I feel like the quality of your animations are getting even better every video. not only nicely explained, but also nice to look at. i realy love your work!
@@Mathologer your videos are quite thought 🤔provoking ..finaminal and was part of my inspiration for this theory I've been working for the past few years... and I was wondering if you could check out my video's "best veiw black hole/ partical wave duality" and "talking of [Universal Emulator]" I'm not looking for a math war.. ...I admit I'm atrocious with my notations.... but I'm sure you will under stand everything I was trying to convey and I think I may have found something extra some thing tesla once spoke of and..... you actually just recently uploaded a video about almost exactly the same thing (19th.dd/02.mm/22.yy)..I just dabble with it some times and use the best math I can to describe it.. I find it fun to work large thought equations. 🙂🙂👍
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!
I must say, I really like your way of presenting the material. Your methods of explaining is superior to almost everything else I've encountered. The calmness and naturaleness of the video really reflect unto the viewer. Keep up the good work :).
Same would work for galaxies, right? So now what can we infer about the origin of the universe? God version (which type 'God' not solved): Immaterial** A joins with immaterial B and as a result Material** C exists. Wash rinse repeat. Non-god version: Your video, with math playing the god role. Cuz clearly it's immaterial and clearly is the cause of the material, as its parental marks are everywhere. So: how does immaterial joining immaterial CREATE material? I don't know yet. By indirections, find directions out. *defined as that which is real but neither mass nor energy, so a stasis, set of rules, etc. **defined as that which is real, finite even if iterative, and is somewhere on the energy-mass spectrum.
@@brainoutyakabrainout Maybe I am just too stupid to understand what you are saying, but my takeaway from what you said is that it seems like you were presenting a bunch of gibberish or word salad or whatever you want to call it. Maybe it isn't meaningless, but I can't pull anything meaningful from it.
You brilliant Teutonic madman! Well done, sir. I have been told ever since childhood that Fibonacci sequences could be seen in flowers, but I never understood what they meant.
Very beautiful proof at the end! About the explanation: This might be because I am quite tired, but I had a hard time following some of your points, which usually isn't the case.
Very nice proof indeed! In my linear algebra class (1st year college), I usually talk about Fibonacci numbers when discussing the solution of linear difference equations using eigenvalues. This proof is certainly accessible to my engineering students.
Mathologer Exactly. I hope to prepare them for the solution of second order differential equations, which they learn a little later in the analysis class. It helps them understand that the term "characteristic polynomial" in both topics is _not_ a misnomer. It really is the same thing. So many more parallels: two basic solutions, two initial conditions determine how the have to be linearly combined.
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.
Thank you for your work. As a dad of two boys who home school, I appreciate the help of your you tube content. These sequences of "1" and "2" make "3" are in the DNA of living creatures as practical purpose, inventive uprising, and ideal envisioning.
How do you made your animations?? You are amazing!! I am a teacher of math, and I want to make some videos for my students with animations like this! Congratulations for your awesome work! Greetings!
I use a whole range of applications for making these videos: Adobe Illustrator, Photoshop, Premier and After Effect, Mathematica. Having said that a lot of the animations use Apple's Keynote presentation software and in particular the "Magic Moves" transitions. It's quite remarkable how much you can achieve in this way :)
I feel like if I had access to these types of videos when I was in school I would have done so much better. I learned in the 70's & early 80's before computers and animations. It was so hard for me to get mental pictures from my text books. This is amazing, and every time I watch a video I think, "ooooohhh, so THAT's what WHY that happens". It used to just be rote and remembering the concepts I was taught, but now I have a mental picture and the understanding has taken hold. Unfortunately I'm an old lady now, and my science days are behind me. But I'm boning up so I can teach my grandkids the amazing language and life of maths.
I saw an explanation video in the Stanford channel called fibonnaci fact vs fiction. An hour in it explains the math in flower patterns, something about interference and optimal growth rate. I'll resume viewing your own explanation now.
If you haven't addressed this question already in the previous comments, I noted (first around 10:37-38 or so) that the green nodes don't exhibit quite the same perfection of symmetry as the red: there are only 3 forming the "left side" of the right most leaf, with four nodes on the other highlighted green spiralsl. On the other side, the red spirals have two sets of 5 nodes and 4 nodes respectively. Perhaps this is a trivial point as the proof stands nevertheless, but sometimes such curiosities can lead to interesting consequence. Many thanks! Your videos are a delight to watch, and, if I have raised the question redundantly, my apologies.
The numbers of red and green spirals that you start with is not important for all this to work. In this case there are 15 green spirals and 18 red spirals overall. The numbers of that jump out on the path that I follow are really my choices. I could change direction earlier or later as long as I end up with a closed path the proof will go through as I showed. But as you say, it's important to pay attention to such details as you are often lead to interesting facts this way :)
I must say, I didn't expect much from this video as ViHart (among others) has 3 excellent videos explaining why plants grow in such a way to produce patterns relating to the Fibonacci numbers, but I have to say that that proof you gave at the end is something I've never seen before and it's just beautiful. I've never thought of it that way, so thank you!
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.
Vihart also did a really nice explanation, also delving into why plants do this, even though they can't do math. It's been a while since I watched it, but I think it had to do with the optimal amount of space per surface area, given that they grow from the center out.
