There are a number of models of hyperbolic geometry. I will only explore one of them: the Poincaré disk. The concept behind the Poincaré disk is that a plane is not infinite. The plane is the region inside (but not including) a bounding circle. Lines are arcs which are perpendicular to the circle at both ends. The big catch is in how distances are measured. The distance between two points is based on standard distances and logarithms.
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Look at the lines l , m , & n. You can see that n and m are arcs which terminate into the side of the circle. Yet l looks like a "normal" line. This is because l includes the center of the Poincaré disk. In order to include the center of the disk and be perpendicular to the circle on both ends, l must be a diameter of the circle. |
One of the most interesting things about hyperbolic geometry is how so many properties differ from Euclidean geometry. For starters, the angles of a triangle measure less than 180 degrees!! Even more exciting is the fact that rectangles do not exist!!! But don't take my word for it. Try some drawing yourself with Rice University's "NonEuclid software on-line! "NonEuclid" is a Software Simulation offering Straightedge and Compass Constructions in Hyperbolic Geometry (a geometry of Einstein's General Relativity Theory and Curved Hyperspace) for use in High School and Undergraduate Education.
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