Nature by Numbers

By Phil Plait | January 13, 2011 7:00 am

Don’t think math is beautiful? Watch this:

Make sure you set the resolution to 720p!

Tip o’ the Fibonacci series to Dan Durda, who sat next to me in countless math classes in college.

CATEGORIZED UNDER: Cool stuff, Science

Comments (48)

  1. ‘Maths,’ surely?

    I’ve never been able to get my head around the subject. I only got through GCSE Maths by coping Chris Bailey.

  2. John Paradox
  3. XPT

    Amazing concept, though using only 3D digital models looks a bit cheesy, but I can understand the challenge of mixing 3D and real images. Very nice anyway.

  4. Loc

    This is very fascinating but it is beyond me. Looks really cool…

  5. Wendy

    LOVED it!!! Best example I’ve seen to date. Thank You!

  6. Sam

    Very cool video, but from a little internet digging it seems that nautilus shells do not actually correspond with the ratio of the Fibonacci spiral.

  7. Meh. That whole golden mean thing is in the eye of the beholder. I’ll take a squishy ol’ octopus any day.

  8. Zucchi

    Reminds me why I used to love geometry, when I was younger and my brain worked better.

  9. Sam H

    I had to drop my Math class this year :( – couldn’t keep up with the pace and workload. For me it was always a love/hate relationship – sometimes interesting, but extremely complex and a pain up the arse to do. This is why I’ve basically given up on any careers in Astrophysics or Spaceflight (such as when I imagined I could be the first human on Mars), although I still hope to go into space at least once in my life. It doesn’t matter how, where or when (preferably after peak oil though), so long as I do it.

    Any tips on how to save $200,000 or more in the next ten years? :)

  10. Daniel J. Andrews

    Like that video. I do wonder though if this is something like numerology combined with cherry-picking. That is, you look hard enough you’ll find a particular organism with a particular mathematical relationship built into some aspect of its body plan. Then ignore all the organisms in which you couldn’t find a mathematical relationship.

    Reminds me why I used to love geometry, when I was younger and my brain worked better.

    Ditto, except my brain works better now that I’m older, but I was developmentally delayed so compared to that stage of my life, my old brain now rocks. :) As an aside, I sometimes wonder if being developmentally delayed early in life means everything else will be delayed, included cognitive decline later in life? Hope so–I’d hate to loose out at both ends.

  11. When I saw the preview frame, I was thinking that it was going to be images of “numbers” in nature, with the dragonfly being the number “1”. That didn’t sound very exciting. Fortunately, it was something much better.

    As for math, I’ve always enjoyed it. (I took AP Calculus in 11th grade, and entered college [math / comp sci major] with 3-1/2 semesters of math credit. I admit it. I’m a geek.)

  12. I gotta say that I was really ticked off that those were artificial sunflowers, and an artificial dragonfly and an imitation nautilus, though they did show a real one for a moment at the end of that segment. I also gotta say that as a mathematician, I’m very dubious about any claims of mathematical perfection in nature. That’s not what nature looks like to me, and I’m glad of it.

  13. Jonathan Lubin (12): Perhaps it might behoove you to take a moment to note that this video:

    1) was lovely and well-done,

    2) perhaps wasn’t intended to show real objects, but instead mathematical models of them,

    3) was enjoyed by many people, and

    4) was created by others and posted by me as a means to show that mathematics — your chosen career, if I’m not mistaken — can be beautiful and interesting and worthy of our attention.

  14. Pete Jackson

    *** * * * * * * * *

  15. Pete Jackson


  16. Gary Ansorge

    9. Sam H

    It’s been my experience that math comprehension has two sides to it; pragmatic/applied or theoretical.

    I had little comprehension for the theoretical side, but when I studied differential equations under Dr. Wolfgang Rindler at U.T Richardson, his cookbook approach was easily absorbed. So, if you have a problem with math, get a better teacher. One more in tune with your style of understanding.

    Great movie Phil. Thanks for that.

    Gary 7

  17. Kaptain K
  18. kevbo

    An extremely good read is the book “The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number” by Mario Livio. It goes into why the ratios shown here show up in nature, but in a very non-technical manner. I used to think pi was cool, but no longer…

  19. Sam H

    @Gary Ansorge

    Personally I don’t think the teachers are that bad, and sometimes during one-on-one sessions I can follow their train of thought (although a more practical application would always be helpful). Still, I happen to KNOW that the problem is with me – I just can’t find math interesting enough to motivate myself to work at it (like many other things in life). And I’m still in highschool, so I don’t think I’d fare well under the required workload of university math either. But the future’s still ahead of me, so I may make a turnaround at some point. But like I said, the math required for any career even remotely related to science is probably beyond me, or at least at this stage.

