So, it has been about 2 months since the Golden Moments puzzle with the two sticks was posted. Pat Wilson solved the puzzle nicely in the June 20 comment, but asked about the significance of his answer, which was 0.618... . Graffolio then touched upon the inverse of 0.618 and correctly came up with 1.618..., the Golden Ratio, and stated that the Fibonacci sequence is related to the ratio. Appalchianjonny then discussed the Fibonacci sequence and its relationship to the Golden Ratio as well as the significance of the Golden Ratio in a reply to Pat Wilson.
So, to summarize, the Golden Ratio is 1.610342... with the decimal places going on forever like another famous number, Pi. Both are "irrational numbers", meaning they cannot be expressed as the ratio of two integers. The Golden Ratio can be derived by using the quadratic equation as per Pat Wilson's comment, and can be approached, but never reached exactly, by using the interesting Fibonacci sequence. What is special about the Golden Ratio? Well, it shows up in nature all over the place, from the patterns of seeds on flowers:
To the galaxies in the universe, including our own Milky Way:
The spiraling arms of the flower seeds and the galaxy, as well as many other parts of nature, are patterned after the Golden Ratio. To learn more about the Golden Ratio, just google it and be amazed at how it shows up in nature in so many ways. And to think that this ratio can be derived by wondering about two sticks lying innocently on the forest floor!