Ting-Chang Hsien was a dear friend when I was in grad school. At the time China was beginning to send a few scholars to work with American counterparts and Ting-Chang came to Stony Brook for a few years to work in an area of particle physics the school was noted for. Near the end of his stay he decided to see America. He flew to my parent's home in Montana and, renting a car, spent several weeks traveling through the West. The day before he flew back to New York my parent's neighbor, Lloyd Erickson, stopped by to chat. Extroverts by nature, Lloyd and Ting-Chang dove into conversation and homemade ice cream. Ting-Chang mentioned as a teenager he was a student in Shanghai with an interest in music as well as math. The school was run by missionaries - his mother was a devout Christian - and his favorite teacher told him he had real talent for math and should give up the piano. He said "she was from Minnesota .. one of her brothers was technical and worked for a company there." Lloyd sat up, "I had an aunt who was a missionary-teacher in China.. she left before the war and and got married." Ting-Chang mentioned the name of his teacher. She was in her late 80s and still alive. Lloyd brought out family photos. That night old teacher and the physicist talked on the telephone for a long time.

We all know it's a small world. We find common connections with total strangers even if we don't consider ourselves well connected. I tend to be shy and don't have that many friends, but continue to be surprised. Of course there's the Six Degrees of Kevin Bacon.

Steven Strogatz does applied math at Cornell. In the mid 1990s he and his student Duncan Watts were interested in the problem of synchronization. How is it that some insects like beetles manage to synchronize their sounds at night? Why is it that types of glowworms blink in unison? Why is it .. there are many of examples of this in nature ... They weren't trying to solve the underlying science, but rather were looking to see if there was a deeper mathematical underpinning. Cornell has lots of beetles in the Summer. They could make measurements.

They considered two extreme cases. One is very regular. Think of a checker board. Each square on a checker board filled has an immediate group of neighbors - small neighborhood cluster. To "talk" to square on the other side there are about eight degrees of separation. Each square is only directly connected to the nearest neighbors. This kind of model is well-studied in physics where a force may only have a very short range. None of us have connections that limited. Next consider complete random connections. If you know 150 people, each of them would know a similar number so the total number of people connected by two degrees of separation is about 150^{2} - 22,500.^{1} Most of us know that's wrong (except for some early models of Internet connectivity:-)

Their gut feeling was the real answer would le somewhere in between. Technically they were looking for the phase transition .. the point at which most of the beetles were suddenly in synchronization, or everyone seems to have these seemingly random connections. The small world problem. What they found was a fairly simple mathematical description that said you only need a small bit of randomness. In your circle of friends an outlier or two.. or knowing someone with such connections.

Their short paper showed examples ranging from power plant distributions to noise making bugs and is one of the one hundred most cited papers in all of science and math. They also noted that given connections afforded by travel these days pandemics could spread rapidly. It's very difficult preventing that phase transition.

The small world problem is very beautiful. It's one of those topics that should be taught in a course that doesn't seem to exist. In college you get music and art appreciation. I'll never do good art or music, but 101 level non-major courses gave me a better appreciation for those very human endeavors.

Math is important enough that students in high school get either working on mindless problems they'll never see again or pre-STEM work that only a small percentage will use. People are properly recommending and even building 'data science' courses.^{2} These seem like a great idea - familiarity with tools, presentation and interpretation - but I'd like to see a math appreciation path.

Math appreciation would have history from a number of civilizations. Was math the first example of external memory? Why did people use different number systems? Why was double-entry accounting so important to Italy? What about zero? Imaginary numbers? The Fibonacci sequence in nature. Why does a slice of pizza fold the way it does? What is infinity, are there different kinds? How are coffee mugs and doughnuts similar? What is the fairest way to vote in a democracy? It's easy to think of dozens of lecture topics that might even inspire a few students and create links to what they finally go into.

Just an idle dream I guess..

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^{1} Apologies to Robin Dunbar - he'd say the number is almost always used out of context anyway).

^{2} I hate the term. Generally anything that adds 'science' to a label isn't science.