Dreamlike animation illustrating Fibonacci sequence, Golden Ratio, and more

In "Nature by Numbers," filmmaker Cristobal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci sequence. Then it moves on to the Golden and Angle Ratios, the Delaunay Triangulation and Voronoi Tessellations. This would be math-class gold, and it's awfully sweet even if math class is years behind you.

Nature by Numbers (Thanks, Peacelove)


  1. Man, that’s beautiful.

    So, presumably, the reason that these shapes and sequences appear over and over again in nature is that they’re energy efficient to construct/deploy (feel free to poke my logic there if I’m missing something in the biology.) My question is: Are they aesthetically pleasing because they’re cheap, or because they’re plentiful?

    1. I love this question and I often ponder it myself.

      My own theory (though I have absolutely NO proof of it’s validity) is that since we humans are all composed of the same matter that makes up the rest of the universe – and all this matter (as far as we understand by our current theories) seems to adhere to the same universal constants. So our visual cortext must, at some level, ‘perceive’ a sort of kinship or connection with these ‘aesthetically pleasing patterns.’

      I personally find the more I look at art, nature, and the work of artists who are aware of this very phenomenon, I find it all the more impossible to ignore. Like a door in the mind, now open, impossible to close.

      For example, look at the work of Hokusai or Kyosai, two very famous Japanese ukiyo-e painters and very keen observers of nature. Though their work proceeds the work of mandelbrot and modern fractal theory, it is very clear that in their representations of plant and animal life they were both clearly aware of the patterns in those forms. (Hokusai’s arguably most famous work, the Great Wave Off Kanagawa is another example of this – like all his waves, seemingly random, but an ordered chaos adhering to those same ‘pleasing’ mathematical patterns) I am sure that such examples in art exist in other cultures as well.

      If there is anyone else out there reading this who has any more insight into this question or has any suggested reading material on the subject, I would be eternally greatful for your linkage.

      Beautiful math visuals are the first things that attracted me to the subject in my youth – especially the lithographs of Good ol’ M.C. Esher. I think material like this for budding students is indispensable… visuals were the biggest thing missing from math curriculum when I was a kid.

      1. There is some literature regarding how the human visual system geometry is related to patterns that occur spontaneously. For example this Plus article and the links within:

        Uncoiling the spiral: Maths and hallucinations

        It’s not clear why the visual system is organized this way, but there is speculation why many things in nature, including the human visual system, might use similar organizing principles.

      2. As far as nature goes, I consider this: “Everything would be symmetrical were it not for everything else.” Everything lives and grows under different conditions, providing different results. For example, you have a different wind one day, and your sunflower goes from Fibonacci to scatterplot.

        Any time two (or more) vaguely similar things happen, they are mathematically related. Welcome to the universe, it is a very large and complicated multi-dimensional fractal.

    2. Here’s my theory: some people love to debate whether math is the “language of god” or whether it’s “just in our heads.” But what’s in our heads is our primary evolutionary adaptation. If it didn’t fit “reality” well (in some sense), we wouldn’t still be here.

      So I think the fact that we find math beautiful is because we’re designed to seek fundamental patterns in the service of our own survival. Of course, structures of behaviors that evolution designs for one purpose often get co-opted…. :)

    3. I’ve always assumed that they are used in nature because they are easy to encode in DNA — which you have to admit is a pretty compressed data format for what it produces!

      And I assume that we like regular patterns like this partly because, as is true of many mammals, especially omnivores and herbivores, we instinctually favor shapes that look healthy and whole — not decayed or deformed by disease. Animals will look for unblemished fruit, or other food that “looks right.” Predators tend to be more opportunistic, and scavengers even more so. But when your diet is concerned with the freshest, most wholesome *browsing*, evolution favors the animals who are selective about the health (and ergo, the mathematically generated encoded shapes) of their food.

      I have no particular proof for this, though.

      Another graphic of math-as-god-in-nature that completely struck me through the heart as a young woman:


      The Fern.

      It may sound odd, but I have remembered that fern shape in times of sheer chaos (of the unmathematical sort) in my life, and felt comforted.

    4. It is because we are built with the same structures and therefore we see the world through a brain that is of like design. It flows through our mind. This is why the world seems to be such a beautiful place.

    1. That was the first thing occur to me too. I saw “Donald Duck in Mathemagic Land” in fourth or fifth grade and it really made an impression on me. I think of it every time I play pool.

  2. Quite liking the new math.

    Another good answer to “Why do we need to know math in the real world?”.

  3. Likewise; the Disney educational cartoon stayed with me as well. (Though I never remember the exact trick for the 3-cushion billiards shot.)

    Re the “math is sexy” image — Also nice. I could see that one as a poster, though I think there needs to be a beefcake equivalent to balance it.

