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.

https://upload.wikimedia.org/wikipedia/en/thumb/3/33/Golden_Ratio_cover.PNG/225px-Golden_Ratio_cover.PNG