How are you 100% certain that they don't do math? They are a living thing, and living things grow. For something to grow, they have to be able to reproduce and communicate. To reproduce is the same as multiplying. If it can multiply then addition is implied. If it can add, it is able to count. If it's able to count it is able to perform calculations and this is math. For something to communicate it must be able to transfer information. In order to transfer information it has to be able to perform calculations. It is my claim that at the least all living things are constantly performing mathematical computations in one way or another. Growing and expanding from DNA instructions, cell reproduction, organ growth, expanding until an age of maturity, reproducing its offspring for the next generation, cycle or iteration. Plants do count! If they couldn't they wouldn't grow! Do you think stars are non living entities or do you think they're living things as well? Within their own environment they are compressing multiple Hydrogen atoms under nuclear fusion and are producing Helium and other heavier elements. They are doing this at an exponential rate governed by at the least quantum mechanics as we currently understand it, and we still don't fully understand it. We are able to count and perform math, yet we are still feeble minded compared to that of the power of the stars. So please tell me again how and why plants can't do math? The Cosmos including our known visible Universe and all that lies beyond I is counting. How and why? Everything is in motion and if something is able to move, it's able to count. And the Cosmos, Universe is moving. It is continuously counting and thus it is a living thing. When I state living, I'm not narrowed to the view that all living things must be "carbon" based. Life here on Earth as we know it within its environment and conditions requires carbon to be considered organic life. Yet we have no understanding of all the possibilities that life can present through the stars. How do we not know that there isn't some arbitrary planet that orbits some arbitrary star has a form of life that is bizarre to us where the conditions there for life as we know it we could not survive its extremely harsh environment for our biological compositions. Take for example, a B or an O blue star that's anywhere from 2-10 times the diameter of our star with 2-100x its mass, with temperatures that are 3-10x that of our star with a higher luminosity, higher radiation, etc. Let's say that this planet is bigger than a super earth but smaller than a gas or ice giant. Let's say it's roughly 15 - 20 times more massive than our earth. It has a slightly higher density and is extremely hot compared to us. The planet in our terms would be baking, even hotter than Mercury or Venus. Let say that its outer layer does have an atmosphere that isn't conventional for us. Let's say that the first 4-5 transitional metals, Scandium, Titanium, Vanadium, Chromium, and Manganese are in a gaseous state, and the next 4-5 transitional metals, Iron, Cobalt, Nickel, Copper, and Zinc are in a liquid state. And something like Germanium which is still in Group 14 with 4 valence electrons acts similar within this star and planets environment as Carbon and Silicon does for us. Let's say it does have a rocky surface where it's lands or mountains are of the heavier rare earth metals that would normally be radioactive to us. The planet's atmosphere is 10x greater than ours, the temperature is 20x hotter than ours, the pressure is 5x greater, the gravity is 7-12x greater, the radiation is 30x greater and the brightness or luminosity due to its bright blue star is 15x brighter than here on earth. There are rivers and oceans of liquid liquid iron similar to but more fluid than our earth's lava. And how do we not know that there aren't growing living things within that environment? That environment can not support Carbon Based Life as Carbon would melt and then boil away it could never take solid except for maybe in diamonds where those diamonds due to the immense temperature and pressure would be metamorphic between a solid and liquid almost as how mercury acts here on Earth... We think we know life, but we barely understand 10% of it. Hell, I'd argue that we are lucky if we only understand 1% it even with all of our marvelous inventions and technology. Take the carnivorous (meat eating) plants here on earth such as the venus fly trap and others. They are designed by their organic chemical makeup to be lethal predators in the form of a trap. They lure their victims in according to the preference of the type of insect or small creature they prefer... Now tell me that's not calculated precision? Life is everywhere! The Universe is alive and it is always counting!
@@skilz8098 so you wrote all of that to be obtuse in response to a 5 year old comment? Only skimmed it, but no, I don’t think plants “do” math, but that natural selection pressured them into a configuration that we can describe mathematically because it was optimal for its survival. Plants are amazing organisms but they don’t have anything comparable to a brain. Why? Because brains are resource expensive and they don’t need them. They can do everything they need just vibing in the sunlight. I’m jealous honestly.
@@jondreauxlaing You're stating that plants don't have a brain. I think our brains are kind of limited to that of all plant life in some ways. Just look at their root systems! You don't think they have the ability to feel pain, or even emotions? Yes they are different than we are and even animals, but they are very much alive! And all living things have a consciousness and are self aware. They may not ponder questions like we do as to how and why 1+1 = 2. Or they may not exactly try to solve a problem such as y = 3x + 2 solve for x when y = 8. No, but the ability to enumerate or count is built within them all the way down to cell structures and DNA. They take in water, nutrients from the ground, CO2 and Sunlight and from that they sprout from seed, grow, mature and reproduce. We can not even fully replicate photosynthesis that plants do within their chlorophyll that turns the energy of sunlight into food. And yet chlorophyll from plants is vital to our bodies. Even non living things can count when they are instructed to. What do you think your computer is doing? What do you think it is made of? It uses insulators, semiconductors and conductors. The semiconductor of modern practical use is Silicon which comes from either pure silica or from sand and sand comes from rocks and salt. We are carbon based and our semiconductor is primarily carbon. Carbon and Silicon are different but also have very similar properties. They both have 4 valence electrons in their outer shells. They can be arranged in many various configurations and patterns, and they can bond with many other elements depending on the state of matter in which they are in. Even non living things can count. What do you think is smarter, an Apple Tree or your computer? I'd put my money on the Apple Tree as your computer can not do anything unless it is instructed to. An apple tree can grow a new branch, blossom, or fruit at will over the course of time and it can reproduce. Do you see computers having offspring on their own? I don't bound my mind to deterministic boundaries or limitations by the "status quo". I don't think inside nor outside of the box. I create and destroy that box at will. There are no limitations within my mindset as I believe that anything and everything is possible! If one can imagine it, even if it doesn't make sense to others, it becomes manifested by the thoughts of it alone!
The way these TH-camrs 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!
is the proof in maths? what's the definition of proof here? I have no idea what difference it makes that he proved it if it's just like, true, is he the first person to notice it?
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!! ;)
Has anyone wondered why this pattern exists at all? The math we develop is a structural form of understanding of material configuration, and the math allows for prediction etc. , but it does not explain the cause of the phenomenon. Could it be that mass: gravity and spin Earth's rotation acting together produce the pattern? Could then the Fibonacci sequence predict mass from rotation and shed light on the gravitational force of galaxies etc? Or has this already been done partially by astrophysicists when calculating trajectories of satellites sent into space? As you can guess I am not a mathematician, I studied philosophy and still ask questions at the age of 72.
The reason that you most likely see a Fibonacci or a Lucas number of spirals is that these are the two simplest of the family of Lucas Number Sequences. All adjacent numbers in a generalized Lucas sequence converge to the golden ratio, since they can be determined by the generalized Binet Formula. Since nature has chosen the obtuse Golden Angle as an an iterative angle for distributing plant structures around a radial point of origin in phyllotaxis, two adjacent recurrence sequence numbers are going to best approximate the golden ratio in q band, and they will appear prominently at different bands, depending on the number of plant structures in that band. The issue you have explored is really only a consequence of the iterative application of the golden angle. The real question, at least to me, is why is the obtuse golden angle the iterative angle of choice in phyllotaxis. First, check out the equi-distribution theorem, which states that the sequence θ, 2θ, 3θ, 4θ, ... will distribute a point to all points on a circle only if θ is an irrational number of turns of a circle. obtuse Golden angle, check.
I must say I greatly enjoy watching your videos. My only complaint is that you're fairly quiet in almost all of your videos, sometimes I can hardly hear through my speakers, even at maxed volume. Other than that, your videos are exhibit superb quality and you are exceptionally good at explaining your content. :)
So could this mean that flowers (and other things in nature/universe) who inherently follow the “least effort / least energy / least action?” method of filling up space from a central point of certain shape -> causes Fibonacci / Lucas or similar number series ( or spirals such as golden ratio) to appear, is that it? In short, my question is: Could a rule such as: “Minimize energy use” have causality with -> Fibonacci/Lucas number series or certain Spirals appearing?
