14
Aug

# The fabulous Fibonacci flower formula

[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]

#### 100 Comments

• Christian Gingras says:

Have a look at "spiral enya" where I present two example of animated rotating spiral. I wrote this application using LabView from National Instrument, a programming language using graphical objects.
The kind of curves demonstrated here are smoothly appearing as the spiral rotate.

• belalakaperal says:

As a German, you should be more cautious with your spirals.

• Stefan Reich says:

Dear Mathologer! No longer do machines have to stay out of mathematics. I have taught Fibonacci numbers to the machine using the English language. http://tinybrain.de/1007674 Cheers

• ProfessorBeautiful says:

Outstanding!

• Lan Mcfly says:

Amazing ! I'm an artist and i'm trying to execute a sunflower so this is very helpful. Beautiful demonstration, Thank you.

• Hans-Georg Lundahl says:

11:13 A werü naiß Pruhf in Tiet!

• Tatiana Mello says:

I really love it! Very clear and objective! Even for me, a Brazilian girl with a lot of English difficulties!

• dan alin george says:

un make a green poin in a red point

• Aiden Ocelot says:

Sunflowers are the spawn of Satan!

• Anderson JS says:

Ah but typical mathologist, you only explained how but you did not explain why it happens. It happens simply because it is the most energy efficient use of space and time vis-a-vis growth. The math is only a perfect accident extropolated by us after the fact.

• General Public says:

Only one minor problem at one point in this video: when you say we can't see the green spiral anymore, I can still see it. Other than that, this is great!

• tjzx3432 says:

Its eerie how the growth pattern of a flower matches the flux lines of a magnet.

• bob smith says:

all I know is every time I've gone outside and actually looked at a plant or a pinecone or something and tried to find the fibonacci numbers I failed miserably. I think the supposed prevalence of these numbers in nature are way overblown. It's only a small fraction of all species that are striking examples of them

• Kommandant7 says:

I don't fully understand the whole thing but the drawings are super pretty!

• 7_qz says:

110100th

• Amorphous says:

Very nice! What program is used to make those designs? Awesome presentation!

• J Burrough says:

Amazing! Thank you.

• jennifer siagian says:

wow this is awesome.. thank you (just came up again in my feed)

• Shane Osborne says:

it all starts with.
0 + 1 ….

• Rob_temsis says:

Interestingly put.

• Hong Konger says:

Oh my god best thing I've ever seen, simply amazing

• 22 says:

String Theory numbers exist above the Pascal's Triangle at https://youtu.be/ss7i1ZL5a2Y

• That Guys Bats TGB says:

So nice to see a Fibonacci video that excludes creationism.

• Crucibelle says:

I have a hard time believing order like this just comes about coincidentally or at random.

• fat loss srivatsa says:

https://youtu.be/HUm7_5A_SHI CHECK OUT MY golden ratio fower in CCAD

• i'm anilllusion says:

excellent. thanks

• Robert Theiss says:

Very nice proof. When you were talking about how all the flowers follow the same pattern (a+b=c) but may start with a different number but usually start with 1 I thought of offspring in general (usually just 1, less often more for humans).

• Ethel says:

Thank you! I AM thrilled with your visuals and explanations! I have been looking to find just these answers. I just couldn’t figure it out by myself. I will still be thinking it over a lot as I photograph nature. It will click with me.📱📷💡🤓 Thanks🌺

• Noah Schuler says:

It looks like the Galactic Senate Chamber in Star Wars

• bryangoggin says:

Very nice proof!

• Gabriel Wendel says:

Poderei ter legenda, meu inglês não é tão bom… rsrdrdrs

• Where'stheremote? says:

Your awesome! Thank you for making math fun and intriguing for some one like myself who still struggles with simple math.

• Mary Karen Solomon says:

Clear explanation and killer proof! Thank you, Mathologist!

• Friedolin 2000 says:

I understood it. An I must say its a beautiful proof. 🙂

• gogogeedus says:

humanity has always been subconsciously using nature based math because of the influence of the observation of our environment.

• thatnonchalantguy says:

x*~1.625

• Jerry Shunk says:

JEHOVAH PUT THEM THERE.

• robin17ism says:

Tool….. Lateralus

• Bentley Kennedy-Stone says:

This is an amazingly well made video, and as always you're a fantastic educator. But I don't think you've given a complete answer to why the Fibonacci sequence appears in nature. Almost every living being contains phi somewhere in its design – even DNA is laid out in waves which correspond to the golden ratio. So while this video does explain why flower buds would tend towards Fibonacci spirals, the answer of "close packing" doesn't apply to most of the cases in other living organisms where Fibonacci appears. I know this is an old video and you're not a biologist or a philosopher, but if you were, I know you'd make a damn good video on a more general explanation of phi's role in nature.

