Looking for mathematical perfection in all the wrong places

The Golden Ratio — that geometric expression of the Fibonacci sequence of numbers (1, 1, 2, 3, 5, etc.) — has influenced the way master painters created art and can be spotted occurring naturally in the seed arrangement on the face of a sunflower. But its serendipitous appearances aren't nearly as frequent as pop culture would have you believe, writes Samuel Arbesman at The Nautilus. In fact, one of the most common examples of mathematical perfection — the chambered nautilus shell — actually isn't. Even math can become part of the myths we tell ourselves as we try to create meaning in the universe.

Image: Golden Ratio, a Creative Commons Attribution (2.0) image from ernestduffoo's photostream


  1. Rule of thumb: whenever you pick up a book that involves the author drawing curves and lines on top of photos of buildings or works of art, you should put it right back down. Something about irrational numbers makes people get all, you know, irrational.

    1. First it’s the curves, then the chanting, and before you know it you’re elbow deep in a fishman sacrifice to some god you can’t even pronounce.

  2. “Even math can become part of the myths we tell ourselves as we try to create DISCOVER meaningFUL PATTERNS in the universe.”

    There I fixed it for you.

      1. That’s the point. He’s right: the goal is to discover meaningful patterns in the universe. We end up creating them – or the fiction of them.

        Pedantic comment on a pedantic comment’s comment. Ugh.

        1. That assumes meaning is outside ourselves, waiting to be discovered. What is the meaning of a stone?

    1. All science deals with the creation of models that approximate the real world. These don’t “exist” out there for people to “discover” — that’s simply Platonic nonsense. Scientific models are created, just like works of art.

      1. My point exactly, but counting up the likes, it looks like the existentialists are badly outnumbered by the Platonists.

      2. Natural science – the development of testable models of nature – is indeed created rather than discovered, but most mathematicians would disagree that the same holds for mathematical truths…they typically have the impression that they are discovering, rather than creating, mathematical truths. The Fundamental Theorem of Calculus, Godel’s Incompleteness Theorems, Fundamental Theorem of Galois Theory, etc., all hold true even in universes devoid of mathematicians to discover them.

        1. They don’t “hold true even in universes devoid of mathematicians.”  They hold true under the axioms of set theory.  The incompleteness theorems are an interesting exception because they seem to set limits on any axiomatic systems (even the ones not conceived of by mathematicians) if those systems are powerful enough to represent the real numbers.

          Informal surveys suggest most working mathematicians are Platonists, at least with respect to mathematics itself, but there are many mathematicians and philosophers of mathematics who are not and there are philosophical approaches to mathematics that do not require the assumption that mathematical results are true in and of themselves (e.g. it could be that mathematical truths are constructed rather than discovered).  At best this is still an open question.

          1. “They hold true under the axioms of set theory.”

            Yes, but the axioms of set theory are independent of the mind who’s come up with them – different mathematical communities might come up with different axiomatic systems, but the mathematical truths are one and the same.

            Models in natural science are, however, dependent on the context in which they’re created – the technology available at the time, the “paradigms” of science (in Kuhn’s sense), what’s considered acceptable for scientific study, etc. Pure mathematics lies at one level above this and I don’t see how someone can reasonably claim that it is invented.

      3. All science deals with the creation of models that HOPEFULLY approximate the real world. You’re right, the models don’t exist out there. But you just nonsensically assume that the real world DOES exist out there and that what your senses tell you proves it.  Think again! 

  3. good(!) he makes mention of the Parthenon.  how many math books, math museums, Disney math toons, we’ve seen that start with: the Parthenon is entirely based on the  Golden Ratio; until someone bothers to actually take measurements.  Its overall lengths, and sub-units, aren’t even very close at all.  …from there it’s off to the cosmic ratios build into the great pyramids.  if one demands to get cosmic just stare at Euler’s identity for a while:   e^(i*pi) = -1

  4. What is the name of that neurological phenomenon where the human brain attempts to see patterns in chaos? It’s the reason we see patterns in certain star groupings and call them constellations, and give them special stories.  

      1. I think it’s a little from column A (Apophenia) and column P (Pareidolia). While some artists may indeed have used the Golden Ratio in their pieces, I think it’s referenced too often. I find it hilarious when people say they use it for web design too.

        1. From what I understand, apophenia is perceiving any kind of pattern in random data, and paredolia is seeing familiar forms and figures in random data, like faces in clouds, and the EVP.  So paredolia is a subset of apohenia.