  20. Jonathan Lubin –

    I find it somewhat sad that, as a mathematician, you don’t see the world, nay, the universe, in math.

    I don’t mean to say that I see nature as a dry set of concepts, but I see it as beauty, as depicted by this video and others from the same site. Mathematics is dry only to those who see it that way.

    I see math in the moon landings, geese flying, the Orion Nebula, street lights at night, a well-thrown curve-ball (and the swing of the bat that misses it), and so much more. It allows me to appreciate both math and nature, as well as the human mind, which is capable of understanding all that.

    I loved this video so much that I am passing it on, through MY blog, to my clients and friends throughout the world that haven’t seen it yet.

    Thanks one more time, Phil!

  21. Sam H –

    I understand your pain. I have the ability to “see” the answers to equations up to four dimensions. I can (or could, at least through college and somewhere beyond) actually visualize what the equation was telling me. Unfortunately, when the number of dimensions goes beyond that, I struggle, mostly because I have trouble simply memorizing things. That usually put the answers to college calculus questions out of my reach.

    Between that and my eyesight, I knew I’d never be a pilot, much less an astronaut. As a kid reading Heinlein’s space opera books, I’d wanted to be an astrogator. Ah, well.

    Hopefully, you will, as I did, find your passion elsewhere and continue to love the concepts of this film as an avocation.

  22. Gary Ansorge

    19. Sam

    Ah, high school, I remember it,,,sorta,,,has it changed much in the last half century?

    We’ve discovered that our brains prefrontal lobes don’t become fully mature until we’re in our early 20s and since they are responsible for the highest levels of our thought, you will likely enjoy significant intellectual growth over the next five years or so.

    HAve fun. It only gets weirder ,,,

    Gary 7

  23. XPT


    Uh? I was under the impression that flowers pack their seeds that way to maximize density. It’s very much compatible with evolution.
    I don’t think anybody is claiming nature is mathematically perfect.

  24. Robin

    Simply Beautiful. Reminded me of my Descriptive Geometry class from almost fifty years ago.

    I studied maths through calculus and found the best teachers always included a demonstration of practical application rather than just teaching equation solving. Sadly, math teachers with the ability to do this were few and far between back then. I hope things have changed.

  25. Steve E.

    I think this video is wonderful. The mathematical models are a wonder in and of themselves, but what is really great about a video such as this one is that it bridges so many gaps — and it shows, explicitly, that mathematics is the foundation of such bridges. Sometimes, Jonathan Lubin, some of us may forget what mathematics looks like to someone who does not study the subject. Anything that can effectively illustrate the intimate connections between mathematics and how we describe the world around us should be applauded.

  26. Wayne H

    Expected a bit better from a skeptical blog than a video of pseudoscientific mumbo jumbo!

    Here’s more info on the Fibonacci rubbish: (Possibly one of the best scientific skeptical sites out there)

    Put simply: phi and the Fibonacci series aren’t everywhere in nature. They’re nowhere in nature! You need to pull up pseudoscience and cherry picking, exactly as done in the video.

  27. mu

    Nice but I like the Dimensions math videos more:

    I’m still waiting for Dimensions II.

  28. Precisely why the one tattoo I have is this based on this…

  29. katwagner

    Beautiful! Should be a viewing requirement in our high school math classes, or middle school too in both tech and math classes. A few years ago our snowball spirea bloomed in the same way that sunflower seeds are all packed together on the stalk. I was amazed the spirea would do that; it must have been pruned just right.

  30. Randall

    The Fibonacci series’s role in nature is highly overstated.

  31. Sam H

    22. Gary Ansorge:

    That certainly sounds like something to look forward to. Apparently the prefrontal cortex helps with decision making – maybe that’ll help me with my procrastination, like the kind I’m doing right now! :)

    Aie…exam pressure. :roll:

  32. Addressing the Fibonacci issue, I believe Phil spoke to that in comment #13.

  33. Did I miss some narrated version? Where in the video is the pseudoscientific mumbo-jumbo that Wayne H (#26) objects to? All I see are some (unassailable) claims about the structure of the Fibonacci sequence and its relationship to phi, and a suggestion that these numbers remind us of some of the patterns we see in the physical world.

    It is easy – perhaps unavoidable – to read our own expectations and biases into a video like this, with its complete lack of explanation. And that’s fine. I’m heartily glad there are skeptics out there to take note when speculation falls under the bus of fact.

    But it’s hardly helpful to take a speculation that’s innocently strolling along the sidewalk and throw it under the bus, just to prove a point.

