DescriptionIn an abstract setting, the hyperbolic plane may be defined using Euclid's axioms for plane geometry but with a different parallel axiom. Indeed, it was discovered by geometers investigating whether the parallel axiom was a consequence of the other axioms (and so confounded Immanuel Kant's assertion that there could be no geometry other than Euclidean geometry). We can naturally define a notion of hyperbolic distance.In practice there are several useful models of the hyperbolic plane. These can be used both for calculation and for drawing pictures. Using tiles that have the same size when measured with the hyperbolic distance function but not with Euclidean distance, we can produce tessellations of the hyperbolic plane. The symmetry groups of these tessellations include the fundamental groups of almost all surfaces and the symmetry groups of several of Escher's drawings . PrerequisitesThe tools necessary for this investigation include simple linear algebra, complex analysis, group theory and geometry. This project offers some insight into the connections of these subjects and may be slanted towards any one of them. It would be useful for students to have taken relevant pure options.The project will be supervised by John Parker. ResourcesWe will use the book James W Anderson Hyperbolic Geometry Springer 1999. Another useful book is Svetlana Katok Fuchsian Groups Chicago UP 1992. See also the wikipedia entry hyperbolic geometry for a good description or hyperbolic tessellations for some nice pictures. |
email: J R Parker
