Professor Knuth,
Thank you for an excellent overview of the curious role that patterns of numbers have played in the history of mathematics and science, with an emphasis on questioning the depth to which these patterns are fixed within what we observe.
I will readily confess to being both a bit of a tolerant pragmatist regarding most such patterns. I am both a pragmatist in the skeptic that the more broadly a pattern is found in diverse types of data, the more likely it is to be attached deeply within the infrastructure behind that data. Thus words in Europe lead back "only" back to Proto-Indo-European, while the spectral element signatures of elements on the other side of the visible universe lead all the way back to the shared particle and space physics of our universe. In many ways, what we really seem to be doing there is (as you note) not so much looking for "laws" as we are looking for points of shared origins in space and time of such patterns.
I am a skeptic in the sense that it's exceedingly unwise both in everyday life and in data analysis to assume that just because you see a pattern that it's necessarily meaningful or even real. We humans are particularly prone to seeing exactly what we want to see, a phenomenon that itself is a pattern that emerges from our need to make fast, efficient use of relatively slow and limited-capacity neural circuitry. Our brains take a lot of shortcuts.
The delicacy of the fine-structure constant is just the tip of the anthropic tuning mystery. Lee Smolin estimates that when you take the product of all of the tunings needed create a life-as-we-know-it tolerant universe, the odds fall to 1 in 10229. For some perspective, that is almost as low as the odds of our President saying just the right words in a multicultural sensitivity training session.
I was not aware of the Koide pattern. Since I am currently working on a paper with strong geometric implications for fermions when represented in the right space, such a vaguely geometric pattern might well prove relevant (or not!) So, my thanks for alerting me to it.
Cheers,
Terry