  34. Some people just can’t see a pretty thing without wanting to smash it.

    Regardless of whether the makes of the video are overstating the prevalence of fib and phi [1], this video makes me wish I knew more about math. There is nothing that can be bad about that.

    [1] And it is WAY above my pay grade to determine that.

  35. Eugene

    Don’t worry Tim, Wayne just needs some time to grieve after realizing that, even though Math is infinite, the Universe rounds up…and Nature rounds up even higher. :p

    The point here, is that nature *aspires* toward Mathematical ideals. It, like us, endeavors to be something greater.

  36. Nigel Depledge

    Eugene (35) said:

    Don’t worry Tim, Wayne just needs some time to grieve after realizing that, even though Math is infinite, the Universe rounds up…and Nature rounds up even higher. :p

    Did you read any of the site to which Wayne linked?

    I did (well, about the first third of that page, as it’s quite long and I have limited lunch-break), and it was a bit of an eye-opener. I, too, had previously – and blithely – accepted assertions about the prevalence of the “golden spiral” in nature, without investigating them at all.

    It turns out that most of the examples used to show how prevalent that spiral is really don’t fit that spiral at all. Instead, they fit a different type of logarithmic spiral. The website author even borrows a picture from a “Fibonacci spiral” fan-page that superimposes the Fibonacci spiral on a chameleon’s coild tail – and it is very, very obviously not a fit at all.

    Spirals occur all over the place in nature, so it is not surprising that some of them fit – at least loosely – a spiral that has especially interesting properties.

    So, until someone shows me some evidence that nature really does use the Fibonacci spiral more often than any other kind of spiral, this is going into my “urban legend” box.

    The point here, is that nature *aspires* toward Mathematical ideals. It, like us, endeavors to be something greater

    And this is just animism. Nature doesn’t aspire to anything. It is the way it is through an evolutionary game of trial and error – that which works lasts, and that which fails dies out. This process is intrinsically incapable of looking forward – there is absolutely no evidence whatever for teleology in nature.

  37. Nigel Depledge

    The barber of civility (32) said:

    Addressing the Fibonacci issue, I believe Phil spoke to that in comment #13

    Not really.

    The video seems to imply that the Fibonacci spiral occurs in many, many places in nature. The counter-point is that really it doesn’t. Phil’s comment (13) makes the same mistake. He agrees that the mathematical models are representations of things that occur in nature. The point made by several commenters is that actually the mathematical models use the wrong type of spiral to represent those particular things.

  38. Thespis

    Watched an episode of “Mathnet” on SquareOne!TV as a kid- they had a parrot squawk “ONE! ONE! TWO! THREE! FIVE! EIGHT! EUREKA!” as part of the plot. Still remember that years later, whereas I can’t remember a lot of the formulas I memorized by rote that same year.

    I hope that maybe someone will see the video, and research further to discover that not all nature is perfect- that it is still evolving and changing; and not all math is cut and dried, that there is beauty in both and more to be learned.

    Thanks for posting this, Dr. Plait.

  39. I have now read the page Wayne linked to (here). Very interesting – I’m glad I read it, because it is important to be careful in how far we take our fun little musings.

    I accept and acknowledge that there is a lot of numeromancy out there. It may even be that this lovely, well-animated, well-scored video was inspired by some mistaken beliefs about the prevalence and meaning of the golden ratio in the physical world.

    I still think that it is going to far to say that the video is “pseudoscientific mumbo jumbo”, or that it “[implies] that the Fibonacci spiral occurs in many, many places in nature.” It suggests that it occurs in exactly 3 places: the nautilus, sunflower, and dragonfly wing that are illustrated. If it is wrong there, then feel free to condemn those mistakes. You are right, and I am glad you pointed it out. While I remain delighted at the mathematical connections between the Fibonacci sequence, phi, and various geometrical constructions, I would not want to base that enjoyment on a falsely-inflated picture of its role in reality.

    But what do you think is communicated when, in condemning a work as inaccurate, you do so inaccurately? Do you at least appreciate that it presents the Fibonacci sequence, and its relationship to phi?

  40. jess tauber

    Actually, Fib-like numbers are everywhere. Last year I discovered, while researching a tetrahedral model of the periodic table, that Fibonacci numbers patterned nonrandomly on the table, when taken as atomic numbers. The s,p,d,f blocks are internally patterned so that single electrons must fill the spaces (the left half of any row in a block) before they start pairing up (in the right hand of the row).

    Up to 89 (all the Fib numbers within the table), ALL the odd Fib numbers fall onto the leftmost spot in the left half row- one electron in one lobe, and ALL the even Fib numbers fall onto the leftmost spot in the right half row, the first pair in one lobe.

