Tuesday 5 September 2023

The Shoemaker's Knife Cuts Beautiful Math Across the Centuries

The term "arbelos" means shoemaker's knife in Greek, and an example is shown at the top of the blog.   The term is also applied to the shaded area in the figure below which resembles the blade of a knife used by cobblers. 


The height of the line segment HA is the geometric mean of the segments r and 1-r. The area of the arbelos (blue) is equal to the area of a circle with diameter AH.  

Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. 

One of Archimedes famous results is shown here. When the two circles drawn on each side of AH and tangent to it and the inner and outer circle,   he showed in his Book of Lemmas (proposition 5) that no matter how the larger diameter semi-circle was divided to produce the two smaller ones, the area of the two smaller circles were equal to each other. The circles are known as the Archimedean circles, Archimedean twins, and other similar names.

Then it got quiet for awhile... a long while.
But in 1954 a Los Angeles dentist (you read that right) named Leon Bankoff found a triplet for the two twins (A Mere Coincidence, Los Angeles Mathematics Newsletter, Nov. 1954).  
Often called the Bankoff triplet circle, it can be found by drawing a third circle tangent to all three semi-circles of the arbelos. Then the triplet emerges from the common points of tangency of this new circle.

Bankoff was not just any dentist, Along with his interest in dentistry were the piano and the guitar. He was fluent in Esperanto, created artistic sculptures, and was interested in the progressive development of computer technology. Above all, he was a specialist in the mathematical world and highly respected as an expert in the field of flat geometry. Since the 1940s, he lectured and published many articles as a co-author. Bankoff collaborated with Paul Erdős in a mathematics paper and therefore has an Erdős number 1.  
After 2000 years, the dam had broken: In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to Scientific American's "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to Professor Clayton Dodge of the University of Maine in 1996. The two were planning to write an article about the Arbelos, in which the Schoch circles would be included; however, Bankoff died the year after.
Schoch's paper can be found here with images of his dozen additions to the Archimedean circle clan.
Then, in 1998, Peter Y. Woo of Biola University, published Schoch's findings on his website. By generalizing two of Schoch's circles, Woo discovered an infinite family of Archimedean circles named the Woo circles in 1999.
And today, well you can see an Online Catalogue of Archimedean circles maintained by Floor van Lamoen, who has a few geometric objects named after himself as well. .

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