A brief look at the evolution of modeling hyperbolic space

摘要

Now considered one of the greatest discoveries of mathematical history, hyperbolic geometry was once laughed at and deemed insignificant. Credited most to Lobachevsky and Bolyai, they independently discovered the subject by proving the negation of Euclid’s Fifth Postulate but neither lived long enough to see their work receive any validation. Several models have been constructed since its acceptance as a subject. I’m going to discuss three graphical models: the Klein Model, the Upper Half Plane Model and the Poincaré Disk Model. I will also discuss and construct three tape and paper models: the Annular Model, the Polyhedral Model and the Hyperbolic Soccer Ball Model. However, I will spend the most time discussing the construction and uses of crocheted models as well as some brief information on how these models are being used by Margaret and Christine Wertheim to promote global environmental effects on coral reef.