An out-of-the-way discussion of an out-of-the-way topic, the golden ratio. As many of my readers likely know, the "golden ratio" is the number found by dividing a line into two parts such that the longer part divided by the smaller part is equal to the whole length divided by the longer part. The very definition suggests the nesting of golden ratios inside one another, an activity that produces neat aesthetic effects. The number itself is an irrational one, 1.6183... Here's a further discussion: https://www.livescience.com/37704-phi-golden-ratio.html But the more out-of-the-way discussion I had in mind, my real gift for you today dear reader, is: https://www.math.cuhk.edu.hk/~pschan/uged1533/markowsky.pdf Geek out there.