A centroid is the path a point on the end of a rolling wheel takes.
When I was in school I saw a picture of a bicycle wheel rolling and noticed that the spokes looked blurred in an interesting way.
I wondered if you placed a marker on each spoke where the speed was the slowest if it would make a pattern.
I talked to my math teacher. Next day he invited me into his office and showed me how to solve the problem. It turned out to have a cool solution.
The minimums along the spokes inscribed a circle within the wheel. The circle was half the radius of the wheel and sitting on the bottom.
The math was interesting and elegant. It wasn't a huge deal; it bordered between cute and cool. My teacher said that he had never seen the problem before and thought it was really something. I showed a couple other teachers and they had never seen it either. The problem just seemed textbook perfect like it should have been known... but nobody had seen it.
Tonight I googled and found that this year (2008), they published a paper on it where they experimentally verified the math. I believe the "new observation" happened in 1996.
I thought that was cool. I think it was in 1990 or so that my teacher and I worked that.
PS: I tried to read the article but they wanted me to PAY to read it! I'll have to derive and post it later on.
PPS: I was also humiliated in front of about 400 people when I was unexpectedly asked to explain it at a math ceremony. Must have been some European in the audience :)
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