“I’ve actually just made this up in a drawing program.” Some of your drawings are clearly made with a mathematical drawing program. Which program? Mathematica? Maple? I have been playing with visualising Fermat’s spiral and Vogel’s model in Mathematica with some success. But your growth animation looks a lot more difficult to produce. P.S. The visual proof for GREEN + RED = BLUE was delightful.
Your proof is so easy when you think about it logically, all blue point are the intersection of the spirals, count the number of intersections, and intersections which are due to red/green and the sum is the blue
Great video, thanks a lot ! Its Euclidean-style satisfying to note that each spiral our gestalt brain sees emerging is just the next iteration in the recursive function. The pattern used for the proof at the end seems to relate to your cardioids etc in the multiplication table video.
The third sequence, 3,4,7,11,18,29, is just the fibonacci sequence plus the fibonacci sequence shifted over twice. The double was shifted 0 times, and one shift would be the fibonacci sequence but with only a single 1. Neat
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).
Great video. I was about to ask about Lucas numbers, but you answered my question in the video :) It would be neat to see a timelapse video of a plant growing like this. Do you know if the mathematical sequence depends on the species, or the individual? For example are there plant species which only exhibit the Lucas numbers? Awesome proof by the way. Keep up the great work.
The 34 and 55 spiral arms going in opposing directions in your flower head diagram occupy the same region, that is they completely overlap. Therefore together each set must have the same total quantity of seeds, though each spiral arm in the 34 must have more seeds than one in the 55. How much more? Well, in your diagram each 55er seems to alternate from 6 to 7, and each 34er from 11 to 12, Is this based on empirical observation or just how you happened to draw things? Anyway, let's assume a consecutive pair of those Fibonacci numbers that most closely approximate: 13 for the 34ers and 8 for the 55ers. So the total seeds in the 34ers would be 34*13=442 and in the 55ers we have 55*8=440. So pretty close, eh? Is this a result specific to the Fibonacci sequence would you say? Moreover the average of 442 and 440 is 441, the exact square of 21 which is the Fib number between 8 13 and 34 55.
Very good thinking. Basically all spot on. And your last observation is actually a general property of the Fibonacci numbers: f_n f_{n+2} - f_{n+1} = plus or minus 1 with the plus and the minus alternating. 5 x 13 - 8^2 = 1 , 8 x 21 - 13^2 = -1 :)
I'll look into versions of that formula for sequences companion to the Fibonacci (such as OEIS A000930 which I mention in a comment to your video on infinite fractions). In the meantime I like your phrase about spirals "jumping out at you" because they certainly do, but what jumps out in your examples is the opposing directions of the spirals, and also the way which all spirals snuggle comfortably and efficiently in with each other. A website which shows when this happens, and - crucially - when it doesn't, is Mathsisfun which has an interactive animation in which you can change the sunflower spiral patterns by varying the rotation ratios. Another important word you use is "asymmetrical" since I think this is the key to Fibonacci growth and proliferation. In Fibonacci cell division for example, some cells in a given generation don't reproduce, but just move aside and make room for those that do. The family tree branches lopsidedly, though can regain its balance.
There is a Rubik's cube themed video coming up next. After that I'll do one exclusively on dealing with all sorts of infinite expressions. That one will also cover the 1=2 puzzle. In the meantime maybe browse some of the great answers to the puzzle that were contributed in the comments section of that video :)
very nice and inspiring video! you mention about starting a 'Fibonacci rule' from small numbers: 1, 2 and also that it is reasonable nature to start small. I assume that for the same reason, we find a 'rule of 2' more frequent than 'rule of 3, or 4' example: rule of 2: 1 1 2 3 5, ... Fibonacci. rule of 3: [1 (1 3] 4 ) 8, ... sum last 3 elements. what is your opinion about the variants of Fibonacci? i am not sure, maybe nature is plenty of them...
You've dumbed it down nicely for a non maths guy like me, but I'm wondering if this geometric depiction of spirals and their diagonal intersections is as simple? I reckon this wouldn't just work with any two random numbers but numbers of a SET A, where each number starting the 3rd number can be defined by Y= x+(x-1) or Y= 2x-1. Doesn't this look something from the Complex algebra thing that was used to plot the Mandelbrot set (or was it called something else. My memory fails me). The Mandelbrot set (or something similar) has a lot of colored points (where supposedly the result is not easily determined and can only be gauged by it's rate of deviation or something.) I may be thinking nonsense right now. I guess I'll just sleep.
This is really cool and the description of the relationship between the spirals is super nice. There is one thing I am just not so clear on. When you are describing the growth from a single bud, what determines the precise location of the new bud and the rate the buds move outwards radially? I saw in the diagram that each but gains a "halo" which increases in thickness with each new bud but what precisely is going on there?
Mathologer, I was looking for a list of square roots and tried googling "square root sequence", but near the top of the list is the wiki page for the Golden Ratio. Is this because of the Fermat/Vogel calibration of the spiral for equal density?
I do know that for any Fibonacci-like sequence (each element is the sum of the previous 2), regardless of starting numbers (so long as they aren't both zero), the ratio of any element of the sequence to its predecessor approaches the golden ratio the further into the sequence you go.
I know I expressed my skepticism about that grand importance of the Fibonacci numbers and the golden ratio with regards to biology (and science) in general, but I do still hold it. Nonetheless, it does seem to be useful as an abstract solution to this particular problem of seed dispersion, and I thank you for making a video on this topic. What is always entertaining, however, is that many people (the Mathologer not included here) go way overboard with the Fibonacci stuff so that it turns into numerology, finding the golden ratio in any physical measurement between 1.5 and 1.8 as evidence of the golden ratio in the "optimum". Upon closer inspection almost none of the claims of the golden ratio appearing are borne out by actual measurements and end up being cherry-picked examples.
I think the book 'Foucault's Pendulum' by Umberto Eco would be something for you www.goodreads.com/book/show/17841.Foucault_s_Pendulum The whole book/story is making fun of numerology, esoterics and conspiracy theories of all kinds.
Viheart made a similar video explaining the golden ratio and how it is the angle that each new petal comes out at. If I recall it's somewhere around 136*. I highly suggest checking it out.
