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The Golden Ratio

Several recent efforts to popularize mathematics highlight profound connections with the golden ratio found in nature. We will discuss and explore these and other mathematical and natural wonders of the golden ratio, also known as the golden section, and even called "the world's most astonishing number" in a recent book. Mathematically, the golden ratio is (1 + sqrt(5))/2: it is the ratio of two segments whose sum has the same ratio to the first segment as the first is to the second: (a+b)/a = a/b. We will explore the relationship between the golden ratio and fibonacci numbers, golden spirals, and other mathematicals facts about this most popular of irrational numbers. I have three pine cones that I will bring in for analysis. Please bring other natural objects that exhibit the relationship between fibonacci numbers and the golden section. We will discuss if this is magic or provable mathematics.

Our discussion will be based on this 1¾ hour video of Keith Devlin (plus some supplementary materials below) which surveys the subject quite well:

http://www.youtube.com/watch?v=4oyyXC5IzEE


George Hart has an excellent short 2m video on what a Nautilus shell would look like if it were a golden spiral:

http://www.youtube.com/watch?v=_gxC8OjoQkQ

George's daughter, Vi Hart, did three exquisite videos (totaling 18 minutes) on spirals which explore the deep mathematical relationships between the golden ratio, fibonacci numbers, spirals, and phyllotaxis:

http://www.youtube.com/watch?v=ahXIMUkSXX0

http://www.youtube.com/watch?v=lOIP_Z_-0Hs

http://www.youtube.com/watch?v=14-NdQwKz9w

Finally, for those who want to read a book on the subject, in the Keith Devlin video, he recommends Mario Livio's book "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number". However, this review (and Devlin as well) strongly criticize parts of Livio's treatment: http://www.ams.org/notices/200503/rev-markowsky.pdf. Nonetheless, Devlin suggests that Livio's book is the best that has been written on The Golden Ratio so far. I got a copy from the library and will skim it to supplement the discussion.

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  • CJ F.

    Keith Devlin is teaching the course "Introduction To Mathematical Thinking" on Stanford's MOOC platform starting Feb 3rd: http://online.stanford.edu/courses/mathematical-thinking-winter-2014

    December 18, 2013

  • CJ F.

    I was impressed that Jay had a copy of Livio's book. I wish I had had more presence of mind to ask more about his impressions and take away from the book.

    As I mentioned, George Markowsky's review is very critical of Livio's book (http://www.ams.org/notices/200503/rev-markowsky.pdf). I found and enjoyed his excellent article "Misconceptions about the Golden Ratio" (http://www.cdlmadrid.org/cdl/htdocs/universidaddeotono/unioto/matematicas/markowsky.pdf).

    But Markowsky's review is a bit unfair. Livio writes on p. 200 "In spite of the Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up its application as a fixed standard for aesthetics, either in the human form or as a thouchstone for the fine arts." Livio is making Markowsky's point. True, he may belaud The Golden Ratio more than skeptical Markowsky would like, but enthusiasm is the time-honored way to make a point in print. Math needs it heros!

    November 25, 2013

  • CJ F.

    The main video I referenced for tomorrow's discussion on "The Golden Ratio" is Keith Devlin's "The Golden Ratio & Fibonacci Numbers: Fact versus Fiction": http://www.youtube.com/watch?v=4oyyXC5IzEE

    It is a wonderful debunking session exploring how "when real mathematics meets the popular cultures strange things can occur".

    In addition to fact checking, Devlin includes some profound commentary on the golden ratio in balancing maximization and minimization. He relates it to the representation of φ as a continued fraction (1+1/(1 + 1/(1 + ...) and says "when you look at all numbers as represented by continued fractions. When everything is a 1, that's sort of the number where everything is the farthest away from everything else and so it is the most irrational in some sense." Let's spend some time tomorrow unpacking the meaning in that statement. I am not sure I understand it!

    Here are my notes on Devlin's 1h 43m video: https://plus.google.com/104222466367230914966/posts/6Rawus9m9k6

    November 22, 2013

  • CJ F.

    Bonus video: Arthur Benjamin discusses the Fibbonacci numbers and their relationship to the golden ratio in this nice short 6 minute TED Talk: http://www.ted.com/talks/arthur_benjamin_the_magic_of_fibonacci_numbers.html

    Benjamin's closing quotable is great: ""Mathematics is not just solving for x, it's also figuring out why"!

    But why did he include the spurious image by Jake Garn Photography of the lime woman in a golden rectangle / spiral? I tracked down the source at http://jakegarn.com/the-rule-of-thirds/ and it isn't the kind of mathematically inspiring destination that I was hoping for. The last paragraph is clearly spurious nonsense. At least it cites Mario Livio's book.

    Anyway: more grist for Saturday's mill!

    November 21, 2013

  • CJ F.

    Here is an overview of Keith Devlin's 1:43 (= 1h 43m = 103min) video "The Golden Ratio & Fibonacci Numbers: Fact versus Fiction": http://www.youtube.com/watch?v=4oyyXC5IzEE

    Devlin's video is the main source for the discussion on Saturday (I am searching for better explanations for some of the "facts" he presents, please bring your insights). In the first 24 minutes of the video Devlin presents a number of claims about the golden ratio. Then from about minute 25 to 1:09 he sifts fact from fiction among the plethora of claims about φ = phi = the golden ratio. From 1:09 until about 1:29 the class explores φ in nature, in pine cones, and other natural forms. I will bring several pine cones with me on Saturday so that we can repeat this exercise (if people are interested). The last 14 minutes of the lecture discuss the origin of algebra which is interesting but irrelevant to the golden ratio topic.

    Watch Devlin's video on the golden ratio here: http://www.youtube.com/watch?v=4oyyXC5IzEE

    November 20, 2013

  • CJ F.

    In the main video that I'm using to understand the basics of the golden ratio (this one: http://www.youtube.com/watch?v=4oyyXC5IzEE), Keith Devlin mentions the work of Stéphane Douady & Yves Couder. Their first paper on the subject was published in March 1992 in Physical Review Letters Vol 68, No 13. I found a copy at http://www.sccs.swarthmore.edu/users/08/bblonder/phys120/docs/douady.pdf

    Can someone summarize that paper? Devlin gives a nice vague overview. I get that they did a mathematical model and some simulations. But why is Devlin convinced that they have proven that phyllotaxis (the arrangement of leaves on a plant) tends to use Fibonacci numbers whose ratios converge to the golden ratio? Are you convinced? Is this a valid and a strong mathematical argument?

    November 19, 2013

    • CJ F.

      I found a good summary of the models for phyllotaxis in the paper "A History of the Study of Phyllotaxis" by Adler, Barabe and Jean in Annals of Botony (1997) 80 (3):[masked]: http://aob.oxfordjour...­

      They write: "The basic assumptions in Adler's model (that is maximization of the minimum distance between primordia), in Douady and Couder's model (that is the principle of minimal energy, a transposition in physical terms of the biological hypothesis of Hofmeister or Snow and Snow), in Jean's model (that is the minimization of an entropy-like function under some constraints), and in Levitov's model (that is the maximization of the energy of repulsion), are likely to be mathematically equivalent."

      Sat's discussion will only go into these depths if participants so desire. My preference is for a less in depth discussion like the videos cited in the event description. But some members have expressed an interest in mathematical depth, so here it is if you want it!

      November 20, 2013

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