The Shoemaker's Knife Cuts Beautiful Math Across the Centuries

Someone posted this puzzle on Twitter yesterday, and it reminded me of a beautiful math idea I had written about before.  I'll let you take a shot at it if you want, and then go on.  So here is the problem. 

The question is , given the three semicircles above, and the vertical height from the junction of the smaller two to a point on the larger is 2 units,  what is the area of the yellow region?   

When you're ready, read on.  Many students have difficulty with this problem because you don't have any other dimensions, and their really is no one to find any of them with the given information.   For me, that is a signal to the problem solver that it doesn't matter, anything that works with the given information is OK.  Of course you want to make a couple of different examples to convince yourself, but it really doesn't matter. 

What we do know from Geometry is the chord divisor theorem.  If two chords intersect in a circle the product of their segments will be equal.  If we complete the rest of the three circles, we know that the  entire chord length of the red line would be 4, so the product of the two diameters for the smaller circles must have a product of 4. 

But so many choices, which do we choose?  I'm going for the easy one, 1x4.  So the smaller semicircle has a diameter of 1, and the larger has a diameter of 4.  So, we can solve given this assumption.  The largest semi-circle has a diameter of 5, or a radius of 2.5, so it must have an area of  \( 1/2*(2.5)^2 * \pi\) , or 3.125 pi square units.   The larger of the two semicircles along that diameter has a  radius of 2, so it's area will be  \( 1/2*(2)^2 * \pi = 2 \pi\).  The smaller has a radius of 1/2 so it will have area  \( 1/2*(.5)^2 * \pi = .125 \pi\)   The two smaller areas add up to 2.125 pi.  So the area shaded is 3.125 pi - 2.125 or 1 pi. 

But we could also use the trusty 2x2 = 4 to make the semi-circles, with the two smaller circles meeting in the middle of a 4 unit diameter circle.  So the smaller semi-circles would have a radius of 1 each, and their area would be 1/2 pi each, or 1 pi added together.  And the largest semi-circle will have a radius of 2, so its area will be 2 pi, and when we subtract, oh joy, 1 pi again. 

Ok, what if you tried diameters of 1/2 and 8 units for the two smaller diameters, and 8.5 for the largest diameter.  Go ahead, try your luck.  You know you want to. 


And now the post from the archives, which is mostly about the history of the beautiful construction called the arbeloss.  Hang on , we're starting all the way back with Archimedes. 

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

Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The shape is made up of three semi-circles, two of which have diameters that sum to the diameter of the third. 

One of Archimedes famous results is shown here.  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 areas of the two circles were the same. 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. (They could have at least named it the Wooo Wooo circles!)
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. .