DescriptionGiven a line in the plane and a point in the plane not on the line, how many lines are there through the point and parallel to the line? The answer is that it depends on which geometry you are thinking about!Euclid's fifth postulate says that there is only one such line. For around 2000 years people tried to show that this was a consequence of his other postulates. Immanuel Kant went as far as asserting that Euclidean geometry is the "inevitable necessity of thought". A few years after that Janos Bolyai (and others) discovered hyperbolic geometry. Bolyai wrote his father: I have discovered things so wonderful that I am myself astonished at them.. out of nothing I have created a strange new world. This "strange new world" - the hyperbolic plane - is the subject of this project! A good model of the hyperbolic plane is the unit disc with a distance function that, makes the boundary circle infinitely far from any point inside. Using tiles that have the same size when measured with this distance function but not with Euclidean distance, we can produce tessellations of the hyperbolic plane. All the tiles have the same hyperbolic size, but to our Euclidean eyes sem to get smaller and smaller towards the edge of the disc. Some of Escher's drawings are hyperbolic tessellations. PrerequisitesThe tools necessary for this investigation include the easy bits from 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, but not absolutely necessary, if you have taken relevant pure options.ResourcesWe will use the book James W Anderson Hyperbolic Geometry Springer 1999. This book is in the main library but you may like to buy a copy.See also the wikipedia entry hyperbolic geometry. |
email: J R Parker
