NonEuclid

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The NonEuclid website from Joel Castellanos, a graduate student in the Computer Science Department at the University of New Mexico, provides a Java Software Simulation download with “Straightedge and Compass Constructions in both the Poincar� Disk and the Upper Half-Plane Models of Hyperbolic Geometry (a geometry of Einstein’s General Relativity Theory and Curved Hyperspace) for use in High School and Undergraduate Education.”In addition to this Java download, users will find a guide to using the appelet and a series of exercises that include adjacent angles, angles, general triangles, isosceles triangles, equilateral triangles, right triangles, congruent triangles, rectangles & squares, parallelograms, rhombus, polygons, circles, and tessellations of the plane.Discussions of the following are also included:

• What is Non-Euclidean Geometry. Euclidean Geometry, Spherical Geometry, Hyperbolic Geometry, and others.
• The Shape of Space. Curved Space, Flatland, Ourland, and Mercury’s Orbit.
• The Pseudosphere. A description of the space of which NonEuclid is a model.
• Parallel Lines:. In Hyperbolic Geometry, a pair of intersecting lines can both be parallel to a third line.
• Axioms and Theorems. Euclid’s Postulates, Hyperbolic Parallel Postulate, SAS Postulate, Hyperbolic Geometry Proofs.
• Area. Examination of A=�bh and A=s� in Hyperbolic Geometry, Properties Necessary for an Area Function, Altitudes of a Hyperbolic Triangle, Defect of a Triangle, Defect of a Polygon, and an Upper Bound to Area.
• X-Y Coordinate System. A description of how an x-y coordinate system can be set up in Hyperbolic Geometry.
• Disk and Upper Half-Plane Models. An informal development of these two models of Hyperbolic Geometry.