[Subtitles contributed by: Zacháry Dorris]
You're watching a Mathologer video, and that probably means you know that nature is crawling with
Fibonacci Numbers.
So they're in flower heads, in pineapples, in pine cones like that, but have you ever heard a really nice,
accessible explanation for why they're there?
Well, for the past three weeks, I've been trying to come up with an explanation like this, that really
gets to the mathematical core that makes this happen.
And I think I've found it, so let me know how I went with this at the end of this video.
There's quite a bit more nice maths to all this that I'm not going to talk about in this video; in particular,
there's a nice connection with the golden ratio (φ) - for that, check out part 2!
So I'll focus on flower heads like this, and let's just have a close look- what jumps out at you, of course, are the spirals
So there's 55 going this way, and 34 going the other way, and there's 21 if you focus in on the middle, and
even further in, there's 13. And of course all Fibonacci Numbers.
Alright, now before we move on, I just want to emphasize that these different numbers are visible in
different parts of the flower head, so the smaller, the further in. So 13 is visible here, 21; further out,
But there's always this region where they overlap, so consecutive numbers, when you see them in the plant,
occupying different regions, but they're always overlapping.
Here, with the next two, 21 and 34. Okay, now plants like this grow, so does the Fibonacci sequence. Starting
with the two seeds, 1 and 1, we grow them like this; 1+1 is 2, 1+2 is 3, 2+3 is 5, 3+5 is 8, and so on.
Now the plants that exhibit these spirals all have something in common, they all grow from a central point,
so there's more and more of these buds being pushed here in the middle, and as they appear in the middle,
they push everything out to the boundary, and that gives this really nice homogeneous... flower head, in this case.
So a bit of a more detailed look, so here we've got the first guy sitting, just sitting there waiting for the second one,
the second one squeezes in like that, and then the third one has to squeeze in above or below,
there's a bit of asymmetry, so he goes for the top here, and then, well, there's this gap here, that's where
the next one is going to squeeze in, there's a gap there, where the next one is going to squeeze in
and as you can also see, these... seeds or buds are growing as they're being pushed out, so all this together
establishes a very nice pattern, very very quickly, very robust, and what that leads to is basically
every seed, or bud, playing the same role inside the flower head. So some consequences of all this:
When you have the plant growing, all the buds are being pushed out radially, so they actually move
along pretty much straight lines. Another thing is, if you focus in on part of the flower head, and take snapshots
you basically always see the same thing; even if you kind of turn around like this, you always see the same thing.
Then, when you have a close look, again, you see that everything here is packed very very densely.
Okay? So things are being squeezed into the middle, and everything gets kind of pushed out, and you're really
packing things as densely as you can.
Now, if this was absolutely optimal, the densest packing of circle-like things like these,
these buds, would really be this pattern here, you don't quite get there but you get fairly close,
so you've got these layers here, and they're interleaving like this, and then you also get these
circles aligning in certain ways, and you've got another one going the other way.
Now let's see where this sort of packing comes up in a real plant, there it is, you can pretty much take any part of
the plant, you'll be able to fit this pattern in there.
A closer look here, now where are our two families? There's the first one, that's a family of spirals,
equally spaced, going around the center of the flower head. And there's a second one going the other way.
Comes about very naturally, just from this little stable pattern being established in the center of the flower,
and then everything being packed as closely as you can.
Um, you get these two families of spirals happening. Now if we focus in on this plant, we can actually see
another family of spirals, there it is. It doesn't jump out at you like the other two, but it's there, and actually, it does
jump out if you extend them out further, into this part of the plant, up there.
Let's have a close look, so the first two families of spirals, they make these diamond cells, and
the third type of spiral, they form diagonals cutting through those diamonds, so I call them the diagonal spirals.
Okay? And these three spirals being connected like this actually translates into the mathematical core that makes
Fibonacci numbers appear in flower heads like this. So what is it? Well, if you have
a family of spirals twirling this way, equally spaced like that, and another one twirling that way equally spaced,
and you look at the diagonal family like this, and you count the number of spirals in these families,
you'll always find that the number of green ones plus the number of red ones is exactly equal to the number of blue ones.
And you can already see the connection, right, so there's two numbers, and they're being added up to
give a third number, just like the way the Fibonacci sequence grows.
I'm actually going to prove this to you, at the end of this video, and it's my own proof, very proud of it!
[MATHOLOGER: I have a brilliant proof of this, but this part of the video is too short to contain it. Sorry :)]
So I have to do it!
So let's just run with this, so we've got one number visible, that's the greens, and we've got another
number visible, that's the reds. We don't have the blue ones yet, so how does the blue one come up as
the next number in the sequence, visually? Well, let's have a look here. I'll highlight one of the green spirals,
I highlight one of the red spirals, and I'll also draw in one of the blue ones, there.