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): link.springer.com/article/10.1007/s00004-015-0285-1
This type of structures are also benificial for the organism because after a rotation (i.e. an operation of ±1 cells) the surface stays almost the same as before. It is more dynamically space efficient than in a static manner. The same principle can be seen in that chocolate illusion where when the pieces are rearranged it seems like we have the same chocolate bar plus one piece. I have seen a puzzle that is like that of the chocolate but it is arranged in spirals like these plants and you can rearrange the pieces in a way that you seemingly have the same surface but with ±1 a piece.
Yes, there is a lot more that one can and has been said about all this. Amazing insights wherever you look. I'd be interested in seeing that spiral-based variation of the disappearing piece illusion. Not sure whether you've seen the version where you cut up a square piece of chocolate with sides a Fibonacci number arranging the cuts based on the previous two Fibonacci numbers. E.g. in the version that I sometime use an 8x8=64 is cut up into four pieces that then get rearranged into a 5x13=65 rectangle. All this is based on the fact that f_i f_{i+2}=±f_{i+1} :)
Maybe you are right, I can't find it either. Akio Hizume is the name of the guy who made that puzzle. This is the bottom part of the puzzle (gallery.bridgesmathart.org/sites/default/files/styles/large/public/bridges2014/akio-hizume/cimg2988.jpg?itok=xEYB667j) and he has also made this cool clip (th-cam.com/video/PDK8RonFBe4/w-d-xo.html).
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.
Great video! I like the theoretical explanation behind the whole pattern process. Do you think it would be possible to draw the spirals from sketch? The idea is to draw a sunflower from scratch but not sure how to start 🙂
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.
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?
Seems like the blue line between buds is the vector sum of the red and green vectors (if you call them vectors, the distance between the buds). So could you prove it by a sum of vectors? I love your videos! Keep them coming :)
at 6:22 ish with the top diagram, if you treat the green red and blue lines like vectors, and add the green and red, it looks like you would get the blue. just an observation that supports the green+red = blue idea
I love this proof. In general, I'm in favour of using visual proofs in mathematics, because they are often the quickest and the most intuitive method of explaining concepts, particularly relating to real world subject matter.
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!
The proof reminded me a bit of the graphical proof of quasi convexity of the work function for the k-server problem (i.e., `The k-server problem' Koutsoupias).
QUESTION; Mr Polster. I'm searching the web without success for a mathematical method that a surveyor could use to pinpoint around 60 intersection points of an 8-13 double spiral that would be drawn on a 30 hectares (75 acres) land. Would you happen to know one formula that would make it possible to calculate the coordinates ? Either angle:distance or XY graph. Thanks for your time.
In 2:10, what is the position of the 1st guy (for eg. why it is not in the center)? Later, what is the position of the next guys? Any suggestion how to draw these spirals X(n)=? and Y(n)=? .
@mathologer Your animation of the buds growing into the gaps and pushing out the older buds which themselves are growing cuts off too soon. It would be *much* better to let the animation run long enough to see the spirals form. It's currently good enough to convey the idea to someone who already understands but for someone who isn't sure where it's going, probably not enough. Reminds me of when I heard a lecture by Ken Ribet describing Wile's Fermat Proof. He was describing elliptic curve patterns and stopped the description before the pattern he was illustrating had formed leaving me wondering "what? What pattern are you talking about?" Nonetheless, it's a great video. I'll be showing it to my students.
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?
Looked again. Things get skewed further from the center. Exactly what would the non-euclidian space have to look like for vector addition to work here?
I‘ve got a question . I study machenical engineering and in higher mathematics we learned the proof through complete induction. In your proof u have one Example for your statement. But how do you know, That its like That for every number of Spiral. Like what is the True mathematics proof. Not Visual. I am very curious
It was Pingala the great Hindu Indian mathematician ,in 200 AD first used this series in sanskrit syllables..which later expressed as Fibonacci sequence...please mension the history whenever you use this sequence...as it will be honour for actual discovery..
Such an elegant an simple proof. What a thing of beauty.
yes!
Instablaster.
Is joy forever
maths made simple
EXTREMELY BEAUTIFUL & ARTISTIC MATHEMATICAL VIDEO. KUDOS TO YOU SIR!
I feel like the quality of your animations are getting even better every video. not only nicely explained, but also nice to look at. i realy love your work!
Glad you think so and thank you very much for the compliment :)
+Mathologer yes please keep up all your wonderful work!
one of the best mathematical proof i have ever seen, congratulations
Nobody beats 3Blue1Brown in that area, but this is still fantastic.
@@Mathologer your videos are quite thought 🤔provoking ..finaminal and was part of my inspiration for this theory I've been working for the past few years... and I was wondering if you could check out my video's "best veiw black hole/ partical wave duality" and "talking of [Universal Emulator]" I'm not looking for a math war.. ...I admit I'm atrocious with my notations.... but I'm sure you will under stand everything I was trying to convey and I think I may have found something extra some thing tesla once spoke of and..... you actually just recently uploaded a video about almost exactly the same thing (19th.dd/02.mm/22.yy)..I just dabble with it some times and use the best math I can to describe it.. I find it fun to work large thought equations. 🙂🙂👍
Wow. I love that proof. That's ace. Very elegant. That's what proofs should be. Great explanation altogether.
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fat loss srivatsa These are mandalas though.
That's a beautiful proof at the end. I love it Mathologer.
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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!
I must say, I really like your way of presenting the material. Your methods of explaining is superior to almost everything else I've encountered. The calmness and naturaleness of the video really reflect unto the viewer. Keep up the good work :).
That proof was very beautiful! It was very easily understood even for me!
Great, mission accomplished :)
Same would work for galaxies, right? So now what can we infer about the origin of the universe?
God version (which type 'God' not solved):
Immaterial** A joins with immaterial B and as a result Material** C exists. Wash rinse repeat.
Non-god version:
Your video, with math playing the god role. Cuz clearly it's immaterial and clearly is the cause of the material, as its parental marks are everywhere.
So: how does immaterial joining immaterial CREATE material? I don't know yet. By indirections, find directions out.
*defined as that which is real but neither mass nor energy, so a stasis, set of rules, etc.
**defined as that which is real, finite even if iterative, and is somewhere on the energy-mass spectrum.
Brian Outy that is like 20 other videos, not to mention immaterial physics and the existence of indirect effect and maybe even time travel. WHY?
@@brainoutyakabrainout Maybe I am just too stupid to understand what you are saying, but my takeaway from what you said is that it seems like you were presenting a bunch of gibberish or word salad or whatever you want to call it. Maybe it isn't meaningless, but I can't pull anything meaningful from it.
@@alexandertownsend3291 its dunning kruger
Congratulations on 100,000 subscribers! Here's to many more!