          Unless I’m totally wrong.  In that case please correct me.

    1. Reading the Dark Tower series at the moment…according to Steven King the number we are looking for is 19…

        1. I dusted off hyperpi, and calculated the first 10million digits, then tossed it into n++ cut out all the CR-LFs and spaces then tried to do a search for it.
          That’s when n++ crashed.  which is hilarious.  It can search through a 10000000+ character file to find and snip out formatting, but looking for a six-character string crashes it.

          Anyhow, threw the file into Vim and had a look,
          Very clever, the sequence actually repeats four times “24242424” and starts at the 242423rd character including “3.”

  5. Things always get mucked up by “experts’ and wannabes. (Some) designers understand proportions. The eye wants to see clean whole number proportions on objects that are man made. This tiny truth gets all bent out of shape by well-intentioned people working to make it more complicated than it needs to be.

    Astrology tells the same tale.

    1. if people want to see whole number proportions, why is the golden ratio seen … it isn’t a whole number, yaknow.

      1. yes it is, in base φ, the golden ratio is displayed as 1, and multiples of φ are all integers.

        What counts as a round number is as arbitrarily defined as how many licks it takes to get to the center of a tootsie pop.

        We just happen to almost always use base 10.  It’s our arbitrarily chosen number system, and it’s not special.  We just happen to have ten fingers, so it only seemed natural to base our numbers off that.

        Pardon me, φ in base φ is 10.

  6. This was very good reading. I’m glad I persevered despite the error in the first two sentences:

    Out of all of the infinite numbers in the world, there are precious few that are given their own letter from the all-too-finite Greek alphabet. The golden ratio, also known by the letter φ, or phi (usually pronounced “fie” in English), is one of those few.

    It’s not infinite. It’s got an infinite expression as a continued fraction. It’s got an infinite decimal expansion. But it’s just as finite as any other real number.

    (Plus I keep thinking, “Out of all the infinite numbers, in all the towns, in all the world, she walks into mine.”)

    1. I think she meant out of all the infinitude of numbers, or out of all the enormous amount of numbers to choose from. 

      1. I’m morally certain you’re right. It would have been nearly impossible to write that piece that informatively without knowing that. But it’s very clumsy writing, especially for a lead. Consider these alternative phrasings:

        “Out of all of the infinitude of numbers in the world” (which you said, basically)

        “Out of the infinity of numbers in the world” (which is accurate but sounds a little clunky to me)

        “Out of all the countless numbers in the world” (I like that one because it invokes uncountability)

        “Out of all of the infinite number of numbers in the world” (which has a nice rhythm to it)

        I might even recast it to remove the leading “Out”.

        So yeah, I was probably being overly critical to call it an “error”. A “poor phrasing” is better. I blame the copyeditor.

    2. perhaps it’s just me, but i read that assertion as: Of the set of infinite numbers (“in the world”) there are only a few awarded a Greek letter designation.   rather than assuming the author didn’t know the distinction between transcendentals and irrationals.

    3. The infinite numbers are all given either Greek or  Hebrew letters, because symbols are the only way *to* write them down. Aleph, omega, theta. Thanks, Cantor!

    4. I think the author intended to state that the set of real numbers was infinite, not any of the reals themselves. 

      Since you mention the continued fraction expansion, it’s worth mentioning that the expansion for phi=(1+sqrt(5))/2 does have the distinction that it’s the slowest converging continued fraction; it’s all 1’s.

      1. The story touched on that, but not as directly as you did:

        Phi is sometimes called “the most irrational number,” meaning that it is the hardest to approximate with a ratio of rational numbers.

      2. and, there are as many sizes of infinity as there are real numbers (which is, itself, larger than the infinity of integers.

        At least in terms of some meanings of the words  “size” and “set.”

        1.  I tell people this and they scoff.  Then I show them the diagonalization proof and they accuse me of “cheating.”

  7. You don’t think that “infinite” was referencing the cardinality of the set of numbers as opposed to phi itself?  Don’t get me wrong, that’s a terrible opening pair of lines.

  8. I object somewhat to the claim that the golden ratio is the geometric expression  of the Fibonacci Series. The golden ratio was known about for well over a thousand years before Fibonacci ever started working with the series.