  4. From Wikipedia:

    “Some specific proportions in the bodies of many animals (including humans[69][70]) and parts of the shells of mollusks[4] and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in specific individuals, and the proportion in question is often significantly different from the golden ratio.[69] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[70] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;[71] however, measurements of nautilus shells do not support this claim.[72]”


    So, while a nice animation, it is a common misconception that sunflowers, etc, have these ratios inside them. I suggest attempting to count the seeds in a spiral sometime, you will be surprisingly not close at all to these ratios.

  5. “structures OR behaviors”

    also: wow. thanks so much for this. I’d actually never caught the connection between phi and the sunflower before, though I’d heard there was one.

  6. Watching that made me think of the moviePi and the hypothesis the main character recited: Math is the language of nature; everything around us can be understood through numbers; through graphing said numbers, patterns emerge.

  7. This was nicely animated. The connection between the golden ratio and plant growth is a good one. It has to do with packing efficiency, although I haven’t actually found the mathematical proof of why.

    I think the nautilus might be a stretch, though. It definitely relates to logarithmic spirals, ones where the curve keeps a constant angle to its center. But the golden ratio relates to a specific angle, and I haven’t seen anyone discuss how close this is to the one in the shell.

    Donald Duck in Mathemagic Land does this too, although in most respects it is brilliant.

  8. I was just teaching my daughter how to calculate the Fibonacci Sequence the other day. This will serve as a great illustration. Thanks!

  9. &#9658 Obligatory music video with the band Tool and their song Lateralus

    This video is probably one of the best in showing how they integrate Fibonacci.

  10. i don’t particularly enjoy math class, but this was an incredible short film. I knew that some of these patterns were in nature in some examples, but i didn’t think that there were this many.

  11. @cymk: And me as well, for a different reason:

    “If you want the number you can find it everywhere: steps from your street corner to your front door, seconds you spend in the elevator. When your mind becomes obsessed you filter everything else out and find that thing everywhere.”

    > Watching that made me think of the moviePi.

  12. The universe is no narrow thing and the order within it is not constrained by any latitude in its conception to repeat what exists in one part in any other part. Even in this world more things exist without our knowledge than with it and the order in creation which you see is that which you have put there, like a string in a maze, so that you shall not lose your way. For existence has its own order an that no man’s mind can compass, that mind itself being but a fact among others.

  13. I am taken by the beauty of your imagination. And the depth of your creativity, leaves me still, swimming in a sea of visionary excellence…

  14. Definitely into the artfulness as well as the technical skill that went into this little production. Very cool visual experience and nice score as well.

    As for the claims, the somewhat technical assertions that come in along the lines of “Well actually, according to …”, (and don’t we all have friend like that?), one might be mindful to exercise caution even with the cautious. No one proposes that there’s any singular example in all of nature that adheres utterly precisely to the Fibonacci sequence. However, the extent to which that model is so often and so closely approximated (plus OR minus) leaves one comfortably confident that on balance, yes, the series is a fit. So no, it’s not about scrutinizing over the seed patterns of a single sunflower (for example). It’s about accepting the fact that any sample flower you inspect would, yes, approximates Fibonacci. This alone is a remarkable thing. Extrapolating by inference that phi (Fibonacci) is the fundamental underlying principle does not require anything other than acknowledging that singularly remarkable fact. What would truly be remarkable, alternately, is that it wasn’t at issue and yet always loosely appeared to be. What says Occam’s Razor? Go with the simpler explanation. Fibonacci is a fit.

    For the empirically driven, however, it must be admitted that it would be interesting to see someone take individual snapshots of, say, a thousand sunflowers and then use 3D Studio Max and maybe Wolfram Mathematica to work up a definitive average. My bet’s on the obvious relation to phi.

    1. should have been soundtracked to Lateralus by Tool

      Here ya go. Butchered together here, but it “works” in parts… need much more video and/or loops, editing as Lateralus is a very long song.

  15. Beautiful video! Unfortunate linking the nautilus shell to the golden spiral — they are not the same shape.

    Golden ratio, and Fibonacci numbers, are *definitely* in sunflower spirals and many others. Here’s an excellent paper by Naylor that’s easy to follow about the mathematics involved:


    The golden ratio is the “most irrational” number, making it the best choice for organizing leaves and seeds and other things in plant. “From chaos comes order” — mind-blowing ideas!

  16. Es fantástico. Pero tiene un fallo. Las inflorescencias de la margarita que sale en el minuto 2,5 deberían estar dispuestas en espirales logarítmicas dextrógiras y levógiras y la cantidad de ambas tienen que ser dos números consecutivos de Fibonacci. Yo también intenté al principio hacer un dibujo similar con la misma estructura en sentido levógiro y dextrógiro y no comprendía por qué me salian figuras diferentes a las de la naturaleza. Al final fue Martin Gardner quien me lo hizo ver en su libro “Izquierda y derecha en el cosmos”.
    De todas formas, enhorabuena! y … gracias.

  17. I am taken by the beauty of your imagination. And the depth of your creativity, give me again, swimming in a sea of Excellence visionary … | ffxiv gil

Comments are closed.