Yes, we've had a little party last week :)
This was a very good and interesting way to show the link between Fibonacci numbers and flowers
This is why I like math so much. There are so many amazing relationships between different patterns and things we came up with, and it’s just so neat.
wow. thank you! I wish you were my math teacher when I was in school. I'm still learning SO MUCH from you! Thank you!
That's great :)
Me too!!!! Thank you
You brilliant Teutonic madman! Well done, sir. I have been told ever since childhood that Fibonacci sequences could be seen in flowers, but I never understood what they meant.
I never had the chance to do math and find your explanations fascinating. I'm addicted. Thank you.
I love how you present a different approach to classic maths problems in such an elegant way. Can't wait for part 2!
Very beautiful proof at the end!
About the explanation: This might be because I am quite tired, but I had a hard time following some of your points, which usually isn't the case.
Very nice proof indeed! In my linear algebra class (1st year college), I usually talk about Fibonacci numbers when discussing the solution of linear difference equations using eigenvalues. This proof is certainly accessible to my engineering students.
Talking about linear algebra, check out 3blue1brown
Glad you like it. You are proving Binet's formula for the nth Fibonacci number using eigenvalues, right ? :)
Mathologer Exactly. I hope to prepare them for the solution of second order differential equations, which they learn a little later in the analysis class. It helps them understand that the term "characteristic polynomial" in both topics is _not_ a misnomer. It really is the same thing. So many more parallels: two basic solutions, two initial conditions determine how the have to be linearly combined.
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.
youtubers are better at teaching math than high school teachers
Some are :) But, to be fair high school teachers simply don't have the two weeks it takes to put something like this together.
We are lucky to be alive now. Just think, a person is able to find so much information just with a few clicks.
You would think they would, after teaching the same course for multiple years.
mathologer are you a teacher in a physical institution, mathologer?
I teach math at Monash University in Australia :)
Thank you for your work. As a dad of two boys who home school, I appreciate the help of your you tube content. These sequences of "1" and "2" make "3" are in the DNA of living creatures as practical purpose, inventive uprising, and ideal envisioning.
How do you made your animations?? You are amazing!! I am a teacher of math, and I want to make some videos for my students with animations like this! Congratulations for your awesome work! Greetings!
I use a whole range of applications for making these videos: Adobe Illustrator, Photoshop, Premier and After Effect, Mathematica. Having said that a lot of the animations use Apple's Keynote presentation software and in particular the "Magic Moves" transitions. It's quite remarkable how much you can achieve in this way :)
I feel like if I had access to these types of videos when I was in school I would have done so much better. I learned in the 70's & early 80's before computers and animations. It was so hard for me to get mental pictures from my text books. This is amazing, and every time I watch a video I think, "ooooohhh, so THAT's what WHY that happens". It used to just be rote and remembering the concepts I was taught, but now I have a mental picture and the understanding has taken hold. Unfortunately I'm an old lady now, and my science days are behind me. But I'm boning up so I can teach my grandkids the amazing language and life of maths.
I saw an explanation video in the Stanford channel called fibonnaci fact vs fiction. An hour in it explains the math in flower patterns, something about interference and optimal growth rate. I'll resume viewing your own explanation now.
at about 6 min in I couldn't help but see the green and red lines as vectors with the blue line being their addition. Great video!
If you haven't addressed this question already in the previous comments, I noted (first around 10:37-38 or so) that the green nodes don't exhibit quite the same perfection of symmetry as the red: there are only 3 forming the "left side" of the right most leaf, with four nodes on the other highlighted green spiralsl. On the other side, the red spirals have two sets of 5 nodes and 4 nodes respectively. Perhaps this is a trivial point as the proof stands nevertheless, but sometimes such curiosities can lead to interesting consequence. Many thanks! Your videos are a delight to watch, and, if I have raised the question redundantly, my apologies.
The numbers of red and green spirals that you start with is not important for all this to work. In this case there are 15 green spirals and 18 red spirals overall. The numbers of that jump out on the path that I follow are really my choices. I could change direction earlier or later as long as I end up with a closed path the proof will go through as I showed. But as you say, it's important to pay attention to such details as you are often lead to interesting facts this way :)
I must say, I didn't expect much from this video as ViHart (among others) has 3 excellent videos explaining why plants grow in such a way to produce patterns relating to the Fibonacci numbers, but I have to say that that proof you gave at the end is something I've never seen before and it's just beautiful. I've never thought of it that way, so thank you!
That's great. I usually try to come up with something that you won't find anywhere else :)
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.
Mathologer ist nicht umsonst einer meiner absoluten Lieblingschannels auf TH-cam!
:)
Vihart also did a really nice explanation, also delving into why plants do this, even though they can't do math. It's been a while since I watched it, but I think it had to do with the optimal amount of space per surface area, given that they grow from the center out.
How are you 100% certain that they don't do math? They are a living thing, and living things grow. For something to grow, they have to be able to reproduce and communicate. To reproduce is the same as multiplying. If it can multiply then addition is implied. If it can add, it is able to count. If it's able to count it is able to perform calculations and this is math. For something to communicate it must be able to transfer information. In order to transfer information it has to be able to perform calculations. It is my claim that at the least all living things are constantly performing mathematical computations in one way or another. Growing and expanding from DNA instructions, cell reproduction, organ growth, expanding until an age of maturity, reproducing its offspring for the next generation, cycle or iteration. Plants do count! If they couldn't they wouldn't grow!
Do you think stars are non living entities or do you think they're living things as well? Within their own environment they are compressing multiple Hydrogen atoms under nuclear fusion and are producing Helium and other heavier elements. They are doing this at an exponential rate governed by at the least quantum mechanics as we currently understand it, and we still don't fully understand it. We are able to count and perform math, yet we are still feeble minded compared to that of the power of the stars. So please tell me again how and why plants can't do math?
The Cosmos including our known visible Universe and all that lies beyond I is counting. How and why? Everything is in motion and if something is able to move, it's able to count. And the Cosmos, Universe is moving. It is continuously counting and thus it is a living thing. When I state living, I'm not narrowed to the view that all living things must be "carbon" based. Life here on Earth as we know it within its environment and conditions requires carbon to be considered organic life. Yet we have no understanding of all the possibilities that life can present through the stars. How do we not know that there isn't some arbitrary planet that orbits some arbitrary star has a form of life that is bizarre to us where the conditions there for life as we know it we could not survive its extremely harsh environment for our biological compositions.
