Department of Mathematics Education
EMT 669
We began by attempting to solve the age old problem of squaring a circle.
We constructed the lune found by Hippocrates of Chios in the 5th century,
and we were unable to "square the circle" but we became interested
in the curves called lunes which are squarable. In fact there are exactly
five ratios of lunes which are squarable.
We had access to the first ratio, 2:1 and began our investigation here.
The first lune, is generated from an isosceles
triangle inscribed in a semicircle. In order to "square the lune"
you use the fact that the area of the isosceles triangle is exactly that
of the area of the lune. We then convert the triangle to a rectangle.
After we have the rectangle we can do some manipulations to create the
square.
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.
