Anne and I just did a section hike on the Appalachian Trail, walking the New Jersey portion with several friends. More (probably much more) on this later, but one of the things that struck me was how meditative just hiking along can be — songs, other earworms, and various musings would take up a lot of the mental landscape.
I’d built a QGIS project before our trip, with info about our section of trail, and put it on my phone. It had some distances in miles and others in kilometers, and I was mentally converting back and forth when it struck me: a kilometer is about 0.6 miles, while a mile is about 1.6 kilometers. In other words, their reciprocals differ by one. Is that true, and if not, what number would it be true for? I worked out the quadratic equation:
![\[x^2 - x -1 = 0\]](https://donkelly.net/wp-content/ql-cache/quicklatex.com-06fbf1a4aae77507bc92484a628e8ebd_l3.png "Rendered by QuickLaTeX.com")
(hmmm, seems familiar)… and I found the answer to be:
![\[1 + \sqrt{5}\over2\]](https://donkelly.net/wp-content/ql-cache/quicklatex.com-3680945ba68d62ff2ab474691733078a_l3.png "Rendered by QuickLaTeX.com")
otherwise known as φ (phi), the golden ratio. Interesting, and a fine bit of mental bubblegum to chew on as I walked along…
When I got home, I broke my internet fast with some Reddit, and the first thing I see there is a photo of a tee shirt, with the golden ratio written on it — meaning, in this case, the “most irrational” number, as in “I am most irrational.” Anyway, just a coincidence, but a strange and striking one.
(You can see the shirt by clicking the image.)
The other thing I carried with me in my head? “The Alley Cat,” a perfect earworm for my hiking rhythm, though I noticed that I changed the tempo as the terrain changed.