Take for example, a B or an O blue star that's anywhere from 2-10 times the diameter of our star with 2-100x its mass, with temperatures that are 3-10x that of our star with a higher luminosity, higher radiation, etc. Let's say that this planet is bigger than a super earth but smaller than a gas or ice giant. Let's say it's roughly 15 - 20 times more massive than our earth. It has a slightly higher density and is extremely hot compared to us. The planet in our terms would be baking, even hotter than Mercury or Venus. Let say that its outer layer does have an atmosphere that isn't conventional for us. Let's say that the first 4-5 transitional metals, Scandium, Titanium, Vanadium, Chromium, and Manganese are in a gaseous state, and the next 4-5 transitional metals, Iron, Cobalt, Nickel, Copper, and Zinc are in a liquid state. And something like Germanium which is still in Group 14 with 4 valence electrons acts similar within this star and planets environment as Carbon and Silicon does for us. Let's say it does have a rocky surface where it's lands or mountains are of the heavier rare earth metals that would normally be radioactive to us. The planet's atmosphere is 10x greater than ours, the temperature is 20x hotter than ours, the pressure is 5x greater, the gravity is 7-12x greater, the radiation is 30x greater and the brightness or luminosity due to its bright blue star is 15x brighter than here on earth. There are rivers and oceans of liquid liquid iron similar to but more fluid than our earth's lava. And how do we not know that there aren't growing living things within that environment? That environment can not support Carbon Based Life as Carbon would melt and then boil away it could never take solid except for maybe in diamonds where those diamonds due to the immense temperature and pressure would be metamorphic between a solid and liquid almost as how mercury acts here on Earth... We think we know life, but we barely understand 10% of it. Hell, I'd argue that we are lucky if we only understand 1% it even with all of our marvelous inventions and technology.
Take the carnivorous (meat eating) plants here on earth such as the venus fly trap and others. They are designed by their organic chemical makeup to be lethal predators in the form of a trap. They lure their victims in according to the preference of the type of insect or small creature they prefer... Now tell me that's not calculated precision? Life is everywhere! The Universe is alive and it is always counting!
@@skilz8098 so you wrote all of that to be obtuse in response to a 5 year old comment? Only skimmed it, but no, I don’t think plants “do” math, but that natural selection pressured them into a configuration that we can describe mathematically because it was optimal for its survival. Plants are amazing organisms but they don’t have anything comparable to a brain. Why? Because brains are resource expensive and they don’t need them. They can do everything they need just vibing in the sunlight. I’m jealous honestly.
@@jondreauxlaing You're stating that plants don't have a brain. I think our brains are kind of limited to that of all plant life in some ways. Just look at their root systems! You don't think they have the ability to feel pain, or even emotions? Yes they are different than we are and even animals, but they are very much alive! And all living things have a consciousness and are self aware. They may not ponder questions like we do as to how and why 1+1 = 2. Or they may not exactly try to solve a problem such as y = 3x + 2 solve for x when y = 8. No, but the ability to enumerate or count is built within them all the way down to cell structures and DNA. They take in water, nutrients from the ground, CO2 and Sunlight and from that they sprout from seed, grow, mature and reproduce. We can not even fully replicate photosynthesis that plants do within their chlorophyll that turns the energy of sunlight into food. And yet chlorophyll from plants is vital to our bodies. Even non living things can count when they are instructed to. What do you think your computer is doing? What do you think it is made of? It uses insulators, semiconductors and conductors. The semiconductor of modern practical use is Silicon which comes from either pure silica or from sand and sand comes from rocks and salt. We are carbon based and our semiconductor is primarily carbon. Carbon and Silicon are different but also have very similar properties. They both have 4 valence electrons in their outer shells. They can be arranged in many various configurations and patterns, and they can bond with many other elements depending on the state of matter in which they are in. Even non living things can count. What do you think is smarter, an Apple Tree or your computer? I'd put my money on the Apple Tree as your computer can not do anything unless it is instructed to. An apple tree can grow a new branch, blossom, or fruit at will over the course of time and it can reproduce. Do you see computers having offspring on their own? I don't bound my mind to deterministic boundaries or limitations by the "status quo". I don't think inside nor outside of the box. I create and destroy that box at will. There are no limitations within my mindset as I believe that anything and everything is possible! If one can imagine it, even if it doesn't make sense to others, it becomes manifested by the thoughts of it alone!
@@skilz8098 k
@@jondreauxlaing It's just food for thought!
This is the most wholesome channel in existence.
This elegantly covers and expands on ViHart's series on plants and the fibonacci sequence. The proof at the end is very simple and clever, too.
The REALLY beautiful part is at 10:18 where you draw a flower in your proof of why the Fibonacci sequence appears in flowers.
😁🌻
The way these TH-camrs 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!
is the proof in maths? what's the definition of proof here? I have no idea what difference it makes that he proved it if it's just like, true, is he the first person to notice it?
Very nice explanation, and very well illustrated. Have wondered why these patterns appears, and this makes it intuitively and mathematically clear :)
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!! ;)
Has anyone wondered why this pattern exists at all? The math we develop is a structural form of understanding of material configuration, and the math allows for prediction etc. , but it does not explain the cause of the phenomenon. Could it be that mass: gravity and spin Earth's rotation acting together produce the pattern? Could then the Fibonacci sequence predict mass from rotation and shed light on the gravitational force of galaxies etc? Or has this already been done partially by astrophysicists when calculating trajectories of satellites sent into space? As you can guess I am not a mathematician, I studied philosophy and still ask questions at the age of 72.
Searching for phyllotaxis ended here. What a clarity in explanation..
Thank-you..
Again an awesome video. Thank you.
I hope you'll keep up with the top-notch quality videos
I love that the closed captions capture the *delighted giggle* at the end of his proof.
Very well explained. I also like how your proof is very visual. Keep making these, I am starting to share them!
Brilliant!! The best explanation. Watched lots of videos but only here it’s explained why plants form like this. 👍🏻👏🏻
The reason that you most likely see a Fibonacci or a Lucas number of spirals is that these are the two simplest of the family of Lucas Number Sequences. All adjacent numbers in a generalized Lucas sequence converge to the golden ratio, since they can be determined by the generalized Binet Formula. Since nature has chosen the obtuse Golden Angle as an an iterative angle for distributing plant structures around a radial point of origin in phyllotaxis, two adjacent recurrence sequence numbers are going to best approximate the golden ratio in q band, and they will appear prominently at different bands, depending on the number of plant structures in that band.
The issue you have explored is really only a consequence of the iterative application of the golden angle. The real question, at least to me, is why is the obtuse golden angle the iterative angle of choice in phyllotaxis. First, check out the equi-distribution theorem, which states that the sequence θ, 2θ, 3θ, 4θ, ... will distribute a point to all points on a circle only if θ is an irrational number of turns of a circle. obtuse Golden angle, check.
