Leonardo da Vinci could perceive worlds in the stains and markings on old, weathered walls, and he eagerly counseled other painters to study walls so as to shape their own imaginings of landscapes, figures in motion and facial expressions.

His arcane advice struck me recently while I was looking up other things in his notebooks. Not an hour later, I read the same bit of Leonardo’s crafty guidance on a friend’s Instagram feed posted from 3,000 miles away. Odd. Later that same day, I came across the curious word “metis” twice in quick succession—first on a map and then on a construction sign. Metis is an ancient Greek term often associated with Leonardo da Vinci, connoting wisdom and magical cunning. Leonardo, again: Was I sensing a meaningful pattern?

I was certainly witnessing...