After we have the rectangle we can do some manipulations to create the
After creating the first squarable lune, we began to investigate further.
We constructed what we thought was a second
lune using the isosceles triangle inscribed in the semicircle, and
an arc that runs through two of the verticesof the triangle and the circumcenter
of the triangle.
After some time we discovered, to our dismay, that this lune had arcs with the same 2:1 ratio.
We created another lune with the 2:1 ratio using an inscribed rectangle in a circle, the rectangle itself had a constructed ratio of 2:1 and allowed for the creation of a squarable lune.
Next we investigated the lunes created from an inscribed equilateral triangle and its respective arcs. Here we found a lune from the ratios 3:2. In this construction we were able to physically demonstrate a triangle with the same area, but were unable to identify the precise point the constructed arc must pass through.
We investigated n-gons from n = 3 - 17 in various ways, we found several lunes that appeared to be squarable from arcs of a pentagon, but our investigation continued to run aground. The 17-gon also had a constructed lune that appeared squareable, and we discovered that it in fact can not be squared using elementary methods. (proved in 1929 by the Bulgarian mathematician Lyubomir Chakalov (1886 - 1963)).
There are exactly five ratios for a and b that will create squarable lunes. Let 2a and 2b be the angles subtended by the arcs of the lune with respect to the centers of their respective circles. Constructions for the cases a:b = 2:1, 3:1, 3:2 5:1 and 5:3 are the only ratios that establish squarable lunes by elementary methods. (Scriba, Christoph J. Welch Kreismonde sind elementar quadrierbar? Die 2400jahrige Geschichte eines Problems bis zur endgultigen Losung in den Jahren 1933/1947. [Which lunes are elementarily squarable? The 2400-year history of a problem until its final solution in the years 1933-1947])
We investigated thoroughly the 2:1 ratio, and to some extent the 3:2 ratio, which leaves ratios 3:1. 5:1 and 5:3 to the reader for further perusal.
In closing, we would like to demonstrate using Hippocrates lunes the attempt to "square the circle" which was the beginning of this "luney" essay.