I must say I greatly enjoy watching your videos. My only complaint is that you're fairly quiet in almost all of your videos, sometimes I can hardly hear through my speakers, even at maxed volume. Other than that, your videos are exhibit superb quality and you are exceptionally good at explaining your content. :)
videos exhibit* the "are" was accidental.
I love this kind of Math. I could watch this sort of thing for ages.
So could this mean that flowers (and other things in nature/universe) who inherently follow the “least effort / least energy / least action?” method of filling up space from a central point of certain shape -> causes Fibonacci / Lucas or similar number series ( or spirals such as golden ratio) to appear, is that it?
In short, my question is:
Could a rule such as: “Minimize energy use” have causality with -> Fibonacci/Lucas number series or certain Spirals appearing?
That's quite an elegant proof.
I wonder if visual, elegant proofs would help young students learn the idea of "proof".
“I’ve actually just made this up in a drawing program.” Some of your drawings are clearly made with a mathematical drawing program. Which program? Mathematica? Maple? I have been playing with visualising Fermat’s spiral and Vogel’s model in Mathematica with some success. But your growth animation looks a lot more difficult to produce. P.S. The visual proof for GREEN + RED = BLUE was delightful.
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?
Your proof is so easy when you think about it logically, all blue point are the intersection of the spirals, count the number of intersections, and intersections which are due to red/green and the sum is the blue
Great video, thanks a lot ! Its Euclidean-style satisfying to note that each spiral our gestalt brain sees emerging is just the next iteration in the recursive function.
The pattern used for the proof at the end seems to relate to your cardioids etc in the multiplication table video.
The third sequence, 3,4,7,11,18,29, is just the fibonacci sequence plus the fibonacci sequence shifted over twice. The double was shifted 0 times, and one shift would be the fibonacci sequence but with only a single 1. Neat
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).
Great video. I was about to ask about Lucas numbers, but you answered my question in the video :)
It would be neat to see a timelapse video of a plant growing like this.
Do you know if the mathematical sequence depends on the species, or the individual? For example are there plant species which only exhibit the Lucas numbers? Awesome proof by the way. Keep up the great work.
The 34 and 55 spiral arms going in opposing directions in your flower head diagram occupy the same region, that is they completely overlap. Therefore together each set must have the same total quantity of seeds, though each spiral arm in the 34 must have more seeds than one in the 55. How much more? Well, in your diagram each 55er seems to alternate from 6 to 7, and each 34er from 11 to 12, Is this based on empirical observation or just how you happened to draw things? Anyway, let's assume a consecutive pair of those Fibonacci numbers that most closely approximate: 13 for the 34ers and 8 for the 55ers. So the total seeds in the 34ers would be 34*13=442 and in the 55ers we have 55*8=440. So pretty close, eh? Is this a result specific to the Fibonacci sequence would you say? Moreover the average of 442 and 440 is 441, the exact square of 21 which is the Fib number between 8 13 and 34 55.
Very good thinking. Basically all spot on. And your last observation is actually a general property of the Fibonacci numbers: f_n f_{n+2} - f_{n+1} = plus or minus 1 with the plus and the minus alternating. 5 x 13 - 8^2 = 1 , 8 x 21 - 13^2 = -1 :)
I'll look into versions of that formula for sequences companion to the Fibonacci (such as OEIS A000930 which I mention in a comment to your video on infinite fractions). In the meantime I like your phrase about spirals "jumping out at you" because they certainly do, but what jumps out in your examples is the opposing directions of the spirals, and also the way which all spirals snuggle comfortably and efficiently in with each other. A website which shows when this happens, and - crucially - when it doesn't, is Mathsisfun which has an interactive animation in which you can change the sunflower spiral patterns by varying the rotation ratios. Another important word you use is "asymmetrical" since I think this is the key to Fibonacci growth and proliferation. In Fibonacci cell division for example, some cells in a given generation don't reproduce, but just move aside and make room for those that do. The family tree branches lopsidedly, though can regain its balance.
Great video. Thank you for keeping them free of any subscriber-hunting phrases and channel promotion.
:)
Very nice! Maybe I will do a simulation of "plant growth"...
But I was sort of expecting the answers for last "paradox" of 1=2
There is a Rubik's cube themed video coming up next. After that I'll do one exclusively on dealing with all sorts of infinite expressions. That one will also cover the 1=2 puzzle. In the meantime maybe browse some of the great answers to the puzzle that were contributed in the comments section of that video :)
Your content is really high quality and I'm looking forward to whatever's coming next. Thanks a lot for all of these!
very nice and inspiring video! you mention about starting a 'Fibonacci rule' from small numbers: 1, 2 and also that it is reasonable nature to start small. I assume that for the same reason, we find a 'rule of 2' more frequent than 'rule of 3, or 4' example:
rule of 2: 1 1 2 3 5, ... Fibonacci.
rule of 3: [1 (1 3] 4 ) 8, ... sum last 3 elements.
what is your opinion about the variants of Fibonacci? i am not sure, maybe nature is plenty of them...
You've dumbed it down nicely for a non maths guy like me, but I'm wondering if this geometric depiction of spirals and their diagonal intersections is as simple? I reckon this wouldn't just work with any two random numbers but numbers of a SET A, where each number starting the 3rd number can be defined by Y= x+(x-1) or Y= 2x-1. Doesn't this look something from the Complex algebra thing that was used to plot the Mandelbrot set (or was it called something else. My memory fails me). The Mandelbrot set (or something similar) has a lot of colored points (where supposedly the result is not easily determined and can only be gauged by it's rate of deviation or something.)
I may be thinking nonsense right now. I guess I'll just sleep.
This is really cool and the description of the relationship between the spirals is super nice. There is one thing I am just not so clear on. When you are describing the growth from a single bud, what determines the precise location of the new bud and the rate the buds move outwards radially? I saw in the diagram that each but gains a "halo" which increases in thickness with each new bud but what precisely is going on there?
Thanks for these elegant little lectures, so confidently-delivered, Dr Polster
Mathologer, I was looking for a list of square roots and tried googling "square root sequence", but near the top of the list is the wiki page for the Golden Ratio. Is this because of the Fermat/Vogel calibration of the spiral for equal density?
I do know that for any Fibonacci-like sequence (each element is the sum of the previous 2), regardless of starting numbers (so long as they aren't both zero), the ratio of any element of the sequence to its predecessor approaches the golden ratio the further into the sequence you go.