• honkyhanky says:

I am happy. This might give me the answer to something that stumped me in 1990, and I had put on the shelf. I wanted to build an interactive display/toy that required fiber optic terminals, arranged in a circular pattern on one side. I also commented tonight on your 'Heart of Math, Multiplication Tables' video, which also tied into my previous artistic work, which now enjoys a renewal of interest, thanks to that clip.

• Matthew Cross says:

FANTASTIC job Burkard! Thank you for your super video and ingenious Proof. As a token of esteem, a link to my TEDx talk on the Golden Ratio & Fibonacci Sequence: https://youtu.be/0tAZe6pP-FM

• Jose Villegas says:

Wish there was more a i love this video button. That proof is so beautiful.

• Rocio Guevara Obando says:

gracias

• Jay Blue says:

Where is part 2?

• Jay Blue says:

Very interesting exposition. How does the golden angle fit into the RGB spiral pattern?

• alan black says:

I love this kind of Math. I could watch this sort of thing for ages.

• sirnukesalot24 says:

From 6:00 to 6:54 also shows signs of nature teaching us how to do vector addition. Is that relationship dead-on for cartesian coordinates in euclidian space or am I just seeing things?

• patrick carney says:

This is informative and entertaining

• Rathin P A says:

That's an awesome proof

• Dominic Sloth says:

Please create a part 2!! :>

• Diego Imbert says:

1:41 deceiving

• rana t says:

Why isn't this taught in Government schools, oh I get it!

• T. Augustus Romer says:

That was beautiful!

• sageinit says:

The classic golden angle (which is what this video is essentially about) in the best floating point number approximations:

degrees: 137.50777
radians: 2.3999631

However, there exist two more generalized golden angles, the so called morphic golden angles:

For circles:
degrees: 76.34541
radians: 1.3324789

For parabola:
degrees: 126.869896
radians: 2.2142975

I wish Mathologer had addressed those.

Paper from which I've derived those numbers (unfortunately the approximations given by the paper don't cut it for computer graphics, so I had to calculate the optimal floating point approximations myself):
https://link.springer.com/article/10.1007/s00004-015-0285-1

• Casper YC says:

Do you still have the Mathematica file for the flower seeds animation? Thanks !

• John Adams says:

Mathologer by coincidence, I found this very related video, watch this video from the link i give you, until 10 seconds in, only the video matters, not audio: https://youtu.be/Uz7_vD91OXc?t=1m22s

• Jay Lorenzen says:

Did anyone else notice that there were different numbers of points in the spirals in the proof? Was that intentional? Part of the proof?

• George R says:

Thanks for these elegant little lectures, so confidently-delivered, Dr Polster

• polymath says:

The use of word "The" in the Thumbnail is not so mathematical. How about replacing it with "Fucking" ? 😉

• Dyslexic Artist Theory on the Physics of 'Time' says:

In this theory we have the Fibonacci Spiral in nature because of a process of spherical 4π symmetry forming and breaking. This process forms a geometrical and therefore mathematical base for Darwin’s theory of evolution. The spontaneous absorption and emission of light forms the momentum and therefore the driving force for the imperfect spiral symmetry of life! We see and feel this process of energy exchange as the passage of time. Energy ∆E slows up the rate that time ∆t flows as a universal process. With the future unfolding photon by photon relative to the energy and momentum or actions of each object and life form!

• tokotokotoko3 says:

Seems like the flower does essentially the calculation of the remainder of the fraction, as seen in the "infinite fractions" video. A single new seed comes in, and always "seeks" the biggest remaining space… fills it up with 1 and then the next seed will do the same ad infinitum.

• Persian Amazon says:

That's why Zodiac/astrology works….it's fascinating

• Mitchell Marquez says:

Couldn't you also use the Pythagorean Formula on the little diamonds to prove that red plus green equals blue or that the size increases in some other crazy predictable way or maybe the spirals are related to the Archimedian Spiral (r=theta) or something…
Oh wait, that's different. It's gotta mean something, though.

• Luigi Luigi says:

The proof is extremely awesome!

• Minnie Venkat says:

Virahanka numbers, and the hrudaya kamalam.

• Numinosan says:

That little giggle at the end! 😀

• Trying2 Understand says:

Nicely done. although one could make the argument that F(0)=0 and F(1) =1, and I wonder if doing so would not have made you proof a little easier to follow. After all when a plant starts to grow in starts with 1 + 0 cells, doesn't it?

• Meryl Slabbert says:

Such a fabulous presentation of the profound power of math, especially as a tool to understand our world!

