# The Golden Ratio jumping the shark

Yesterday evening I saw a post on twitter about a new “Golden Ratio” video:

My first reaction while watching the video was to check if the upload date was April 1st.  Sadly (and surprisingly), it wasn’t.  Patrick Honner’s comment on twitter summed it  up best:  “I am so tired of the Cult of the Golden Ratio.”    Indeed.

For reasons I don’t completely understand, I found it hard to stop thinking about this video out last night.  It bothered me more than most contrived math stuff usually does.  Here’s my best attempt to explain why.

The so-called “Golden Ratio” is the number $(1 + \sqrt{5} ) / 2$, which is about 1.618.  My first introduction to this number actually had nothing to do with a classroom setting at all.  I was a kid – either in junior high or in high school, I don’t remember – just playing around on a calculator.  I’d enter any positive number, hit “1/x,” then add 1, then hit “1/x,” and repeat.  No matter what number I started with, I always eventually ended up at 1.618.  It was both strange and compelling.  What was special about this number?

Later, this time definitely in high school, my math teacher, Mr. Waterman, introduced several different fun computer models involving iteration.  We played with the Mandelbrot set, Julia sets, and also studied these interesting bifurcations:

http://en.wikipedia.org/wiki/Bifurcation_diagram

http://mathworld.wolfram.com/FeigenbaumConstant.html

From those bifurcation diagrams I learned that the convergence that I’d first seen on my calculator didn’t always happen.  Sometimes an iterative process converged to one point, other times there was some sort of “convergence” to more than one point, and other times there was absolutely no structure at all.   It was amazing to learn about this stuff, and quite possibly the first time I’d seem math that wasn’t either just plain vanilla high school textbook math or math contest math.
Around the same time, one of the legendary former students from our high school, Anita Barnes, came back from Wash. U in St. Louis and gave a lecture about recurrence relations.  She explained how you could take a simple recurrence relation such as the one for the Fibonacci numbers: $x_1 = 1$, $x_2 = 1$, and $x_{n+1} = x_n + x_{n - 1}$

and write down an exact solution for each x_n.  It really blew me away.  After her lecture half of the board had a list of Fibonacci numbers and the other half showed a way to calculate each one of them using an amazingly complicated formula involving the square root of 5.   Right in the middle of that messy little formula was my old friend 1.618 . . . Wait, what??

As an aside, even today I see neat stuff about these type of iterated processes.  This past summer, for example, Steven Strogatz of Cornell posted the following fun question on twitter:

Anyway, back to high school and another fun topic we studied:  continued fractions.  Looking at one of the simplest continued fractions, the Fibonacci numbers and the golden ratio pop up again.  It is such a fun example (and not that hard) that I used it with my older son when we talked about the quadratic formula a few weeks ago:

The main point, of course, is the really interesting math, not the specific numbers that come out of that math.   Pretty sure that the words “golden ratio” do not appear in the video.

Thinking about that fun talk with my son got me thinking about another one we did a few months ago.  This one was on pentagons and tied together lots of seemingly unrelated topics that we’d run across over the years.  This talk showed how you could use quadratic equations to calculate that cos(72) = (-1 + Sqrt(5)) / 4.   The same math shows that cos(144) = (-1 – sqrt(5) ) /4 which I guess was yet another missed opportunity to mention the golden ratio:

Luckily, “Donald Duck in Math Magic Land” connected the golden ratio to pentagons, so I don’t have to feel too bad about my missed opportunity.

At the end of the day, I guess what bothered me about the new Khan video is more than it being another example of the  “Cult of the Golden Ratio.”  It is the missed opportunity that Feynman gets to in this interview:

There are many interesting and surprising places where the “Golden Ratio” pops up in mathematics.  I’ve mentioned a few above, and I’m sure there are 10x as many that I’ve failed to mention.   There are so many twists and turns you can take from these specific examples, too.    However, the fact that the radius of the Earth to the radius of the Moon happens to have some contrived relationship to the golden ratio just doesn’t seem that interesting to me.  Putting the focus on these odd coincidences rather than on other interesting math where the golden ratio also appears puts knowing the name of something ahead of actual learning, and that’s really a shame.