    Then I looked at the Lucas numbers: 2,1,3,4,7,11,18,29,47,76…. THEY pattern to the RIGHTMOST positions in the half rows, where all lobes have one electron and the orbital is half full in the left half row, and where all lobes have pairs, in the right half, for a full orbital. This works perfectly up to 18, then 29Cu and 47Ag are misplaced one step to the left of where they should be for a full orbital. BUT, their electron system is altered so that one electron from a full lower s (2 electrons) is taken and donated to the nearly filled d orbital.Since half and full orbitals are both Lucas, this works out perfectly.

    76, Osmium, ends up acting like 54Xe, a noble (but reactive) gas with a full p6 orbital, with reinterpretation of its d6 orbital (same count of electrons).

    The NEXT Fib-like series (no name), 3,1,4,5,9,14,23,37,60,97… also trends coherently on the periodic table, again with exceptions. Its preferred positions are the midpoints within the half rows.

    The tetrahedral model, as well as the Left Step periodic table of Charles Janet, utilize the tetrahedral diagonal of Pascal’s Triangle to delimit periods f,d,p,s in that order. Fibonacci and similar series get their values (among many other ways) by summing samplings across Pascal diagonals. All this stuff is related to very basic math, almost cellular automata at the atomic scale. Others have found very strong relations to the Golden Ratio in the genome, and now people are finding it in language structure and usage.

    Pretty videos are nice, but may mislead as far as the complexity of such systems, which can be multifactor- and because the Fiblike sequences themselves are periodic (look upstream you’ll find multiples of these numbers, plus increments), in real life (such as with seed heads, leaves, spin-orbit couplings in the periodic table which I’m now investigating, and the equivalent in celestial bodies) we might want to look at these upstream values to help pin down reality.

  41. Wayne H

    Good point Timothy, so I put a cut out of a nautilus (from Wikipedia) and superimposed a fibonacci spiral. It didn’t agree very well even by the first 270 degrees of rotation. Then I tried a hurricane (Rita) and a galaxy (M51) and these too did not describe the same spiral that the mathematics did. They’re all based on the logarithmic spiral, but the Fibonacci spiral is a special case which we don’t see in nature.

    Like Phil’s said many times, nature’s awesome enough without us making up stuff about it.

  42. One Eyed Jack

    The major point of the video seems to be missed by many posters.

    It’s not to say that structure in nature is perfectly described by simple mathematics. It’s to say that at the heart of their structure, there is math. A biological organism is more complex than a single formula, so it’s no surprise that the “in-vivo” might show some variation from the perfect geometry.

    Fractals are very common in living organisms. The structure of many plants or the nervous and circulatory systems of vertebrates are good examples. They are never as “clean” as a pure Mandelbrot Set, but the essence of the architecture is there.

    Math can be beautiful, or maybe it’s more accurate to say “We see beauty in nature that demonstrates mathematical symmetry.”

  43. I started to read the comments here, but my eyes started to hurt and my brain got sad.

    This was gorgeous, a lovely use of CGI and must have been a labour of love for Mr. Vila. Speaking as a 3D animator and visual effects artist, I have to say – the computer does a hell of a lot less for you than the layman may think. Most of this was done by hand, or the mathematics to drive the procedures was hand-written, hand-tuned and then the textures, light, rendering and post-processing for glows, flares et cetera…. guess what? Done by hand. I’m sure someone will pooh-pooh and say “well, you click a button and you get a lens-flare!” to which I shall reply by arching an eyebrow, sipping my wine and playing this lovely little video again.

  44. Nigel Depledge

    One-eyed Jack (42) said:

    It’s not to say that structure in nature is perfectly described by simple mathematics. It’s to say that at the heart of their structure, there is math. A biological organism is more complex than a single formula, so it’s no surprise that the “in-vivo” might show some variation from the perfect geometry.

    But the point isn’t that real in-vivo systems show variation from the Fibonacci spiral. They really fit a different type of spiral altogether. It is that logarithmic spiral – not the Fibonacci spiral – around which they exhibit natural variation.

    Fractals are very common in living organisms. The structure of many plants or the nervous and circulatory systems of vertebrates are good examples. They are never as “clean” as a pure Mandelbrot Set, but the essence of the architecture is there.

    True perhaps, but not relevant, as I point out above.

    Math can be beautiful, or maybe it’s more accurate to say “We see beauty in nature that demonstrates mathematical symmetry.”

    Agreed. But what does this have to do with the claim that the Fibonacci spiral crops up all over nature?

