The Golden Ratio (why it is so irrational, but better than sacred ;) - courtesy of Numberphile

Very cool video that does a wonderful job of explaining what’s so special about that Golden Ratio that pops up throughout nature.
I think some of you might really enjoy the computer program he uses to illustrate what he’s talking about.
https://www.youtube.com/watch?v=sj8Sg8qnjOg

Love Numberphile. I also recommend Mathologer’s channel. He has a related video, even: https://www.youtube.com/watch?v=CaasbfdJdJg

As a musical aside, there is a fair amount of speculation that many classical composers, especially Mozart, deliberately incorporated golden ration proportions into their works. It’s a neat idea, but the composers themselves didn’t really document any deliberate attempt as far as I know. That’s not to say that they did not - many of them simply didn’t write essays about their compositional processes. I think it’s certainly possible that some composers (like Bach and Mozart) who were exceptionally bright in many areas of intellectual pursuits were hip to some of the developments in mathematics in their day, and played with the ideas.
For context, Leonhard Euler, the most published mathematician in history, was alive from 1707-1783. Euler’s publications included high quality educational books on mathematics, which is probably one of the reasons why he has so many iconic mathematical tools named for him (aside from his ridiculous genius). Bach was alive 1685-1750, so if he ever encountered Euler’s works it would have been late in his life; however, that would put Euler contemporary with Bach’s influential children, Johann Christian and Carl Phillip Emanuel. Joseph Haydn lived from 1732-1809, and Mozart lived from 1756-1791. Both of them would have lived in a time when Euler’s books were widely available and (as such things are, at least) popular.

Thanks for link

As a musical aside, there is a fair amount of speculation that many classical composers, especially Mozart, deliberately incorporated golden ration proportions into their works. It's a neat idea, but the composers themselves didn't really document any deliberate attempt as far as I know. That's not to say that they did not - many of them simply didn't write essays about their compositional processes. I think it's certainly possible that some composers (like Bach and Mozart) who were exceptionally bright in many areas of intellectual pursuits were hip to some of the developments in mathematics in their day, and played with the ideas.
Now that sounds like a heck of a video/documentary waiting to be made. Explaining the thing and then stepping up and showing how some believe it was incorporated in certain musical examples.