Bolyai ended his 1832 introduction to non-Euclidean geometry with a strategy for constructing regular quadrilaterals (squares) and circles of the same area. In this article, we provide the steps for these constructions in the Poincare disk. We come to the surprising conclusion that Bolyai's strategy of building the circle and square separately is the only way to perform the constructions. That is, we cannot in general construct the square from the circle, nor vice versa.
Prof. Mike McDaniel, Professor of Mathematics, Aquinas College
"Squaring The Circle In The Hyperbolic Disk,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 15
, Article 2.
Available at: http://scholar.rose-hulman.edu/rhumj/vol15/iss1/2