want to put it there.
Oh look what do we notice it fits again doesn't it so see 140 degrees
again and what about the angle between three and four? You want to slide it
around and we'll check that one.
That's 140 degrees as well.
OK, now, don't go away. I need you for a lot more yet what we discovered is this special
angle of about 140 degrees. You want to hold that for a minute and perhaps come
around here. That's it.
which two primordia you take, if you take a primordium and the next one that
follows it and you look to where they go and then you measure the angle,
you always get the same number and the answer they got was 137.5 degrees
So your hundred and forty is extremely close.
What we've got to do is find out the significance of that special
137.5 degree angle.
OK.
What does that angle got to do with -
Fibonacci numbers and the sunflower spirals?
What I've got here is a couple of pies and look at this one we've cut the
circle into 8 equal pieces.
Equal slices. 8 that's a Fibonacci number.
And the previous Fibonacci number is 5 so what I've got to do is take away five of
those eight. I'll get rid of those. Now, Marcus, you're the custodian of the angle
can iu just compare it with that angle there let's have a look yeah you wanna
lean over that's super. It's pretty close and in fact if we use the breve brothers
angle which is about here it's even closer okay so three eighths of
Fibonacci fraction of a circle is very very close to one hundred and thirty
seven and a half degrees in fact that 135 degrees it's only two and a half
wrong what about this one okay now here we've got 13 slices that's the next
Fibonacci number and I'm throwing away eight so five are left 5 8 13 let's
compare that angle that's amazingly close as well look at that it's almost
exact in fact this is 138 point four degrees it's almost a degree bigger okay
so we've got an angle that's slightly too small an angle that's slightly too
big both related to Fibonacci numbers and 137 1/2 degree angle in the middle
and if you use bigger and bigger and bit bigger Fibonacci numbers and form
fractions from them you get angles which sandwich in on the angle the Bravo
brothers measured it's called the golden angle very special and it's the one
defined by this process with Fibonacci numbers now we haven't finished yet no
I'm gonna give you the wrong angle and they're going to ask you to make us a
sunflower seed head using the wrong angle and see what we get so you want to
come around here and it what I want you to do we're going to
put in the primordia and to save time instead of putting me in the middle and
moving them to the edge we'll just put them at the edge where they end up and
suppose we used an angle of 135 degrees to put them in
what pattern are we going to get okay so you want to use that to measure and
let's see where the next one goes I guess okay so over there it's going to
be there is it okay so I'll put that one there now perhaps you want to pop them
in so the next one will be there great and then the next one after that it's
going to be over there and then next one after that all that century I'll do that
that's over there and the next one after that here we go does that go it goes
there and then after that comes one over there and now one over here okay and
then where would it go after that oh I'm guessing that's it well it would go sort
of inside it there what you notice is it's always going to be on one of these
lines because what are we doing we're using three slices of the pie to measure
it so it always just moves round three yeah and if we kept on doing this it's
gonna move around like this and like that and like that and like that and I
won't bother putting any more in you're beginning to see what will happen okay
Marcus thanks a lot that's great
so what we're getting here is a pattern of eight spokes would that make a good
sunflower seed head no he wouldn't why wouldn't it because it's got big gaps in
it it will fall to bits it will break apart it wouldn't make this nice compact
circular seed head it's got a Fibonacci number of spokes but their spokes now
the same sort of thing would happen if we used this one with its 13 radial
lines if I were to use this angle to make the seed head what I would get will
be 13 straight spokes so what we're seeing here is there's an angle that's a
bit too small there's the correct angle the golden
angle and there's an angle that's a little bit too big neither of those make
good sunflower heads but they do make patterns with Fibonacci numbers in them
but instead of spirals they're straight so now what I'm going to do is show you
what happens if we start with this angle small one eight spokes and move slowly
change that angle and keep building different sunflower seed heads pass
through the golden angle in the middle and see what happens when we get there
and then move on a little bit to this angle 5 thirteenth's of a circle where
we will get 13 spokes now to do that we're going to put that on a computer
and computers over here and if you just bear with me for a moment I'll have to
adjust this so that we get just the right angle for the camera to see it
that's about it okay so what we see there is the 8 spokes
that comes from that slightly smaller angle 135 degrees I'm going to change
that angle half a degree time and I want you to watch what
happens so if we go 235 and a half what we see now is still eight spokes but
they're curving they're starting to look a bit like those spirals why they
curving because it misses by half a degree every time and so it bends and if
we go up again another half a degree they've been more at 136 degrees 136 and
a half it's still not making a good sunflower head there are gaps in it 137
still quite big gaps up comes the golden angle 137 and a half and there we see
beautiful patterns and they all fit together it looks exactly like the
sunflower spirals so that's the angle at which it all pecks together perfectly
and because these things are very close to Fibonacci fractions we see all these
Fibonacci numbers
you