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.

Author Bio

Noah Davis is a junior at Aquinas College in Grand Rapids, Michigan. He is a major in mathematics and is currently considering a physics or theology minor. Noah is planning to attend graduate school for mathematics and is looking forward to having no general education classes there. He was born in Southern California and grew up in the Midwest. Noah enjoys baseball, reading, video games, and math research. His favorite author is C.S. Lewis.