Arbelos

My Arbelos Story

1979: I discovered a dozen more Archimedean circles

When I was a student of art and biology teacher I used to read the "Scientific American" magazine, in which the "Mathematical Games" by Martin Gardner always were of particular interest to me. In 1979 Martin Gardner wrote about the Arbelos [4], the "twin circles of Archimedes" and the "triplet circle", which was discovered by the famous Leon Bankoff in 1974 [2].

I instantly felt excited to search for further circles with Archimedean radius in the Arbelos, and for several weeks I incessantly scribbled lots of circles and lines on hundreds of sheets of paper. Finally I had discovered a dozen new Archimedean circles, and I sent a paper with descriptions and proofs, in German, to Gardner. He forwarded my manuscript to Leon Bankoff, and I didn't hear about it for a long time.

What I didn't know: In 1996 Leon Bankoff gave a copy of my manuscript to Prof Clayton Dodge, University of Maine, when they were planning to write an article about the Arbelos. One year later Bankoff passed away.

1998: My circles on Peter Woo's Arbelos website

Twenty years after my discoveries, in 1998, I told a colleague (a mathematician) about the Arbelos. He encouraged me to search the web for any Arbelos topics. To my surprise the Arbelos still seemed to be of interest to the mathematicians. I especially enjoyed the website of Prof Peter Woo of Biola University, California, who had dedicated a whole page to the Arbelos [25]. (That way I came to know that the fourth circle had been discovered in the meantime.)

I sent a description of my findings and some illustrations to him, and he published that stuff on his website [21].

1999: The joint article in Mathematics Magazine

Peter Woo then discovered an infinite family of Archimedean circles (the "Woo circles") by generalizing two of my circles, and wrote a paper on it [5]. Shortly before its publication Woo was contacted by Clayton Dodge, who was led to him by Prof Paul Yiu of Florida Atlantic University, who also took part in the Arbelos research.

That way we four came together and agreed in writing a joint article about our separate findings. It admittedly took nearly one more year till the editors accepted it and published it in the Mathematics Magazine [3].

The ongoing Arbelos research

The mathematician community is still dealing with the Arbelos. Some samples: A generalization of the Archimedean circles of Peter Woo and me [14], and a reflection on the Archimedean twin circles in a "skewed Arbelos" [15], both described by Hiroshi Okumura and Masayuki Watanabe, Maebashi Institute of Technology, Japan (2004). 2006 Floor van Lamoen discovers many more Archimedean circles and several infinite families of Archimedean circles [12] and publishes his "Online catalogue of Archimedean circles" [13].

Since 2013 Hiroshi Okumura publishes several papers presenting stunning new properties of the Arbelos and interesting generalizations, see [17], [18], [19], [20].

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