# The fabulous Fibonacci flower formula

you're watching a mythology 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 found it so let me know

How I went was this at the end of this video there's quite a bit more nice match to all this that I'm going to talk about in this video in particular there's a nice connection with the golden ratio for that check out part 2

So I'll focus on flower heads like this and lets us have a close look what jumps out at you of course 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 Fibonacci numbers all right 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 are occupying different regions but they're always overlapping here was the next to 21 and 34 okay now

Plants like this grow so does the Fibonacci sequence starting with the two seeds one and one we grow them like this one plus one is two one plus two is three two plus three is five three plus five is eight and so on now the plants

That exhibit these spells 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 so the boundary that gives this really nice homogeneous flower head in this case so a little bit more detailed look so here I've got the first guy sitting just sitting there waiting for a second one the second one

Squeezes in like that and then the one has to kind of squeeze in a buffer below there's a bit of asymmetry so it goes for the top here and then well there's this gap here that's where the next one is going to squeeze and there's

A gap there but the next one's going to squeeze and 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 seat or but playing the same role inside the flower head so some consequences of all this when you have the plant growing all the parts 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 you can kind of turn around like this you all 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's kind of pushed out and you're really packing things as densely as you can now if this was absolutely

Optimal at the densest packing of circle like things like these these buds it 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 there 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 plan there it is you can pretty much take any part of the plant you'll

Be able to fit this pattern in there closer look here now there are two families there's the first one that's a family of spirals now 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 will have been packed as close as you can 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 this 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 a plant up there let's have a close look so the first two families of sparrows they make these diamond cells and the third type of sparrow they form diagonals cutting through those diamonds so I call in the diagonal

Sparrows okay and these three spirals being connected like this actually translate into the mathematical core that makes the fibonacci members 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 were equally spaced and you look at the diagonal family like this and you count a number of spirals in these families you 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 it's my own proof very proud of it so I have to do it but let's just run with this so we've got one number visible that's the greens and we'll put another number

Visible it's 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 greens pearls 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 focusing on this point magnified out over here now the spirals correspond to the shortest connections I mean the spirals are not there you're just making

Them up basically and what you do is you're looking for neighboring butts and then extrapolate these connections that you see here and the neighboring bud see are indicated by the green and the red at the moment so

What happens when now everything gets pushed out further let's 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 lengths ratios change in fact this one has now become the longest connection and the other one are shorter and when 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 we 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 four different kind of spirals here already so we kind of start with those two we know that the numbers here add up to the third number needs to 'visible this one

Becomes visible next these two numbers add up to that one here they're 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 where it doesn't quite explain yet is well why do we start with Fibonacci seeds like one and one and one and two or two and three or three and five and not some other

Numbers okay could be some other numbers that pop up here first and once they established everything else is determined by our rule well there's a bit of confession that after make and 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 if you do not see numbers here but they're also like double the

Fibonacci numbers for example and they also disguise here they're called Lukas numbers so all 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 have to do a very detailed analysis there and

Where I linked in a couple of papers in the description it's a lot more complicated 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 nothing else if you just think about it a plant really also starts from very small numbers right it starts from one one two three and so on and since this pattern that we've been talking about is

Established very quickly you'll 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 in these families of spirals right and so it's going to be out of two and three

Or three or five or six and ten four and six 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 to 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 drawing program and another one would kind of twirls the

Other way we overlap them like that and we draw in the family of diagonal spouse and now I claim 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 he's

Not proof so we circumnavigate this ring and we start at this corner and first we follow one of the red spots and we can't go any further then we switch to one of the green spots follow that one for a while

Then we switch to red one again and then to green one red one green one it doesn't really matter how you do it exactly it doesn't matter as long as you make a closed path like this yellow paths again now the points of

Intersection here on the yellow path we highlight so first make those all greens or 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 all right and now you 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 it means there's exactly as many

Red spirals as the red points and the same way there's exactly as many green points as their green spirals on the other hand it is exactly as many points as their blue spirals and that shows at a glance that

Green plus red is blue isn't that nice and that's it for this video so eventually we'll also make part 2 so watch out for this one that's going to be then highlighting the connection with the golden ratio

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