• Karla Bassols says:

the link to the original paper is broken :'(

• Keith Sullivan says:

Something I theorized while watching: It appears there are infinitely many spirals in between the spirals (like with the blue and yellow spirals you drew), with the points making up these spirals being farther away from each other each time you look deeper between two spirals (by what ratio, I wonder, 2?).

This would be interesting to look at as an "orchard problem," and I wonder what would be discovered?

Thank you as always for the elegant, mind-expanding videos! And excellent proof! You are truly amazing! <3

• David Wilkie says:

Great graphics.
Exponential growth makes spirals of spirals.

• 박찬우 says:

KMath+KPop: Do you know 'Fibonacci-Nucha Parabola'? (※ y = xx – x – 1)
I'll show you relationship
about 'Fibonacci-Nucha Parabola' and 'Fibonacci Number' and 'Golden Ratio'
https://www.youtube.com/watch?v=53J4KVmPL3I

• Toronto Bud says:

So far, from what I can glean =

European(namely GB) maths educators = show you how to love maths.
American modern physicists = show you how to love physics and the wonky concepts.
Chemistry = everything teaches you chemistry, silly, that's all that exists!

Will be doing a paper on this and proof for my doctorate in liberal arts.

• Rajendra Misir says:

This is a clever explanation of why sunflowers, pine apples and acorns grow according to the Fibonacci sequence which is a recursive relation. The golden spiral, golden rectangle, golden ratio and nautilus shell are all related to Fibonacci sequence. I like the proof.

• Rajendra Misir says:

This is a clever explanation of why sunflowers, pine apples and acorns grow according to the Fibonacci sequence which is a recursive relation. The golden spiral, golden rectangle, golden ratio and nautilus shell are all related to Fibonacci sequence. I like the proof.

• Venkatesh babu says:

Fibonacci is an additional sequence clustering.

• Vincent 01 says:

which program did you use @Mathologer

• Evgeny Bobkin says:

I do not find the second part(

• aya sh says:

thank you for hilping

• Patryk Unlock says:

i have seen magnolia fibonacci flower then i won the treasure

• Hans Henrik Larsen says:

Is it true that in a flower head where all the buds are organized as Fibonacci numbers, no 3 buds lie on a straight line?

• Gospel of Christ says:

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• GPC™ says:

dead link: https://folk.ntnu.no/jarlet/Douady96.pdf

• Casper YC says:

The link for Mathematica Demo is broken. Could you re-upload it? Or could I get a copy of the Notebook file itself? Thanks!

• Denis Vermeulen says:

To be honest math isn't my cup of tea, but I have to admit that this video and its proof that you made all by yourself really interested me. It was very clear and helpful. Thank you and keep making such nice videos!! 😉

• tamil kings says:

This done 3000 Years
Before… In Tamil
Language…. Bro…
It all *Nature..god
Of murga. Three
Face god…
Six face called
Arumugam…
Four face God Brama…
Two face God
Shivsakthi…..
Five face god
Pancha muga Kaneshan..
Muga means face..bro
Ten face Two myth
Vishnu… God Hero
Good Will god
Opposite
Ravanaa
Ten face..

• Erika Song says:

thank you so much! God bless you 🙂

• Serhii Kyrychenko says:

https://www.youtube.com/watch?v=kkGeOWYOFoA

• manole ioan says:

I think a very interesting problem in Fibonacci flower would be how we can isolate by numbers the cardioid shape composed by some of spirals branches…

• Suzanne Barton says:

We have a designer!

• Jay Bee says:

Brilliant!! The best explanation. Watched lots of videos but only here it’s explained why plants form like this. 👍🏻👏🏻

• Luca Babnik says:

This is pretty helpful to visualize the connection between math and reality .. congratulations you are definitely math-cool !!

• Chris Bova says:

I wonder why it took so long to get this video to find its way into suggested viewing… I have figured some things out, but I have no math background. I wonder if there is anyone to talk to about this.

• Kazi Abu Rousan says:

I can't find the paper …the link in the description for the pdf is not working

• Trudy Conway says:

I've been fascinated by the subject ever since watching Jason Lisle's video (https://www.youtube.com/watch?v=NaDvPeWjBuY&t=1810s). So beautiful. Simple and complex at the same time.

• Ankit Khandelwal says:

The way these YouTubers are showing these visualizations, I am falling in love with maths again 🙂
Somewhere in middle, my textbooks made me wonder why I am not understanding things.
Now I know, thanks!

• Soma Chakraborty says:

Your proof was just wow

• ciceroramosferreira ferreira says:

Vcs poderia explicar em Português BRASIL

• Calvin Hikes says:

Makes sense. And man oh man do I love things that make sense!

• Aparajit Rituparna says:

Wow…Perfect!!!