I know I expressed my skepticism about that grand importance of the Fibonacci numbers and the golden ratio with regards to biology (and science) in general, but I do still hold it. Nonetheless, it does seem to be useful as an abstract solution to this particular problem of seed dispersion, and I thank you for making a video on this topic.
What is always entertaining, however, is that many people (the Mathologer not included here) go way overboard with the Fibonacci stuff so that it turns into numerology, finding the golden ratio in any physical measurement between 1.5 and 1.8 as evidence of the golden ratio in the "optimum". Upon closer inspection almost none of the claims of the golden ratio appearing are borne out by actual measurements and end up being cherry-picked examples.
I think the book 'Foucault's Pendulum' by Umberto Eco would be something for you
www.goodreads.com/book/show/17841.Foucault_s_Pendulum
The whole book/story is making fun of numerology, esoterics and conspiracy theories of all kinds.
Very nice! What program is used to make those designs? Awesome presentation!
Viheart made a similar video explaining the golden ratio and how it is the angle that each new petal comes out at. If I recall it's somewhere around 136*. I highly suggest checking it out.
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):
link.springer.com/article/10.1007/s00004-015-0285-1
This type of structures are also benificial for the organism because after a rotation (i.e. an operation of ±1 cells) the surface stays almost the same as before. It is more dynamically space efficient than in a static manner. The same principle can be seen in that chocolate illusion where when the pieces are rearranged it seems like we have the same chocolate bar plus one piece. I have seen a puzzle that is like that of the chocolate but it is arranged in spirals like these plants and you can rearrange the pieces in a way that you seemingly have the same surface but with ±1 a piece.
Yes, there is a lot more that one can and has been said about all this. Amazing insights wherever you look. I'd be interested in seeing that spiral-based variation of the disappearing piece illusion. Not sure whether you've seen the version where you cut up a square piece of chocolate with sides a Fibonacci number arranging the cuts based on the previous two Fibonacci numbers. E.g. in the version that I sometime use an 8x8=64 is cut up into four pieces that then get rearranged into a 5x13=65 rectangle. All this is based on the fact that f_i f_{i+2}=±f_{i+1} :)
Maybe you are right, I can't find it either. Akio Hizume is the name of the guy who made that puzzle. This is the bottom part of the puzzle (gallery.bridgesmathart.org/sites/default/files/styles/large/public/bridges2014/akio-hizume/cimg2988.jpg?itok=xEYB667j) and he has also made this cool clip (th-cam.com/video/PDK8RonFBe4/w-d-xo.html).
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.
Makes sense. And man oh man do I love things that make sense!
Great video! I like the theoretical explanation behind the whole pattern process. Do you think it would be possible to draw the spirals from sketch? The idea is to draw a sunflower from scratch but not sure how to start 🙂
Which program did you use for the animations?
Mostly Adobe Illustrator, Mathematica and Apple Keynote :)
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.
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?
Seems like the blue line between buds is the vector sum of the red and green vectors (if you call them vectors, the distance between the buds). So could you prove it by a sum of vectors?
I love your videos! Keep them coming :)
at 6:22 ish with the top diagram, if you treat the green red and blue lines like vectors, and add the green and red, it looks like you would get the blue. just an observation that supports the green+red = blue idea
I love this proof. In general, I'm in favour of using visual proofs in mathematics, because they are often the quickest and the most intuitive method of explaining concepts, particularly relating to real world subject matter.
Very nice proof, simple and clear. Your videos are always excellent, thanks!
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!
At first, I was like “What is really going on?” Then, I was like “Wow, that was cool and satisfying.”
The proof reminded me a bit of the graphical proof of quasi convexity of the work function for the k-server problem (i.e., `The k-server problem' Koutsoupias).
Its eerie how the growth pattern of a flower matches the flux lines of a magnet.
QUESTION; Mr Polster. I'm searching the web without success for a mathematical method that a surveyor could use to pinpoint around 60 intersection points of an 8-13 double spiral that would be drawn on a 30 hectares (75 acres) land. Would you happen to know one formula that would make it possible to calculate the coordinates ? Either angle:distance or XY graph. Thanks for your time.
Very interesting exposition. How does the golden angle fit into the RGB spiral pattern?
In 2:10, what is the position of the 1st guy (for eg. why it is not in the center)? Later, what is the position of the next guys? Any suggestion how to draw these spirals X(n)=? and Y(n)=? .
you deserve a nobel price in maths for THAT proove!
Clear explanation and killer proof! Thank you, Mathologist!
I really love it! Very clear and objective! Even for me, a Brazilian girl with a lot of English difficulties!
Beautiful, visual and mathematical explanation. Thank you!
Why the red dots go 4,5 in a closed loop of iteration (4,5,4,5) and then greens only goes 4,3(only once) like (4,3,4,4)?
@mathologer Your animation of the buds growing into the gaps and pushing out the older buds which themselves are growing cuts off too soon. It would be *much* better to let the animation run long enough to see the spirals form. It's currently good enough to convey the idea to someone who already understands but for someone who isn't sure where it's going, probably not enough. Reminds me of when I heard a lecture by Ken Ribet describing Wile's Fermat Proof. He was describing elliptic curve patterns and stopped the description before the pattern he was illustrating had formed leaving me wondering "what? What pattern are you talking about?"
Nonetheless, it's a great video. I'll be showing it to my students.
Hey Mathologer, you should explain
"The integral sec y dy from 0 to one sixth of pi
is log to base e
of the square root of 3
to the 64th power of i"
Fabulous explanation, and great proof at the end!!
Do you intend to add captions to your videos?
My english is not so good :(
There are a couple of viewers who've been helping with that. Maybe check back in a day or so :)
Great graphics.
Exponential growth makes spirals of spirals.
The link for Mathematica Demo is broken. Could you re-upload it? Or could I get a copy of the Notebook file itself? Thanks!
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?
Looked again. Things get skewed further from the center. Exactly what would the non-euclidian space have to look like for vector addition to work here?
I‘ve got a question . I study machenical engineering and in higher mathematics we learned the proof through complete induction. In your proof u have one Example for your statement. But how do you know, That its like That for every number of Spiral. Like what is the True mathematics proof. Not Visual. I am very curious
congrats on the 100K subs!
Heyy can you plzz tell what is the use or importance of this sequence in sunflower??
It was Pingala the great Hindu Indian mathematician ,in 200 AD first used this series in sanskrit syllables..which later expressed as Fibonacci sequence...please mension the history whenever you use this sequence...as it will be honour for actual discovery..
Such a fabulous presentation of the profound power of math, especially as a tool to understand our world!