The Disk Model of Hyperbolic Geometry
Class discussions throughout the course have focused on the role of axioms in a mathematical system, and we considered Euclid's "5th Postulate" (axiom) extensively from several perspectives: aesthetic, practical, and historical. We took a look at the historical development of non-Euclidean geometries that have as their foundation a denial of Euclid's 5th, while maintaining the rest of the Euclidean structure.
The last part of the Seminar is devoted to investigating a particular model of one non-Euclidean geometry, plane hyperbolic geometry. And, we do this by working in a "micro-world" created within Geometer's Sketchpad. As you work in this micro-world, you should be guided by two types of problems:
1. Are there inconsistencies between the microworld and the axioms of hyperbolic geometry? Document your tests of consistency.
2. What theorems hold in this microworld? What theorems of Euclidean geometry still seem to hold in the microworld? Which fail to hold?
For a guided start at investigating the above two types of problems, complete Labs 9 and 10.
The microworld of the disk model is available on the computers in 420 Clapp. Just use Script Tools, the Hyperbolic Tools Folder. Directions are available by opening the Sketch, h-disk, located in the Sketch folder on the hard drive. You can transfer these to another pc by going to the courses/jmorrow folder in Webspace and copying and pasting onto a floppy the HyperTools folder. HyperTools is also available for Macs from JMorrow. Directions for using the script tools may be found at Using Script Tools.