    1. I was thinking the same thing.  The Fibonacci Series approaches the golden ratio as it proceeds higher.   If anything, the Fibonacci Series is an expression of the golden ratio, not the other way around.

        1. What I meant was that the ratio between each number and the previous number in the Fibonacci series becomes closer to the golden ratio as the numbers increase.  1/1 = 1,  2/1 = 2,  3/2 = 1.5, 5/3 = 1.666,  8/5 = 1.6,  13/8 = 1.625  etc.  

          1. I AM talking about the F series. I am comparing each number in the sequence to the previous number.

          2. bb (reply to below), look up the meaning of “series” and “sequence” — You can be excused for confusing the two meanings with reference to the Fibonacci S — a lot of well meaning people use the word “series” in this context. But you are NOT talking about series, not in the way that mathematicians use the word (and since the F S is a mathematical concept …) . In fact, in the second sentence of your post below you say “each number in the sequence”.  The Fibonacci sequence begins 1, 1, 2, 3, 5 and you have used these numbers in demonstrating the sequence of ratios. The series generated from the F sequence could certainly be called the F series. Written as the sequence of partial sums of the F sequence it would begin 1, 1+1=2, 1+1+2=4, 1+1+2+3=7, 1+1+2+3+5=12.  If you were demonstrating the sequence of ratios of the F series then you would have 2/1=2, 4/2=2, 7/4=1.75, 12/7~1.71428, 20/12~1.6666, 33/20=1.65, … . This sequence of ratios does also converge to the golden ratio — a couple differences: the sequence of F-sequence-ratios alternates between going below phi and above phi, whereas in the F-series-ratios they steadily decrease (after the 2nd ratio) toward phi.

          3. You’re right Mick.  I didn’t realize my mistake.  However, this still doesn’t change the fact that the  Golden Ratio is not the geometric expression of the Fibonacci Sequence as stated in the post description.  

    2. I was trying to tie it quickly into the reference that most people were going to remember. I realize the ratio came first, but, in my experience, it’s so closely linked with Fibonacci that that’s the thing that’s going to make people go “Oh, yeah!”

      1.  Yeah, I appreciate that a lot of people these days first learned about the golden ratio from the Fibonacci.

        Euclid had a construction for the A:B::B:A+B ratio ratio pretty early in The Elements. Comes up again in the double-angle isosceles triangle, which is the basis for constructing the pentagon. The diagonals of a pentagon cross each other in a golden ratio. The ratio also comes up in the perfect solids a few times.

  9. I remember reading a musical analysis of James Brown’s Goodfoot that claimed     That it had a golden mean therein – such academic trash!

    1. Did your description “trash,” come before or after your investigating the analysis? 

      1. I read the analysis and could not disagree any more than I do now, especially knowing how JB worked his band and how form changes from performance to performance.  It is very much a case of attempting to impose order where there is none for the sake of academic cleverness.

  10.  Inspired by the classic text “On Growth and Form” by D’Arcy Wentworth Thompson where he devotes a long chapter to the spiral, and pointedly notes that we find the finished shape primarily in the dead matter of living things: shells, horns, claws, I set out to place a calculated logarithmic spiral onto an image of a nautilus shell.

    What I found was no precise match for much of the spiral’s length. There were spiral segments which matched well enough but the mathematics diverged from the shell too much for me to call it a close match.

    It did a better job though matching the M74 galaxy.

    So much for believing someone did all their homework years ago.

    On Growth and Form (The Complete Revised Edition)’ by D’Arcy Wentworth Thompson

  11. Hi All,  In our work (as just presented to ARUP Foresight & Innovation in London yesterday, Fri May 3 2013), the Asynsis-Constructal Law team describe the thermodynamic behaviours in nature that reveal the golden ratio’s geometrical optimisation signatures in both space and time: Cosmomimetic Design in Nature, Consciousness & Culture – Asynsis Principle-Constructal Law Seminar:
    Shanghai University-Nantes L’ecole de Design wp.me/p1zCSP-2i via @ASYNSIS:disqus

  12. q is a positive whole number (so q-1, q, and q+1 are consecutive integers) and (q-1)x^2 – qx – (q+1) is a FACTORABLE quadratic (with the usual meaning of “factorable quadratic”). Every other member (alternating members) of the Fibonacci sequence beginning with 2 (and only those integers) satisfies the above description for q.

  13. Mathematics is discovered by humans, and the fact that it is essential in explaining the natural world is strong evidence in favor of its objective existence.

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