It's not jumping out at you yet? Focus in on this point, magnify it out, over here, now
the spirals correspond to the shortest connections. Now, I mean, the spirals are not there -
you're just making them up, basically, and what you do is you're looking for neighboring buds, and then
extrapolate these connections that you see here, and the neighboring buds here are indicated by the green and the
red at the moment.
So what happens when now everything gets pushed out further, let's just go, so you can see
I mean, we've got the same arrangement all the way throughout, but everything kind of gets spread
along larger and larger rings here, and what you can also see is that
the length ratios change. In fact, this one has now become the longest connection, and the other ones
are shorter, and when I take away the highlighting here, and you close your eyes for a second, and open them
up again, you can actually no longer see the green. But what you can see now, very clearly, is the blue and the red.
So, that's how it goes. And, well, you see the next type of spiral appearing there in the middle, and it will
become dominant further out, as we push things further out. Okay, so we've had like 4 different kinds of spirals
here already, so we kind of start with those two, we know that the numbers here add up to the third number.
These two are visible - this one becomes visible next, these two numbers add up to that one here,
they are visible at the moment, that one's going to be visible next, and so on.
So starting with two seed numbers here, we get a Fibonacci like sequence happening from that point onwards.
Okay, so that's definitely part of the explanation, what it doesn't quite explain yet is, well,
why do we start with Fibonacci seeds, like 1 and 1, or 1 and 2, and 2 and 3, or 3 and 5, and not some other numbers?
[Good question]
M'kay, could be some other numbers that pop up here first, and once they're established, everything else is
determined by our rule. Well, there's a bit of a confession I have to make. I mean, it's often claimed that
the only numbers that come up in these plants are Fibonacci numbers. But that's actually not true at all.
There's actually a lot of different sequences that come up.
So there are the Fibonacci numbers, here, but there are also, like, double the Fibonacci numbers, for example
and there are also these guys here, they're called Lucas Numbers.
So all of these come up quite, quite a bit, but what they all have in common is that they follow our rule.
So, two numbers always add up to the next one in the sequence.
Okay. Well, there's still a bit of a predominance of the Fibonacci sequence, and how do you explain that? Well,
you really have to have a close look at the individual plant, and you have to do very detailed analysis there,
and, well, I link in a couple of papers in the description, it's a lot more complicated, and it goes beyond
what I'm going to explain here, I can just give you one more bit of insight into why, you know, these sorts of sequences should come up
and nothing else. If you just think about it, a plant really also starts from very small numbers, right?
It starts from 1, 1, 2, 3, and so on. And since this pattern that I've been talking about is established very quickly,
you also very quickly see, like a ring in which two of these families are apparent, it's going to be small numbers of spirals
in these families of spirals, right? And so it's going to be either 2 and 3, or 3 and 5,
or, 6 and 10, 4 and 6, one of those guys, and it's going to take off from there. So, you know, it's quite plausible.
Alright, so I'm quite happy with this explanation, so tell me whether you're happy too. Apart from that,
I still want to give you my proof,
for why green plus red is equal to blue.
So we start out with these two families of equally spaced spirals, and I've actually just made this up
in a, in a drawing program, and another one which kind of twirls the other way, we overlap them, like that,
And we draw in the family of diagonal spirals. Now I claimed that, whenever we do something like this,
doesn't matter how this comes about, we always get green plus red is equal to blue. Okay, so here's my proof.
So, we circumnavigate this ring and we start at this corner, and first we follow one of the red spirals until we can't go any further.
Then, we switch to one of the green spirals, follow that one for awhile. Then, we switch to a red one again,
and then to a green one, red one, green one, it doesn't really matter how you do it exactly, doesn't matter,
as long as you make a closed path like this yellow path, okay?
Now, the points of intersection here on the yellow path, we highlight. So first, make those all green,
so green from this corner on up to there, then, in this corner here, we put red, and we go up to here with red,
and then, again, switch to green, and then keep on going like this all the way around.
Alright, and now we can actually see, at a glance, that green plus red is equal to blue. Here we go!
Every red point is exactly one red spiral, and that means there's exactly as many red spirals as red points.
and the same way, there's exactly as many green points as the green spirals. On the other hand, there's exactly as many points
as there are blue spirals.
And that shows, *delighted giggle* at a glance, that green plus red is blue. Isn't that nice?
*Inhale*
And that's it for this video. Um, so eventually we'll also make part 2, so watch out for this one, that's
going to be, then, highlighting the connection with, um, φ.
[Subtitles contributed by: Zacháry Dorris]