    Natural systems, in fact, very rarely show what is technically “symmetry”. Take the simplest kinds of “organism” – viruses – as an aexample. If you see a detailed picture of the structure of a virion (virus particle), it seems to have fivefold icosohedral symmetry. Virologists use the term “pseudosymmetry” because they know that what we observe is not true symmetry, since the protein molecules from which a virion is assembled (and for something like lambda phage it could only be 100 or so individual protein molecules) is asymmetric.

  45. Nigel Depledge

    @ Jess Tauber (40) –
    That’s quite interesting.

    Are you bringing this to our attention to suggest that there is some connection between Fibonacci numbers and the periodic table, or highlighting an interesting coincidence?

  46. All this seems to go back to the Pascal system, and modifications of it. You can generate ANY Fib-like sequence (such as Lucas) merely by changing the values in the outermost edges of the triangle from 1’s to other values (for Luc its 1 on one side and 2 on the other). In the Left-Step Periodic Table by Charles Janet (1929), the rightmost elements are the alkaline earths (the s block is on the right, where it should be from a purely quantum perspective). Every other alkaline earth atomic number, that is 4,20,56,120, which are at the ends of pairs of same length period ‘duals’ as they are known (the traditional table fails to pair the first two elements this way) in the Janet table, is identical to every other tetrahedral diagonal number in the Pascal triangle. Clearly there is something not so coincidental going on here.

    The Fib-like numbers, as mentioned in the last post, are generated by summing across samplings of each diagonal in a straight line through the Pascal triangle. If one realizes that each diagonal represents a dimension (1D for integers, 2D for triangular numbers, 3D for tetrahedrals, 4D for pentatope and so on), then each of these summations generating the Fib-like numbers samples all odd or all even dimensioned diagonals at any one time.

    The Fib-like numbers themselves are organized into triplets of two odd and one even in sequence (that is, an even number of odds, and an odd number of evens). Remember that in mapping to the periodic table the odd Fib numbers map to singlet electrons, and the even ones to paired. So again, a nonrandom result and hard to rationalize as simply coincidental.

    The magic numbers can be found in the Pascal system too- by two diagonals in parallel interacting (either two tetrahedral, or a tetrahedral and triangular) which give the electronic and nuclear magic numbers by strict formulae.

    By the way, the ratio of neutrons to protons in the nucleus converges on the Golden Ratio- known from atomic masses even before the discovery of the neutron, but unrecognized for what it was as far back as 1917. See work by the South African physical chemist Jan Boeyens, who has recently published a book detailing his idea that Golden Spirals motivate the relative counts of the nucleons.

    As for the mismatches from pure positionality for the mappings (for Fib starting at 144, outside known elements, for Luc starting at 29Cu, etc.) these may actually fit reality better. Both the Janet table and my tetrahedral system represent purely quantum theory relations, ignoring known and cumulating effects from relativistic mass increases on the electrons. This causes redistribution of energy levels, and new relations such as spin-orbit coupling. I’m still investigating this, but it appears that the Fib-like number mappings are party to this restructuring. That is, there is a pattern to the exceptions. Thus the behaviors of copper, silver, and osmium are altered to make them fit the prevailing Lucas trend, from what they would have been if just obeying quantum theory. How far this goes I can’t tell, but it would imply a coordination of some sort- meaning that the result is also nonarbitrary, either through external design or as the result of evolutionary convergence.

  47. Something new from after I posted this last- relating to the Pascal triangle again. If one measures the atomic number differences leftwards from any alkaline earth element the intervals are always Pascal triangular numbers to the places where the quantum number m sub l is zero, which are the midpoints of half orbital rows, those same places where the Fiblike series just after the Lucas trends for mapping as atomic numbers. So 120-120 is zero, 120-119 is 1, 120-117 is 3, 120 -114 is 6, 120-110 is 10, 120 – 105 is 15, 120-99 is 21, 120-92 is 28. You can do this from any alkaline earth number, and you will always get triangular numbers- this works no matter how many new quantum numbers l you introduce (new blocks). Remember that every other alkaline earth number is identical to every other Pascal tetrahedral number. Now on to the pentatope numbers, and beyond. Anyone think this is all coincidental, arbitrary, or random? You can follow the development of all this ‘numerology’ at the T3 blog:

  48. Nigel Depledge

    @ Jess Tauber (46 & 47) –
    I think I need diagrams to fully understand the point you are trying to make.

    However, in general, are you saying that certain relationships in certain formulations of the periodic table form patterns that match Fib-like number sequences, or are you saying that Fib-like number sequences dictate properties in the periodic table?

    Because it seems to me that you can equally find any number of patterns in the periodic table that don’t match a Fib-like number sequence. So does it actually mean anything?


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